New Applications of Renormalization Group
Methods in Nuclear Physics
Kai Hebeler (OSU)
In collaboration with:
S. Bogner (MSU), R. Furnstahl (OSU), J. Lattimer (Stony Brook),
A. Nogga (Juelich), C. Pethick (Nordita), A. Schwenk (Darmstadt)

Kyoto, August 31, 2011
Renormalization Group Approachfrom Ultra Cold Atoms to the Hot QGP

Nuclear equation of state
and astrophysical applications

Light andneutron-rich nuclei
UNEDF

Correlations in
nuclear systems

Nuclear equation of state
and astrophysical applications

Light andneutron-rich nuclei
UNEDF

Unified microscopic
description!

Correlations in
nuclear systems

Overview
Overview RG
RG Summary
Summary Extras
ExtrasPhysics
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"
!!

!!
fit
V
rr22 dr
(kr
)) =
## =⇒
VL=0
(k, kk!!)) ∝∝ to
dr jj00low-energy(kr)) VV(r(r)) jj00(k(k!!rrscattering= "k|V"k|VL=0|kdata=⇒ V
Vkk
matrix• constructed
L=0(k,
L=0|kkk matrix
NN interactions
repulsivecore
at small relative distance• “hard”
Momentumunits=
relative
momentum
Momentumunits (!
(! == cccontain= 1):1): typical
typicalrelative
momentum
−1
−1
in
large
nucleus
≈
1
fm
200
MeV
in
large
nucleus≈1fm
200high-momentum
MeV but
but .. .. ..
components• strong coupling between low≈≈and−1−1
Repulsive
Repulsive core
core =⇒=⇒ large
large high-k
high-k (!
(! 22 fmfm )) components
components⇒ nuclear many-body problem non-perturbative, hard to solve!
!!

Dick
Dick Furnstahl
FurnstahlRG
RG inin Nuclear
Nuclear Physics
Physics

Claim:
Problems due to high resolution from interaction!

Overview
Overview RG
RG Summary
Summary Extras
ExtrasPhysics
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"
!!

!!
fit
V
rr22 dr
(kr
)) =
## =⇒
VL=0
(k, kk!!)) ∝∝ to
dr jj00low-energy(kr)) VV(r(r)) jj00(k(k!!rrscattering= "k|V"k|VL=0|kdata=⇒ V
Vkk
matrix• constructed
L=0(k,
L=0|kkk matrix
NN interactions
repulsivecore
at small relative distance• “hard”
Momentumunits=
relative
momentum
Momentumunits (!
(! == cccontain= 1):1): typical
typicalrelative
momentum
−1
−1
in
large
nucleus
≈
1
fm
200
MeV
in
large
nucleus≈1fm
200high-momentum
MeV but
but .. .. ..
components• strong coupling between low≈≈and−1−1
Repulsive
Repulsive core
core =⇒=⇒ large
large high-k
high-k (!
(! 22 fmfm )) components
components⇒ nuclear many-body problem non-perturbative, hard to solve!
!!

Dick
Dick Furnstahl
FurnstahlRG
RG inin Nuclear
Nuclear Physics
Physics

Claim:
Problems due to high resolution from interaction!

Resolution dependence of nuclear forces
QCD

Effective theory for NN, 3N, many-N interactions:Λ ! Λchiralquarks+gluons/partons: Q ! mπ

Λchiraltypical momenta in nuclei: Q ∼ mπchiral EFT: nucleons interacting via pion exchanges andshort-range contact interactions

Λpionlesslarge scattering length physics: Q ! mπ
pionless EFT: unitary regime,
non-universal corrections

n

9Li

n

Basic concepts of chiral effective field theoryNN• choose effective degrees of

freedom: here nucleons and pions• short-range physics captured infew 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

3N

4N

Leading order chiral 3N forcesNN

3N

dominant contribution from !(1232 M∆

4N+ shorter-range parts
long (2π)

intermediate (π)

short-range

tidal effects leads to 3-body forces in

cE term

c1 , c3 , c4 terms cD term

• large uncertainties in 2π coupling
constants at present:

1.5

leads to theoretical uncertainties inmany-body observables• cD and cE have to be determined
in A ≥ 3 systems

Changing the resolution:
The (Similarity) Renormalization Group• goal: generate unitary transformation of “hard” Hamiltonian†
Hλ = Uλ HUλ with the resolution parameter λdHλ= [ηλ , Hλ ]dλ• change resolution in small steps:• transformed wave functions and operators†
Oλ = Uλ OUλ|ψλ ! = Uλ |ψ!

