nuclear physics and cosmology …. from outer space to … to nuclear astrophysics …. why modern...

From outer space to deep inside …

Nuclear physics and cosmology ….

Introduction to Nuclear Astrophysics ….

WHY modern nuclear physics is

so COOL? (a short

introduction)

The nuclear realm

LRP Nuclear Science Advisory Committee(2008)

Res

olut

ion

Complexity

Simplicity Hot and dense quark-gluon matter

Hadron structure

Nuclear structure Nuclear reactions

Nuclear astrophysics

Applications of nuclear science

Hadron-Nuclear interface

Courtesy R.F.Casten (WNSL)

LOOKING INWARD

SCALES and PHASES of NUCLEAR MATTER

NucleonNuclear

StructureNuclear

AstrophysicsNeutron

Stars HypernucleiHot and Dense Matter

OUTWARD LOOKING

The SCALES: first constraint

„If the Lord Almighty had consulted me before embarking upon creation, I would have recommended something simpler.“ King Alphonse X. of Castille and Léon (1221-1284), on having the Ptolemaic system of epicycles explained to him

QCD in the non-perturbative

regime

F. Wilczek “QCD made simple” (http://www.frankwilczek.com/)

http://www.frankwilczek.com/Wilczek_Easy_Pieces/298_QCD_Made_Simple.pdf

UNEDF SciDAC Collaboration Universal Nuclear Energy Density Functional

Modern nuclear physics is about...

➜Linking QCD to many body systems

The SCALES: first constraint

Ab initio Nuclear Structure: an experimentalist point of view

courtesy R. Roth

courtesy R. Rothcourtesy R. Roth

Ab initio Nuclear Structure


Effective Field Theories


Effective Field Theories6NUCLEAR FORCES in CHIRAL NUCLEAR EFT

• expansion of the potential in powers of Q [small parameter]

• explains observed hierarchy of the nuclear forces�Chiral expansion of nuclear forces [based on NDA]

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Korrektur 1. Ordnung



Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

LO (Q0)

NLO (Q2)

N2LO (Q3)

N3LO (Q4)


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Drei-Nukleon-Kraft Vier-Nukleon-Kraft


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N4LO (Q5)

— have been worked out and employed (recently, also parts of the N5LO NN force have been worked out )

— have been worked out but not employed yet [LENPIC, work in progress]— have not been completely worked out yet worked out and applied worked out and to be applied calculations in progress

– Ulf-G. Meißner, Chiral Nuclear Dynamics – SFB 634 Concl. Conf., June 2015 · � C < ^ O > B •


Effective Field Theories

E. Epelbaum et al, PRL 106, 192501 (2011)

6NUCLEAR FORCES in CHIRAL NUCLEAR EFT• expansion of the potential in powers of Q [small parameter]

• explains observed hierarchy of the nuclear forces�Chiral expansion of nuclear forces [based on NDA]


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Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

LO (Q0)

NLO (Q2)

N2LO (Q3)

N3LO (Q4)


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Zwei-Nukleon-Kraft

Führender Beitrag





N4LO (Q5)

— have been worked out and employed (recently, also parts of the N5LO NN force have been worked out )

— have been worked out but not employed yet [LENPIC, work in progress]— have not been completely worked out yet worked out and applied worked out and to be applied calculations in progress

– Ulf-G. Meißner, Chiral Nuclear Dynamics – SFB 634 Concl. Conf., June 2015 · � C < ^ O > B •


The Hoyle State

E. Epelbaum et al, PRL 106, 192501 (2011)

The Hoyle State (…the holy grail)


BB


BBstellar burning


The Hoyle State


Ab initio Nuclear Structure

The PHASES: second constraint

© Jens Rydén

© Corgi Lane

© ryandury

It’s always just water

Each PHASE can exist in a variety of states.

Each STATE is characterized by defined properties

The EOS summarizes the physically possible combination of states.