⇒"ψ| O |ψ# = "ψλ | Oλ |ψλ #Overview RG Summary Extras
Flow Results History Eqs. Problem
Overview RG Summary Extras
Flow Results History Eqs. Problem

Two ways toTwo
decouple
RG equations
ways towith
decouplewith RG equations= [Gλ ,RG
Hλ ]Similarity RG: ηλ SimilarityG• specifying ηλ by “Vgenerator
”
“Vλ ”
k’
k’

low k

k’

k

low k

k

k

Λ2

k’

k

Λ2

Λ1

Λ1

Λ0

λ0Λ

0

λ1

λ2λ

0

λ1

λ2

Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :Changing the resolution:
The (Similarity) Renormalization Group• common choice for generator
relative kinetic energy operator Gλ = T :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-nof 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 systemdescribed explicitlychange of 3N force• three-nucleon forces are crucial at low resolution!
+ shorter-range parts

Equation of state: Many-body perturbation theory
central quantity of interest: energy per particle E/NH(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!
• use chiral interactions as initial input for RG evolution

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= 1.8 fm -1= 2.0 fm -1= 2.4 fm -1= 2.8 fm -1

50

E NN+3N,eff +c 3 uncertainty(2)
E(1)
NN + E NN

15

0.05

0.10

[fm -3 ]0.15 00.05

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= 1.8 fm -1= 2.0 fm -1= 2.4 fm -1= 2.8 fm -1

5

0

15

10

5

0

0

0.05

0.10

[fm -3 ]0.15 00.05

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)Neutron matter:
Symmetry energy
K0
2
(ρ
−
ρ
)
+ S2 (ρ)
E(ρ, α = 1) = −aV +
0
2
18ρ0

p0
S2 (ρ) = a4 + 2 (ρ − ρ0 )
ρ0c1 [GeV]
−0.81
−0.81
−0.7
−1.4

c3 [GeV]
−3.2
−5.7
−3.2
−5.7

a4 [MeV]
31.7
33.7
31.7
34.5

−3

p0 [MeV fm
2.4/2.5
2.9/3.0
2.4/2.5
3.3/3.4]• uncertainties in ci couplings lead to uncertainties in symmetry energy• given the experimental constraint a4 = 30 ± 4 MeV
smaller absolute values of c3 seem to be preferred from our results