PV = nRT


© CERN © NASA

It’s always just nuclear matter

© SciencePhotoLibrary


It’s always just nuclear matter


Modern nuclear physics is about...

➜ Unravelling the phases of nuclear matter

LRP Nuclear Science Advisory Committee(2008)


A heavy nucleus (like 208Pb) is 18 orders of magnitude smaller and 55 orders of magnitude lighter than a neutron star

Yet bounded by the same EOS

The Equation of State of Nuclear Matter

© NASA

© SciencePhotoLibrary

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

18-OM smaller 55-OM lighter … same EOS

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…


Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

Equation Of State

3/12

symmetry energy

slope parameter

curvature parameter

…

does not. Then, we have to conclude that a 3% accuracy inAPV sets modest constraints on L, implying that some ofthe expectations that this measurement will constrain Lprecisely may have to be revised to some extent. To narrowdown L, though demanding more experimental effort, a!1% measurement of APV should be sought ultimately inPREX. Our approach can support it to yield a new accuracynear !!rnp ! 0:02 fm and !L! 10 MeV, well below anyprevious constraint. Moreover, PREX is unique in that thecentral value of !rnp and L follows from a probe largelyfree of strong force uncertainties.

In summary, PREX ought to be instrumental to pave theway for electroweak studies of neutron densities in heavynuclei [9,10,26]. To accurately extract the neutron radiusand skin of 208Pb from the experiment requires a preciseconnection between the parity-violating asymmetry APV

and these properties. We investigated parity-violating elec-tron scattering in nuclear models constrained by availablelaboratory data to support this extraction without specificassumptions on the shape of the nucleon densities. Wedemonstrated a linear correlation, universal in the meanfield framework, between APV and!rnp that has very smallscatter. Because of its high quality, it will not spoil theexperimental accuracy even in improved measurements ofAPV. With a 1% measurement of APV it can allow one toconstrain the slope L of the symmetry energy to near anovel 10 MeV level. A mostly model-independent deter-mination of !rnp of 208Pb and L should have enduringimpact on a variety of fields, including atomic paritynonconservation and low-energy tests of the standardmodel [8,9,32].

We thank G. Colo, A. Polls, P. Schuck, and E. Vivesfor valuable discussions, H. Liang for the densities ofthe RHF-PK and PC-PK models, and K. Kumar for infor-mation on PREX kinematics. Work supported by theConsolider Ingenio Programme CPAN CSD2007 00042

and Grants No. FIS2008-01661 from MEC and FEDER,No. 2009SGR-1289 from Generalitat de Catalunya, andNo. N N202 231137 from Polish MNiSW.

[1] Special issue on The Fifth International Conference onExotic Nuclei and Atomic Masses ENAM’08, edited byM. Pfutzner [Eur. Phys. J. A 42, 299 (2009)].

[2] G.W. Hoffmann et al., Phys. Rev. C 21, 1488 (1980).[3] J. Zenihiro et al., Phys. Rev. C 82, 044611 (2010).[4] A. Krasznahorkay et al., Nucl. Phys. A731, 224 (2004).[5] B. Kłos et al., Phys. Rev. C 76, 014311 (2007).[6] E. Friedman, Hyperfine Interact. 193, 33 (2009).[7] T.W. Donnelly, J. Dubach, and Ingo Sick, Nucl. Phys.

A503, 589 (1989).[8] D. Vretenar et al., Phys. Rev. C 61, 064307 (2000).[9] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R.

Michaels, Phys. Rev. C 63, 025501 (2001).[10] K. Kumar, P. A. Souder, R. Michaels, and G.M. Urciuoli,

http://hallaweb.jlab.org/parity/prex (see section ‘‘Statusand Plans’’ for latest updates).

[11] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda,Phys. Rev. C 82, 054314 (2010).