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 PSR30
ab 30
companionin blueand the emitted
radio beam,
J1614-2230.
Timingresidual—theexcess delay
pointing
towards
Earth,
inyellow.
At
20
20
not accounted for by the timing model—as a orbital phaseof 0.25
theorbital
Earth–pulsar
functionof theturns,
pulsar’sphase. a,line
Fullof sight passes
1010
doi:10.1038/nature09466
nearestto thecompanion(,240,000
magnitudeof theShapirodelay when
all otherkm),
producing
the
sharp
peak
in
pulse
We found0model parameters are fixed at their best-fitdelay.
values.
0no evidence
kind of form
pulseofintensity
The solidline showsfor
theanyfunctionalthe–10
variations,
as
from
an
eclipse,near
conjunction.
Shapirodelay,
andtheredpoints
are
the
1,752
–10
Best-fit residuals
obtainedusing anorbitaltimingb,measurements
in our
GBT–GUPPI
data
set. model–20
RESEARCH
LETTER
The diagrams
this panel
show top-down effects.
–20
that doesinset
not inaccount
for general-relativistic–30
schematics
of case,the binary
orbital phasesof
In thissomesystemof theatShapiro
delay signal
isP.
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
absorbedbycovariantnon-relativistic
model
–40
neutronstar is shownin these
red,delay
the
white
dwarf
Figure1 | Shapiro
measurement
parameters.
Thatresiduals
deviate for PSR
30
a
30
b –40
companion
in
blue
and
the
emittedradio
beam,
Neutron stars are composed of the densest form of matter known long-term data set, parameter covariance and dispersion measure varisignificantly
fromarandom,Gaussiandistribution
J1614-2230.
Timingresidual—theexcessdelay
30
c to2020
pointing
towards
Earth,
inyellow.
At
orbitalphase
exist in our Universe, the composition and properties of which ation can be found in Supplementary Information.
ofnot
zero
mean showsthatthetiming
Shapirodelay mustaccounted
for bythemodel—asa beare20still theoretically uncertain. Measurements of the masses or
As shown in Fig. 1, the Shapiro delay was detected in ourwithof data0.25included
turns,
theEarth–pulsar
line
of
sight
passes
to
model
the
pulse
arrivaltimes
properly,function of the pulsar’s orbital phase. a, Fullradii
10 of these objects can strongly constrain the neutron star matter extremely high significance, and must be included to model
the arrival
10
nearestto the companion (,240,000 km),especiallyat ofconjunction.
additiontheotherredmagnitude
the ShapiroIndelay
whentoall
equation
of
state
and
rule
out
theoreticalmodelsoftheircompositimes
of
the
radio
pulsescorrectly.
However,
estimating
parameter
values
101,2
producingthe sharp peak in pulse delay. We found
tion
.
The
observed
rangeofneutronstar
masses,however,has
anduncertainties
canbedifficult
owing
to
the
high
covariance
between0
GBT–GUPPI
points,
the
454
grey
pointsshow
themodel 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-nucleoniccomponents3–6. 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
datathequality
isorbital
apparent.
c, 1,752
Post-fit
Shapirodelay,
redanpoints
aretheivistic–10 increase in light travel time through the curved space-time significantly curved or otherwise non-Gaussian . To obtain b,
robusterror
Best-fit
residuals
obtained
using
model
7
near
a
massivebody.
Forhighlyinclined(nearlyedge-on)binaryestimates,
we
used
a
Markov
chain
Monte
Carlo
(MCMC)
approachtoresiduals
for
the
fully
relativistic
timingmodel
timingmeasurements
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 thispanel
show
top-downInthethiscase, diagrams
someShapiroof theShapirodelay
signalisa 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
squaredresidual
of
1.1
ms
and
a
reducedx–30
schematics
ofthebinarysystematorbitalphasesabsorbed by covariant non-relativistic model
precision . Here we present radio timing observations of the binary
10,11
Table
1
|
Physicalparametersfor
PSR
J1614-2230
1.4
withdegreesof (from
freedom.
bars,The1s.
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
ParameterValuesignature.
We
calculatethepulsarmass
tobe(1.976
0.04)
M
,
which
[
fromstara 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
neutronis shown
in red, distribution
the white dwarf
30c30 out almost all currently proposed2–5 hyperon or boson con- Ecliptic longitude (l)b rules245.78827556(5)u
of
zero
mean
shows
that
the
Shapirodelay
must
be beam,companion
in
blue
and
the
emittedradio
Ecliptic latitude (b)21.256744(2)udensate equations of state (M[, solar mass). Quark matter
canphasesup- (turns)
Orbital20
Propermotioninl9.79(7)masyrincluded
to
model
the
pulse
arrivaltimes
properly,
portquarks are strongly interacting andpointing towards Earth, in yellow. At orbital phase
20a star this massive only if theProper motion in b
230(3) mas yr
aretherefore not ‘free’ quarks12.
especiallyat conjunction.
Inaddition toredsystemtoberemarkablyedge-on,with anofthe
89.17u
6 0.02u.
Demorestetal.,Nature467,1081(2010)
of 0.25
turns,
theinclination
Earth–pulsar
line
of sight
passes10
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
pointsshow
the10This is the9.6216(9)most 3inclined
pulsarsystem knownat present.
nearest binary
to the companion(,240,000km), The
the2230
highof Astronomy
this detection,
our
MCMC
procedure
derivative and a10ss
with significancethe National Radio
Observatory
Green
Bank Period0
previous‘long-term’
data
set.
The
drastic2
Reference
epoch
(MJD)
53,600producing
the
sharp
peak
in
pulse
delay.
Wewithfound
Telescope x(GBT),
timed to follow
the systemthrough one complete
amplitude andsharpness of the Shapiro delay increase rapidly
fit produce
similaruncertainties.
standard
Dispersion measure*34.4865 pc cm
improvement
in
data
qualityisapparent.
c,
Post-fit–100 orbit with special
8.7-dattention
paid
to
the
orbitalconjunction,
where
maxShapiro! a companion!binary
any kindof pulse
Orbital period
8.6866194196(2)d no evidence
inclination
theforoverallscalingof intensity
the signal isFromthedetectedmeasuremass of increasing
residuals
for andthe fully
relativistic
timing modelthe
Shapirodelay
signal is strongest.
Thesedelay,
data werewe
takenwith the newly
Projected semimajor axis
11.2911975(2)light svariations,
as
from
an
eclipse,near
conjunction.
–20–10
to the Shapiro
mass ofdelay),
the whichcompanion
star.meanThus, the(esin v) linearly proportional
1.1(3) 3 10
built GreenBank Ultimate
Pulsar Processing Instrument (GUPPI). First Laplace(including
have a root
is aparameter
helium–(0.50060.006)M[, which implies that the companion
Second Laplace parameter (ecos v)
21.29(3) 3 10b,Best-fit
residuals
obtainedusing
anorbitalmodel
2
GUPPI coherently removes interstellar
dispersive
smearing
from
the
16
uniquecombination
of
the
high
orbitalinclination
andmassivesquared 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
pulsarsignalandintegrates
thedata
modulothecurrent
apparent
pulse
–20
that
does
not accountfor general-relativistic
effects.
Sine of inclination angle
0.999894(5)1.4 with
2,165
degreesof 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-phaseb0.4 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 Spanlargerthan 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 precisionachievable
the pulsar
89.22
We used the measured arrival times to determine key physical paraparameters.
That theseresidualsfrom
deviate–40
Right
ascension(J2000)16h14min36.5051(5)smeters of the neutron star and its binary system by fitting them to awiththeremarkably
GBT
andedge-on,
GUPPI
provideafrom
verya random,highsignal-to-noise
ratio
significantly
Gaussiandistribution
systemto be
withan inclinationof 89.17u
6 0.02u.
Declination(J2000)to
222u 309 31.081(7)99
parameters,
MCMC
error
estimates,
arerotation
givenofintheTable
1.
Owing
3089.2 withtimingmodel that
accounts
for everyc comprehensive
Orbital eccentricity (e)3 10 Shapiro
measurement1.30(4)of bothdelay
parameters
a singleof zeromean
shows that within
the Shapiro
delayorbit.
must bestar
over the timeof
spannedby the fit. Theour
modelMCMC
predicts atproceduretheneutronhighsignificancethis 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
arrivaltimes
properly,The
standard
Keplerianorbitalparameters,
combined
with
the
known
amplitude and sharpnessof the Shapiro delay increase rapidly withstandard
x fit produce similar uncertainties.
1.2 kpc
rotation and spin-down, astrometric terms (sky position and proper Dispersion-derived distance{
especiallyat conjunction.
additiontocompanion
mass
andorbitalinclination,
fully
describethedynamics
of a89.16
Parallax
distance.0.9
kpc
increasing
binaryinclinationandthe overallscalingofInthesignalis the red
10 theFromdetected
Shapiro time-variabledelay, we measure
a companion
mass ofmotion),binaryorbital parameters,interstellar disperSurface magnetic field
1.8 3 10 G −1
GBT–GUPPI
points,
the
454
grey
pointsshow
3‘clean’binary system—one
two stableobjects—thelinearly
proportionalof the companion
star.compact
Thus, thesion andgeneral-relativistic
effects such as the Shapiro delay (Table 1).
age
5.2to
Gyrthe masscomprising
is a helium–(0.50060.006)M[, which implies that the companionCharacteristic
89.14
0‘long-term’
data massive
set. theThepulsar’s
drastic mass.
1.2 3 10erg s highprevious
We compared the observed arrival16times with the model predictions, Spin-down luminosity
uniquecombinationof theinclination
and
white
under
generalrelativity
andorbital
therefore
also determine
delay
also
the
binarycarbon–oxygen
white
dwarf .byTheand obtained best-fit
parametersx2 Shapiro
minimization,using
theshows
Averageflux
density*
at
1.4
GHz
1.2
mJy
improvement
in
data
qualityisapparent.
Post-fit–10
89.12
in21.9(1)J1614-2230
cause a0.04)M
Shapirodelay amplitude13
2 results Spectral index, 1.1–1.92GHzdwarfWecompanion
measure a 2
pulsarmass of (1.976
is by far thec, high[, which
TEMPO2 software package2. We also obtained consistentresiduals
for other
the fully
relativisticpulsars.
timing model
89.24the original TEMPO package. The post-fit
a usingb228.0(3)rad m thanresiduals, that is, Rotation measureordersmagnitudelargerforstarmost
millisecond
est ofprecisely
measuredneutronmass determined
to date. In contrast–2089.1(including
Shapirodelay),
which
a root meanderived from timing model parameter values (middle) andthe differences between the observed and the model-predicted pulse Timing model parameters (top), quantities
17 from the have
Inproperties
addition,
the inexcellent
timingprecisionachievablepulsar
89.22
radio spectral and interstellar medium(bottom).
Valuesparenthesesmass/radius
represent
the 1s
with
X-ray-based
measurements,
theShapirodelay
pro-ofarrivaleffectively 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
provideavery
highsignal-to-noise
ratio
the89.2
data, andare 0.49
shown in1. We0.52
included1.8
both1.85
a previously
and new GBT–GUPPI vides
spanning
threemonths.
The
new
data
were
observed
no information about1.4
thewithneutronradius.
However,
unlike2,165star’s
degreesof freedom.
Errorbars,the1s.
using an(M800-MHz-wideband 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 byTolman-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