[12] I. Angeli, At. Data Nucl. Data Tables 87, 185 (2004).[13] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000); S. Typel

and B.A. Brown, Phys. Rev. C 64, 027302 (2001).[14] R. J. Furnstahl, Nucl. Phys. A706, 85 (2002).[15] A.W. Steiner, M. Prakash, J.M. Lattimer, and P. J. Ellis,

Phys. Rep. 411, 325 (2005).[16] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. 95,

122501 (2005).[17] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda,

Phys. Rev. Lett. 102, 122502 (2009); M. Warda, X. Vinas,X. Roca-Maza, and M. Centelles, Phys. Rev. C 80, 024316(2009).

[18] A. Carbone et al., Phys. Rev. C 81, 041301(R) (2010).[19] L.W. Chen et al., Phys. Rev. C 82, 024321 (2010).[20] B. A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113

(2008).[21] M. B. Tsang et al., Phys. Rev. Lett. 102, 122701 (2009).[22] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86,

5647 (2001).[23] J. Xu et al., Astrophys. J. 697, 1549 (2009).[24] A.W. Steiner, J.M. Lattimer, and E. F. Brown, Astrophys.

J. 722, 33 (2010).[25] O. Moreno, E. Moya de Guerra, P. Sarriguren, and J.M.

Udıas, J. Phys. G 37, 064019 (2010).[26] S. Ban, C. J. Horowitz, and R. Michaels, arXiv:1010.3246.[27] N. R. Draper and H. Smith, Applied Regression Analysis

(Wiley, New York, 1998), 3rd ed.[28] K. Hebeler, J.M. Lattimer, C. J. Pethick, and A. Schwenk,

Phys. Rev. Lett. 105, 161102 (2010).[29] A.W. Steiner and A. L. Watts, Phys. Rev. Lett. 103,

181101 (2009).[30] D. H. Wen, B. A. Li, and P. G. Krastev, Phys. Rev. C 80,

025801 (2009).[31] I. Vidana, C. Providencia, A. Polls, and A. Rios, Phys.

Rev. C 80, 045806 (2009).[32] T. Sil et al., Phys. Rev. C 71, 045502 (2005).

v090M

Sk7H

FB-8

SkP

HFB

-17SkM

*

Ska

Sk-Rs

Sk-T4

DD

-ME2

DD

-ME1

FSUG

oldD

D-PC

1PK

1.s24N

L3.s25

G2

NL-SV2PK

1N

L3N

L3*

NL2

NL1

0 50 100 150 L (MeV)

0.1

0.15

0.2

0.25

0.3

∆rnp

(fm

)

Linear Fit, r = 0.979Nonrelativistic modelsRelativistic models

D1S

D1N

SGII

Sk-T6SkX SLy5

SLy4

MSkA

MSL0

SIVSkSM

*SkM

P

SkI2SV

G1

TM1

NL-SH

NL-R

A1

PC-F1

BC

P

RH

F-PKO

3Sk-G

s

RH

F-PKA

1PC

-PK1

SkI5

FIG. 3 (color online). Neutron skin of 208Pb against slopeof the symmetry energy. The linear fit is !rnp ¼ 0:101þ0:001 47L. A sample test constraint from a 3% accuracy inAPV is drawn.

PRL 106, 252501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending24 JUNE 2011

252501-4

X. Roca-Maza, at al. Phys. Rev. Lett. 106, 252501 (2011)


Where do the neutrons go?

Nuclear charge radii

One example

Where do the neutrons go?

Pressure forces neutrons out against surface tension

EOS

One example

Pressure forces neutrons out against surface tension

Measures how much neutrons stick out past protons

Constrains the pressure of neutron matter @ low ρ. i.e. calibrate the EOS of neutron rich matter ....

One example

least model dependent method: PV e- scattering

5/12

Mass ≥1.4 M Diameter: 20 km Density: 1018 kg/m3

Neutron skins constraint the EOS[@low ρ] of ...

The most neutron rich matter in the Universe

Surface gravity: 1012 higher Escape velocity: 0.6c Rotation rate: few to many times per second Magnetic field: 1012 Earth's!

Pressure @ low ρ Crust thickness

Find a [NS + White Dwarf] Binary

DPSR J1614-2230


Measure the delay in pulse arrival

PB Demorest et al. Nature 467, 1081-1083 (2010)






Determine the mass of the NS

M = 1.97± 0.04 M






M = 1.97± 0.04 M

J1614-2230 needs enough pressure in the core to support its mass against collapse into a black hole.






M = 1.97± 0.04 M

J1614-2230 needs enough pressure in the core to support its mass against collapse into a black hole.

All soft EOS are ruled out!


M-R Curve for NS and EOS

Measure the mass, assume the radius MR curve and NS matter EOS related by GR hydrostatic equation

A. Ohnishi @ YONUPA, Aug.17, 2015 8

M-R curve and EOS

M-R curve and NS matter EOS has 1 to 1 correspondence

TOV(Tolman-Oppenheimer-Volkoff) equation=GR Hydrostatic Eq.

dP

dr=−G

/c2P /c2M4 r3P /c2r21−2GM /rc2

dM

dr=4 r2 /c2 , P=P EOS

E/A

Density(ρB)ρ

02ρ

0

EOS Mass (M)

Radius (R)MR relation

prediction

Judge

Observation

Tolman-Oppenheimer-Volkoff (TOV) Eq.

The up-to-date picture

Calibrate a point @ low density

Pb Radius vs Neutron Star Radius• The 208Pb radius constrains the

pressure of neutron matter at subnuclear densities. Typel + Brown find sharp correlation between P at 2/3 ρ0 and Rn.

• The NS radius depends on the pressure at nuclear density and above. Central density of NS few to 10 x nuclear density.

• Pb radius probes low density, NS radius medium density, and maximum NS mass probes high density equation of state.

• An observed softening of EOS with density (smaller increase in pressure) could strongly suggest a transition to an exotic high density phase such as quark matter, strange matter, or a color superconductor…

J. Piekarewicz, CJH

Chiral EFT calc. of pressure P of neutron matter by Hebeler et al. including three neutron forces (blue band) agree with PREX results but two nucleon only calculations yield smaller P.

does not. Then, we have to conclude that a 3% accuracy inAPV sets modest constraints on L, implying that some ofthe expectations that this measurement will constrain Lprecisely may have to be revised to some extent. To narrowdown L, though demanding more experimental effort, a!1% measurement of APV should be sought ultimately inPREX. Our approach can support it to yield a new accuracynear !!rnp ! 0:02 fm and !L! 10 MeV, well below anyprevious constraint. Moreover, PREX is unique in that thecentral value of !rnp and L follows from a probe largelyfree of strong force uncertainties.

In summary, PREX ought to be instrumental to pave theway for electroweak studies of neutron densities in heavynuclei [9,10,26]. To accurately extract the neutron radiusand skin of 208Pb from the experiment requires a preciseconnection between the parity-violating asymmetry APV

and these properties. We investigated parity-violating elec-tron scattering in nuclear models constrained by availablelaboratory data to support this extraction without specificassumptions on the shape of the nucleon densities. Wedemonstrated a linear correlation, universal in the meanfield framework, between APV and!rnp that has very smallscatter. Because of its high quality, it will not spoil theexperimental accuracy even in improved measurements ofAPV. With a 1% measurement of APV it can allow one toconstrain the slope L of the symmetry energy to near anovel 10 MeV level. A mostly model-independent deter-mination of !rnp of 208Pb and L should have enduringimpact on a variety of fields, including atomic paritynonconservation and low-energy tests of the standardmodel [8,9,32].

We thank G. Colo, A. Polls, P. Schuck, and E. Vivesfor valuable discussions, H. Liang for the densities ofthe RHF-PK and PC-PK models, and K. Kumar for infor-mation on PREX kinematics. Work supported by theConsolider Ingenio Programme CPAN CSD2007 00042

and Grants No. FIS2008-01661 from MEC and FEDER,No. 2009SGR-1289 from Generalitat de Catalunya, andNo. N N202 231137 from Polish MNiSW.