measurementof bothour
Shapiroparameters
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.
Thelong-term data(forexample spin89.18, from which pulse
times of arrival 0.8(TOAs) were determined.
The
standard
Keplerianorbital parameters, combined with the known
quantities vary stochastically on >1-dtimescales.
Valuespresented here
are the averages forbeing an adequate description of gravity.
only
ongeneralrelativity
down rate
and astrometry)characteristic
longerthan * Theseimage
Figure2 | Resultsof thewith
MCMC
error timescales
analysis.
a,Grey-scale
shows
the
our GUPPI data set.
Orbitalphase(turns)companion
mass andorbitalinclination,
fully describe
the dynamics ofbasedaa89.16few weeks, whereas the new data best constrain parameters on { Shown in parentheses are separate
for theaddition,
long-term (first) andunlike
new (second) datasets.
In
statisticalpulsarmass determinations
on
two-dimensional
posterior probability density function (PDF) in the M2–i pulsarvalues.
distancemodel
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 matterparametrize piecewise
high-density extensions of EOS:p∼ρΓ

• range of parametersΓ1 , ρ12 , Γ2limited by physics!

log 10 P [dyne / cm 2 ]• use polytropic ansatz

37
36

neutron star matter

crust EOS
neutron star matterwith c i uncertainties

2

35

1

34
33
32
31
13.0

13.5

14.0

log 10[g / cm 3 ]1

12

KH et al., PRL 105, 161102 (2010)

3.0

Neutron star radiusconstraints

2.5

use the constraints:
2.0

Mmax > 1.97 M!

M [M ]

recent NS observation

causalityvs (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 105, 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Gravitational wave signals
from neutron star binary mergers
4

!22

2
0

0

1

2

f [kHz]

3.5

fpeak

eosUU

3

4

2.5
2

5

1.5
9

10

11

12

13

14

4

!22

x 10

FIG. 3: Peak frequency of the postmerger GW
emission vs.
2
radius of the maximum-mass TOV solution. Blue cases are
0!21
excluded by [3]. See text for symbols.