[1] Special issue on The Fifth International Conference onExotic Nuclei and Atomic Masses ENAM’08, edited byM. Pfutzner [Eur. Phys. J. A 42, 299 (2009)].

[2] G.W. Hoffmann et al., Phys. Rev. C 21, 1488 (1980).[3] J. Zenihiro et al., Phys. Rev. C 82, 044611 (2010).[4] A. Krasznahorkay et al., Nucl. Phys. A731, 224 (2004).[5] B. Kłos et al., Phys. Rev. C 76, 014311 (2007).[6] E. Friedman, Hyperfine Interact. 193, 33 (2009).[7] T.W. Donnelly, J. Dubach, and Ingo Sick, Nucl. Phys.

A503, 589 (1989).[8] D. Vretenar et al., Phys. Rev. C 61, 064307 (2000).[9] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R.

Michaels, Phys. Rev. C 63, 025501 (2001).[10] K. Kumar, P. A. Souder, R. Michaels, and G.M. Urciuoli,

http://hallaweb.jlab.org/parity/prex (see section ‘‘Statusand Plans’’ for latest updates).

[11] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda,Phys. Rev. C 82, 054314 (2010).

[12] I. Angeli, At. Data Nucl. Data Tables 87, 185 (2004).[13] B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000); S. Typel

and B.A. Brown, Phys. Rev. C 64, 027302 (2001).[14] R. J. Furnstahl, Nucl. Phys. A706, 85 (2002).[15] A.W. Steiner, M. Prakash, J.M. Lattimer, and P. J. Ellis,

Phys. Rep. 411, 325 (2005).[16] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. 95,

122501 (2005).[17] M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda,

Phys. Rev. Lett. 102, 122502 (2009); M. Warda, X. Vinas,X. Roca-Maza, and M. Centelles, Phys. Rev. C 80, 024316(2009).

[18] A. Carbone et al., Phys. Rev. C 81, 041301(R) (2010).[19] L.W. Chen et al., Phys. Rev. C 82, 024321 (2010).[20] B. A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113

(2008).[21] M. B. Tsang et al., Phys. Rev. Lett. 102, 122701 (2009).[22] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86,

5647 (2001).[23] J. Xu et al., Astrophys. J. 697, 1549 (2009).[24] A.W. Steiner, J.M. Lattimer, and E. F. Brown, Astrophys.

J. 722, 33 (2010).[25] O. Moreno, E. Moya de Guerra, P. Sarriguren, and J.M.

Udıas, J. Phys. G 37, 064019 (2010).[26] S. Ban, C. J. Horowitz, and R. Michaels, arXiv:1010.3246.[27] N. R. Draper and H. Smith, Applied Regression Analysis

(Wiley, New York, 1998), 3rd ed.[28] K. Hebeler, J.M. Lattimer, C. J. Pethick, and A. Schwenk,

Phys. Rev. Lett. 105, 161102 (2010).[29] A.W. Steiner and A. L. Watts, Phys. Rev. Lett. 103,

181101 (2009).[30] D. H. Wen, B. A. Li, and P. G. Krastev, Phys. Rev. C 80,

025801 (2009).[31] I. Vidana, C. Providencia, A. Polls, and A. Rios, Phys.