)

1/2
1/2

/Hz

4

log(h+(f)f

peak

[kHz]

3.5

f

33 gles) belong to simulations for the MIT60 and Glendnh3
EoSs, which both have strikingly different M-R relations(dashed 14.4
lines in Fig.
1). Notethat a SQM
14.2
14.6
14.8
15.0 EoS could
15.2
lead to discriminating observationalfeatures, e.g. in the
3
[gthe
/ cm
]
10 butcosmic ray flux [20,log
22],
particularmodel MIT60
is ruled out by
theet
1.97
! NS of [3]. The Glendnh3 EoS
KH
al.,MPRL
105, 161102 (2010)
seems in conflict with theoretical knowledge of EoS properties at subnuclear densities [4]. Ignoring the two outliers, thefpeak − Rmax correlation (crosses only) becomeseven stronger. Already one determination of fpeak could
therefore seriously constrain the M-R relation and consequently the nuclear EoS. Additionally, simulated mergers of 1.2 M! -1.5 M! binaries for selected EoSs (circles)
demonstrate that the relation between fpeak and Rmax is
not very sensitive to the initial mass ratio [11]. Squares
in Fig. 3 display results for 1.2 M! -1.2 M! mergers. For
peakthose fpeak is clearly lower [11] with differences being
larger for smaller Rmax . But also for the symmetric binaries with lower mass a correlation seems to exist. We
stress that the total peakbinary mass Mtot will be measurable
by the GW inspiral signal [43].
fturns out to correlate also with other NS proper-

Bauswein and Janka,
arXiv:1106.1616 (2011)

15

Rmax [km]+FIG. 2: Scaled power spectral density of the GW signal for
the Shen (black solid line) and the eosUU (blue line) EoSs
compared to the Advanced LIGO (red dashed line) and ET
34(black dashed line) unity SNR sensitivities. The inset shows
the GW amplitude of the + polarization at 50 Mpc for theShen EoS.

3

3
2.5

0

5

10

15

20

t [ms]

!23

eosUU

!24

3
2.5

Shen

2
1.5
10

!22

3.5

fpeak

!2

fpeak [kHz]

35

5

PAL
GS1 151020
t [ms]
GS2
PCL2
SQM1
SQM2
SQM3
PS Shenh at 50 Mpc/Hz

log(h+(f)flog 10 P [dyne / cm 2 ]

1/2

36

1/2

)

+

WFF1
!2
WFF2
0
!22
WFF3
AP4
!23 AP3
MS1!24 MS3
GM3!25 ENG

fpeak [kHz]h at 50 Mpc

x 10

!21

3

!25
0
12

14
R1.35 [km]

0.04

1
0.06

0.08

2
0.1

f [kHz]

3

4

2

5

(Mtot/(Rmax)3)1/2• high-density part of nuclear EOS only loosely constrained• simulations of NS binary mergers show strong correlation between between(black dashed line) unity SNR sensitivities. The inset shows
the GW amplitude of the + polarization at 50 Mpc for the
Shen
EoS.
nation for the fact that thepostmerger
GW emissionof the GW spectrum andis R
ofR thecorrespondingEOS
max bydetermined. Numericalcalculations have• measuring f

FIG. 3: P
radius of
excludedmax

shown that for any EoS the frequency of the f-mode,
which generates the GW radiation
at fpeakdepends for the MIT60 and Glendnh3gles) belong
to[42],
simulations
nearly linearly on the square root of the mean density
EoSs, which both have strikingly different M-R relations(M/R3 )1/2 [44]. Since we fix Mtot , the mass-dependence(dashedlines
in Fig.
drops out. Assuming thatthe radiusof theDRO 1).
re- Note that a SQM EoS could

is key step for constraining chiral EOS systematically at large ρ

4

3.5

z]

f

FIG. of2:a 1.35
Scaleddensity of the GW signal for
FIG.!4: Peak frequency vs. radius
M! power
NS (left)spectral
andtheunits
Shen(right)(blackline) and the eosUU (blue line) EoSs
3
Mtot /Rmax
in geometricalwithsolid
Mtot beingvs.
comparedto thethe binary mass. Symbols havesame meaning
as Advanced
in Fig. 3. LIGO (red dashed line) and ET

1.5
9

3

Overview RG Summary Extras

Physics Resolution Forces Filter Coupling

Equation of state of symmetric nuclear matter,
Why is textbook nuclear
physicsso hard?
nuclearsaturation
0
3

Vlow k NN from N LO (500 MeV)¯
lS

Energy/nucleon [MeV]

!5
!10
!15
!20
!25
!30

VL=0 (k, k ! ) ∝!

!1

3NF fit to E3H and r4He "3NF = 2.0 fm

3rd order pp+hh!1

NN only!1

NN only" = 1.8 fm" = 2.8 fm

0.8

1.0

1.2

NN only

1.4

1.6

!1

kF [fm ]2!
!
KH!etmatrixal., PRC(R) 83, 031301 (2011)r“Verydr jsoft
(kr
)
V
(r
)
j
(k
r
)
=
"k|V
|k
#
=⇒
V
0
0
L=0
kk
potentials must be

excluded because they do notempirical nuclear
Momentum units(! = c = 1): typical relative momentum
give saturation;
saturation properties−1
they give ≈too1muchand MeV but . . .
in large nucleus
fm binding≈ 200
nS ∼ 0.16 fm−3
too high density. In particular, a−1
Repulsive core =⇒ large high-k (! 2 fm ) components
substantial tensor force isDick Furnstahl
RG in Nuclear Physics Ebinding /N ∼ −16 MeV
required.”¯lS ∼ 1.8 fm
Hans Bethe (1971)

Overview RG Summary Extras
Overview RG Summary Extras

Physics Resolution Forces Filter Coupling
Physics Resolution Forces Filter Coupling

Equation of state of symmetric nuclear matter,
Why isis textbook
textbook nuclear
nuclearphysics
so
hard?
Why
physicsso
hard?
Nuclear saturation
0
3

Vlow k NN from N LO (500 MeV)

Energy/nucleon [MeV]

!5

¯
l
¯
lSS

!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

0.8

1.0

NN only

NN only
1.2

1.4

1.6
!
!1
!
kF [fm ]!
2!
!
(k,kk! ))∝∝ rr2 dr
drj j0(kr(kr))VV(r(r))j j0(k(k! rr))=="k|V"k|VL=0|k|k! ## =⇒=⇒ VVKHkk!et!matrix
matrixal.,
PRC(R) 83, 031301 (2011)
VVL=0(k,

L=0

0

0

L=0

kk

saturation
delicatetotypical
cancellations
ofmomentum
large kinetic andMomentumunits(!==ccdue1):
typicalrelative
relative
momentum• nuclear
Momentum
units
(!
==1):
−1
−1 ≈ 200 MeV but . . .
potential
energycontributions
inlarge
nucleus≈1
fm
in large nucleus ≈ 1 fm ≈ 200 MeV but . . .

−1 3
4
−1
Repulsive
core
=⇒
large
high-k(!
2
fm)
components
forces are coreessential!
3N interactions
H and He properties• 3NRepulsive=⇒ large
high-k (! 2fitted
fm )tocomponents
Dick Furnstahl
Dick Furnstahl

RG in Nuclear Physics
RG in Nuclear Physics

Equation of state of symmetric nuclear matter,
Nuclear saturation
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

kF [fm ]1.4

1.6 0.8

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

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-evolved NN + 3N interactions in light nuclei

consider ab-initio calculations of light nuclei
based on SRG-evolved interactions

SRG evolution of 3N interactions• So far:

fit intermediate (cD) and short-range(cE) 3NF couplings to few-body systems at
different resolution scales:

c1 , c3 , c4 terms cD termcE term

E3H = −8.482 MeV and r4He = 1.95 − 1.96 fm• Ideal case: evolve 3NF consistently to lower resolution
has been achieved using oscillator basis states, promising results in very
light nuclei; problems in heavier nuclei, not2 suitable for use in infinite systems★

"7.4

Ground-State Energy [MeV]

erformed in the Jacobi coordiHO) basis of the No-Core Shells is a translationally invariant,each A, with a complete set of
xcitation of Nmax !Ω above theation, where Ω is the harmonic
procedures used here build dipresents a one-dimensional imoach along with a general anal-body hierarchy.
in the A = 2 subsystem, which(2)
body matrix elements !Vλ ".
he A = 3 subsystem we deterus-three-body matrix elements.
body matrix elements by sub)" elements in the A = 3 bad the separate NN and NNN
apply them unchanged to any
e to include any initial three-3

NN-only
NN + NNN-induced
NN + NNN

H

"7.6
"7.8
"8.0

3

N LO (500 MeV)

"8.2
"8.4

Expt.
"8.6

1

2

3

4 5 6 7"1

! [fm ]

1020

Jurgenson, Navratil, Furnstahl, PRL 103, 082501(2009)

(b)
(b)

Indications of significant
4N contributions?

NN+3N-full
NN+3N-full
(c)
(c)

44
He

He

EEgsgs [MeV]
[MeV]

!Ω
!Ω =
= 20
20 MeV
MeV

-25
-25

ab initio calculations(no-core shell model) of light nuclei:

-26
-26
-27
-27
-28
-28

exp.
exp.

-29
-29
..
-22
-22

(e)
(e)

(d)
(d)

(f)(f)

6
Li

Li

-24
-24

small resolution dependence

Egs
[MeV]
gs [MeV]!Ω = 20 MeV
MeV

-26
-26
-28
-28
-30
-30

3
exp.
exp.

NN+3N-induced!

umn
range ofof flow
flow parameters:
parameters: αα == 0.04
0.04fmfm (•(•),),
umn headings) for a range
-90 4
44
44
44
0.05
("), 0.08
0.08exp.
fm (!),(!), andand0.160.16fmfm (!).
(!).Error
Error
0.05 fm ( ), 0.0625 fm ("),
fmbars
uncertainties ofof thethe threshold
threshold extrapolations.
extrapolations. The
The
bars-100indicate the uncertainties
bars
of eacheach panel
panelindicate
indicatethe
theresults
resultsofofexpoexpobars at the right-hand-sideright-hand-side of.nential extrapolations of thenentialthe individual
individualNNmax-sequences(see(seetext).
text).
max-sequences
(d)

-100

16

(e)

(f)