Rev. C 80, 045806 (2009).[32] T. Sil et al., Phys. Rev. C 71, 045502 (2005).

v090M

Sk7H

FB-8

SkP

HFB

-17SkM

*

Ska

Sk-Rs

Sk-T4

DD

-ME2

DD

-ME1

FSUG

oldD

D-PC

1PK

1.s24N

L3.s25

G2

NL-SV2PK

1N

L3N

L3*

NL2

NL1

0 50 100 150 L (MeV)

0.1

0.15

0.2

0.25

0.3

∆rnp

(fm

)

Linear Fit, r = 0.979Nonrelativistic modelsRelativistic models

D1S

D1N

SGII

Sk-T6SkX SLy5

SLy4

MSkA

MSL0

SIVSkSM

*SkM

P

SkI2SV

G1

TM1

NL-SH

NL-R

A1

PC-F1

BC

P

RH

F-PKO

3Sk-G

s

RH

F-PKA

1PC

-PK1

SkI5

FIG. 3 (color online). Neutron skin of 208Pb against slopeof the symmetry energy. The linear fit is !rnp ¼ 0:101þ0:001 47L. A sample test constraint from a 3% accuracy inAPV is drawn.

PRL 106, 252501 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending24 JUNE 2011

252501-4

PREX/JLAB

The up-to-date picture


…J1614-2230 with EFT does the rest

Trivial? It is a long winding road …


CAUTION NEUTRON

SKIN AHEAD

...from measurable observables

to the neutron skin


Classical picture

form factor: G(q2) = 1e

� �

0⇥(r)

sin qrqr

4�r2 dr

dipolee.g. proton

form

fact

or |G

(q2 )|

→

gaussiane.g. 6Li

momentum transfer |q| →

oscillatinge.g. 40Ca

Fourier

�

Transform

exponential

char

ge d

ensi

ty ρ

(r) →

gaussian

radius r →

sphere withdiffuse edge

charge distribution: ⇥(r) = e(2�)3

� �

0G(q2)

sin qrqr

4�q2 dq


Non-PV e-scatteringElectron scattering γ exchange provides Rp through nucleus FFs

PV e-scatteringElectron also exchange Z, which is parity violatingPrimarily couples to neutron


N"N"

N"

2"

N"N"

N"

2"

N"N"

N"

2"

N"N"

2"

...since...

...to measure ...N"

....construct ....

The shortest of the roads …

Rn#

Assume#surface#thickness#good#to#25%#(MFT)#

Neutron#density#at#one#Q2#

Small#correcAons#for#

###############MEC#

Weak#density#at#one#Q2#

Correct#for#Coulomb#DistorAons#

Measured#APV#

PHYSICAL REVIEW C88, 034325 (2013)

INFORMATION CONTENT OF THE WEAK-CHARGE FORM . . . PHYSICAL REVIEW C 88, 034325 (2013)

0 0.4 0.8 1.2 1.6q(fm-1)

0

0.2

0.4

0.6

0.8

1

F w(q

)

SV-minFSUGoldPREX

qCREX

48Ca

208Pb

Fw×10

FIG. 3. (Color online) Weak-charge form factors with corre-sponding theoretical errors for 48Ca and 208Pb as predicted bySV-min and FSUGold. Note that the theoretical error bars havebeen artificially increased by a factor of 10. Indicated in the figureare the values of the momentum transfer appropriate for PREX-II(q = 0.475 fm−1) and CREX (q = 0.778 fm−1).

the (absolute value) of the correlation as predicted by SV-min and FSUGold. At small momentum transfer, the formfactor behaves as FW (q) ≈ 1 − q2r2

W/6 ≈ 1 − q2r2n/6 so the

correlation coefficient is nearly 1. Note that we have used thefact that the weak-charge radius rW is approximately equal torn [4]. Also note that, although at the momentum transfer of thePREX experiment the low-q expression is not valid, the strongcorrelation is still maintained. Indeed, the robust correlation is

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

q (fm-1)

SV-min

(b)

no swith s

0.6

0.7

0.8

0.9

1.0

(a)

SV-minFSU

Cr n

, FW i

n 20

8 Pb

PR

EX

-II

PR

EX

-II

FIG. 4. (Color online) Correlation coefficient (9) between r208n

and F 208W (q) as a function of the momentum transfer q. Panel (a) shows

the absolute value of the correlation coefficient predicted by SV-minand FSUGold assuming no strange-quark contribution to the nucleonform factor. Panel (b) shows the impact of including the experimentaluncertainty in the strange-quark contribution to the nucleon formfactor. The arrow marks the PREX-II momentum transfer of q =0.475 fm−1. The first dashed vertical line indicates the position ofthe first zero of F 208

W (q), the second one marks the position of thefirst maximum of |F 208

W (q)| (from which the surface thickness can bededuced).

maintained at all q values, except for diffraction minima andmaxima. Given the similar patterns predicted by SV-min andFSUGold, we suggest that the observed q dependence of thecorrelation with rn represents a generic model feature.