O

!Ω = 20 MeV −1/4
−1/4

−1
2.24 fm−1, 2.11 fm−1−1 , 2 fm−1−1 ,-120mentumscales Λ = αα== 2.24 fm , 2.11 fm , 2 fm ,
−1
−1
exp.
1.88 fm , and 1.58 fm−1 ,, respectively.
respectively.
Forextrapolations
extrapolationstotoFor-140
infinite model space, N
Nmax
→ ∞,
∞, we
we useuse simple
simple exponenexponenmax →tial
last 33 oror 44 data
data points.
points. The
Theextrapolated
extrapolated-160fits based on the lastenergy is given by the average
average ofof thethe two
two extrapolations,
extrapolations,the
the
-180
.
uncertainty
by the difference.
∞ 2 4 6 8 10 12∞
2 4 6 8 10 12difference.
14 ∞ 2 4 6 8 10 12
N
N
N
max
max
max
The
energies obtained
obtained inin IT-NCSM
IT-NCSM calculacalculaThe ground-state
ground-state energies
44

66

(a)

-120 (b)

-26

4

-27

He

!Ω = 20 MeV

-28
. -29
0

-130

16

-140

O

-150
!Ω = 20 MeV

-160
-170

.

-180

significant resolution dependence!
to Nlong-range
part energies of He
FIG. 3:(due(color online)-extrapolated ground-state
and O as function of the flow parameter α for the NN-only (•), theof initial
3N
interaction)
NN+3N-induced( ), andthe NN+3N-full
Hamiltonian (").
0.05

0.1
α [fm4 ]

0.15

max

16

0

0.05

0.1
α [fm4 ]

0.15

4

!

44

-110

-25

Egs,∞ [MeV]

NN+3N-full.. -34-34∞∞∞∞22 44 66 88 10∞22 44 66 88 10
10 12
1012
14
1214
14
121414(c)(a)∞(b)-60 2 4 6 8 10 12 14
Nmax
NNmax
NNmax
max
max
12
C
-70
4
= 20 MeV
FIG.
IT-NCSM ground-state
ground-state energies
energiesforfor4 He
Heand
and
FIG. 1: (color!Ωonline)
IT-NCSM66 -80Li
for thethe three
three types
typesofofHamiltonians
Hamiltonians(see(seecolcolLi as function of Nmax
max forNN only

Egs,∞ [MeV]

-32
-32

Egs [MeV]

-ee
m
m
ff
O
O
ll
rr
dd
ee
yy
ee
nn
00
ee
hh
tt
ee
ss
rr
oo
xx
22

(a)
(a)

-23
-23
-24
-24

Egs [MeV]

gg
yy
--nn
-ss
A
A

NN+3N-induced
NN+3N-induced

NN only
NN
only

ee
rr
yy

tions for the ground states of 12 C and 16 O starting from chiral
NN+3N interactions. By combining the IT-NCSM with the
JT -coupled
schemefor the 3N matrix elements weRothet. al, PRLstorage107, 072501(2011)SRG evolution of 3N interactions• contradicts our nucleonic matter results!
• convergence problems in RG evolution of 3N interactions in oscillator basis?
• current project: evolve 3N interaction in plane-wave basis• similar technology to solving the A=3 Schroedinger (Faddeev) equations• allows systematic investigation of flow of low-energy couplings andprovides matrix elements suitable for finite nuclei and infinite-matter
calculations• makes it possible to study the evolution of operators like densities

Correlations in nuclear systemsb)What is this vertex?
e’
e
q
N
N

A

A!2

Higinbotham,arXiv:1010.4433
correlation studies is to cleanly isolate diagramb) froma).
h final-state interactions, that can produce the same final state as
matics that were dominated by diagram b) it would finally allowof the nucleon-nucleon potential.

q

Short-range-correlationinterpretation:energy and moment transferred, but also the energy andween the transferred
k ! and detected energy and momentum
entum, pmiss , respectively. From the theoretical works on
kmomentum distribution of nucleons in the nucleus [6], it
verage potential motion of the nucleon in the nucleus ofcorrelations.
Electron
Beam(CEBAF) [7], it was possible torel.Facility
momentumk : low
ic kinematics
into the pmiss > 250 MeV/c region. In the!
rel. momentum
k : high3
E89-044He(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 termsof low-momentum interactions?

Correlations in nuclear systemsb)What is this vertex?
e’
e
q
N
N

A

A!2

Higinbotham,arXiv:1010.4433
correlation studies is to cleanly isolate diagramb) froma).
h final-state interactions, that can produce the same final state as
matics that were dominated by diagram b) it would finally allowof the nucleon-nucleon potential.

q

Short-range-correlationinterpretation:energy and moment transferred, but also the energy andween the transferred
k ! and detected energy and momentum
entum, pmiss , respectively. From the theoretical works on
kmomentum distribution of nucleons in the nucleus [6], it
verage potential motion of the nucleon in the nucleus ofcorrelations.
Electron
Beam(CEBAF) [7], it was possible torel.Facility
momentumk : low
ic kinematics
into the pmiss > 250 MeV/c region. In the!
rel. momentum
k : high3
E89-044He(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 termsof low-momentum interactions?
Vertex depends on the resolution!
RG provides systematic
way to calculate suchprocesses at low resolution.

medof the configurations sampled inent
on a for each
Figure 1 shows the resulting ratios integrated over 1:4 <2
nt onwalk.
a
Q2 The< 2:6 GeV
. Thesecross ofsectionratios of(a) scaleini- X
om
large
range
valuesx andtially for 1:5 < xB < 2, which indicates that NN SRCs
toobtainconverged results, especially for 3 He,
S specc calo- large numbers
airlyof points; we used grids ofa)3
used tofor x and X, respectively. We
2.5
reand
used 80 points
maining2
over all pairs
instead of just pair 12.
indows1.5
used
to pp momentum distributions in 3 He, 4 He,and1. Kine4 shown in Fig. 1 as functions of the
Bedriftnuclei areb)orinties.
momentumq3 at fixed total pair momentum Q=0,
SIM, tonding
moving back to back. The
rs, tak- to nucleons
2
in
lnels
errorsdue1 to the Monte Carlo integration areptance3 He, for the
6 pp
pairs; they are negligibly small
nonlyc)
pgsian’spairs. The striking features seen in all cases
AC data4
heHallmomentum
distribution of np pairs is muchC

In He and He with uncorrelated wave functions, 3/4 ofthe np pairs are in deuteron-like T, S=0,1 states, whiler( He/ He)

Scaling in nuclear systems
10p − p = 2q

3

5

6

!NN(q,Q=0) (fm )

4
12

3

r( C/ He)

p! + p = Q = 0

6

!

4

10

3

He
4
He
6
Li
8
Be

3

10

2

10

np pairs

1

10

56

3

r( Fe/ He)

10

0

10

2 pairs for relative momenta in theanthat of pp
ons as a-1−1
pp pairs
10–3.0, and1 ii)1.25for1.5the1.75helium
andlithium
CLAS fm
2
2.25 2.5 2.75n
cross
the node in the pp momentum
distribution is
xB
0
1
2
3
4
5
-1
the np FIG.
one,
which crossinsteadexhibits
2a change ofq (fm )
<sectionQ2 <2.6[see
GeV
1. Weighted1.4
ratiosEq. (2)] of (a) 4 He,−1of x for Q2 >3(c) 56 Feof
to p
He
as1.5
a function(b) 12 C, andvalue
characteristic
#
fm
. The
nodal
B
Schiavilla et al., PRL 98, 132501 (2007)
indicate
the
NN
(1:5
<
1:4 GeV2 . The horizontal dashed lines;(2)
Egiyan
et
al.
PRL
96,
1082501(2006)
8
FIG. 1: (Color online) The np (lines) and pp (symbols) mois muchlessin regions.
Be. At small val2) andprominent
3N (xB > 2:25) scalingxB <mentum distributions in various nuclei as functions of thethe ratios of np to pp momentum distributions
relative momentum
q at vanishing total pair momentum Q.
scalingbehaviorof momentum
function:r to082501-3
thoseof np topp pair numbers,
which in distribution•ρNN (q, Q = 0) ≈ CA × ρNN,Deuteron (q, Q = 0)

at large q• dominance of np pairs over pp pairs• “hard” (high resolution) interaction used, calculations hard!
• dominance explained by short-range tensor forces

Nuclear scaling at low resolution!

"⇒ ψλ |Oλ |ψλ factorizes into a low-momentum structure and auniversal high momentum part if the initial operator only
weakly couples low and high momenta
explains scaling!
key: Uλ (k, q) ≈ K(k)Q(q) for k < λ and q ! λfactorization!

That leads to:!ψλ | Oλ |ψλ "

=

! λ
0

≈

! λ
0

dk dk!

! ∞0dq dq ! ψ † (k)Uλ (k, q)O(q, q ! )Uλ (q ! , k ! )ψλ (k)"!
λdk dk! ψ † (k ! )#dq dq ! K(k)K(q)O(q, q ! )K(q ! )K(k ! ) + IQOQ K(k)K(k ! ) ψ † (k)0with the universal quantity:
IQOQ =! ∞dq dq " Q(q)O(q, q " )Q(q " )λvalid if initial operator weakly couples low and high momenta

Nuclear scaling at low resolution!

"⇒ ψλ |Oλ |ψλ factorizes into a low-momentum structure and auniversal high momentum part if the initial operator only
weakly couples low and high momenta
explains scaling!
RG transformation ofpair density operator(induced many-bodyterms 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
10

8

PRELIMINARY
with tensor
interaction

0.0001

7<!(P=0,q)>np<!(P=0,q)>np/<!(P=0,q)>nn

PRELIMINARY
965

no tensor
interaction

4

-1

-1

" = 1.8 fm-1" = 2.0 fm-1" = 2.5 fm-1" = 3.0 fm

3
2
1
0
1.5

1e-06

2

3
2.5
-1
q [fm ]

3.5

1e-08

4

1e-10
1.5

KH et al., in preparation" = 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

• 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!
High-resolution experiments can be explained by low-resolution methods!
Opens door to study other electro-weak processes and higher-body correlations.

Summary• low-resolution interactions allow simpler calculations for nuclear systems• observables invariant under changes in resolution scale, interpretation can change!
• chiral EFT provides systematic framework for constructing nuclear Hamiltonians• 3N interactions are essential at low resolution• nuclear matter equation of state consistent with empirical constraints• constraints for the nuclear equation of state and radii of neutron stars