Figure 4(b) displays the same correlation, but now we alsoinclude the experimental uncertainty on the strange-quark formfactor. Although the strange-quark contribution to the electricform factor of the nucleon appears to be very small [47],there is an experimental error attached to it that we want toexplore. For simplicity, only results using SV-min are shownwith and without incorporating the experimental uncertaintyon the s quark. We note that an almost perfect correlation atlow-to-moderate momentum transfer gets diluted by about 6%as the uncertainty in the strange-quark contribution is included.Most interestingly, the difference almost disappears near theactual PREX point, lending confidence that the experimentalconditions are ideal for the extraction of r208

n . Finally, given thatthe strong correlation between the neutron radius and the formfactor is maintained up to the first diffraction minima (aboutq = 1.2 fm−1 in the case of 48Ca), the CREX experimentalpoint lies safely within this range (figure not shown).

IV. CONCLUSIONS AND OUTLOOK

In this survey, we have studied the potential impact of theproposed PREX-II and CREX measurements on constrainingthe isovector sector of the nuclear EDF. In particular, weexplored correlations between the weak-charge form factorof both 48Ca and 208Pb, and a variety of observables sensitiveto the symmetry energy. We wish to emphasize that we havechosen the weak-charge form factor rather than other derivedquantities—such as the weak-charge (or neutron) radius—since FW is directly accessed by experiment. To assess correla-tions among observables, two different approaches have beenimplemented. In both cases we relied exclusively on modelsthat were accurately calibrated to a variety of ground-state dataon finite nuclei. In the “trend analysis,” the parameters of theoptimal model were adjusted in order to systematically changethe symmetry energy, and the resulting impact on nuclearobservables was monitored. In the “covariance analysis,” weobtained correlation coefficients by relying exclusively on thecovariance (or error) matrix that was obtained in the processof model optimization. From such combined analysis we findthe following:

(i) We verified that the neutron skin of 208Pb provides afundamental link to the equation of state of neutron-richmatter. The landmark PREX experiment achieved avery small systematic error on r208

n that suggests thatreaching the total error of ±0.06 fm anticipated inPREX-II is realistic.

(ii) We also concluded that an accurate determination ofr208

skin is insufficient to constrain the neutron skin of48Ca. Indeed, because of the significant difference inthe surface-to-volume ratio of these two nuclei, thereis a considerable spread in the predictions of themodels [17]. Given that CREX intends to measurer48

skin with an unprecedented error of ±0.02 fm, thismodel dependence can be tested experimentally [18].

034325-7

INFORMATION CONTENT OF THE WEAK-CHARGE FORM . . . PHYSICAL REVIEW C 88, 034325 (2013)

0 0.4 0.8 1.2 1.6q(fm-1)

0

0.2

0.4

0.6

0.8

1

F w(q

)

SV-minFSUGoldPREX

qCREX

48Ca

208Pb

Fw×10

FIG. 3. (Color online) Weak-charge form factors with corre-sponding theoretical errors for 48Ca and 208Pb as predicted bySV-min and FSUGold. Note that the theoretical error bars havebeen artificially increased by a factor of 10. Indicated in the figureare the values of the momentum transfer appropriate for PREX-II(q = 0.475 fm−1) and CREX (q = 0.778 fm−1).

the (absolute value) of the correlation as predicted by SV-min and FSUGold. At small momentum transfer, the formfactor behaves as FW (q) ≈ 1 − q2r2

W/6 ≈ 1 − q2r2n/6 so the

correlation coefficient is nearly 1. Note that we have used thefact that the weak-charge radius rW is approximately equal torn [4]. Also note that, although at the momentum transfer of thePREX experiment the low-q expression is not valid, the strongcorrelation is still maintained. Indeed, the robust correlation is

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

q (fm-1)

SV-min

(b)

no swith s

0.6

0.7

0.8

0.9

1.0

(a)

SV-minFSU

Cr n

, FW i

n 20

8 Pb

PR

EX

-II

PR

EX

-II

FIG. 4. (Color online) Correlation coefficient (9) between r208n

and F 208W (q) as a function of the momentum transfer q. Panel (a) shows

the absolute value of the correlation coefficient predicted by SV-minand FSUGold assuming no strange-quark contribution to the nucleonform factor. Panel (b) shows the impact of including the experimentaluncertainty in the strange-quark contribution to the nucleon formfactor. The arrow marks the PREX-II momentum transfer of q =0.475 fm−1. The first dashed vertical line indicates the position ofthe first zero of F 208

W (q), the second one marks the position of thefirst maximum of |F 208

W (q)| (from which the surface thickness can bededuced).

maintained at all q values, except for diffraction minima andmaxima. Given the similar patterns predicted by SV-min andFSUGold, we suggest that the observed q dependence of thecorrelation with rn represents a generic model feature.

Figure 4(b) displays the same correlation, but now we alsoinclude the experimental uncertainty on the strange-quark formfactor. Although the strange-quark contribution to the electricform factor of the nucleon appears to be very small [47],there is an experimental error attached to it that we want toexplore. For simplicity, only results using SV-min are shownwith and without incorporating the experimental uncertaintyon the s quark. We note that an almost perfect correlation atlow-to-moderate momentum transfer gets diluted by about 6%as the uncertainty in the strange-quark contribution is included.Most interestingly, the difference almost disappears near theactual PREX point, lending confidence that the experimentalconditions are ideal for the extraction of r208

n . Finally, given thatthe strong correlation between the neutron radius and the formfactor is maintained up to the first diffraction minima (aboutq = 1.2 fm−1 in the case of 48Ca), the CREX experimentalpoint lies safely within this range (figure not shown).

IV. CONCLUSIONS AND OUTLOOK

In this survey, we have studied the potential impact of theproposed PREX-II and CREX measurements on constrainingthe isovector sector of the nuclear EDF. In particular, weexplored correlations between the weak-charge form factorof both 48Ca and 208Pb, and a variety of observables sensitiveto the symmetry energy. We wish to emphasize that we havechosen the weak-charge form factor rather than other derivedquantities—such as the weak-charge (or neutron) radius—since FW is directly accessed by experiment. To assess correla-tions among observables, two different approaches have beenimplemented. In both cases we relied exclusively on modelsthat were accurately calibrated to a variety of ground-state dataon finite nuclei. In the “trend analysis,” the parameters of theoptimal model were adjusted in order to systematically changethe symmetry energy, and the resulting impact on nuclearobservables was monitored. In the “covariance analysis,” weobtained correlation coefficients by relying exclusively on thecovariance (or error) matrix that was obtained in the processof model optimization. From such combined analysis we findthe following:

(i) We verified that the neutron skin of 208Pb provides afundamental link to the equation of state of neutron-richmatter. The landmark PREX experiment achieved avery small systematic error on r208

n that suggests thatreaching the total error of ±0.06 fm anticipated inPREX-II is realistic.

(ii) We also concluded that an accurate determination ofr208

skin is insufficient to constrain the neutron skin of48Ca. Indeed, because of the significant difference inthe surface-to-volume ratio of these two nuclei, thereis a considerable spread in the predictions of themodels [17]. Given that CREX intends to measurer48

skin with an unprecedented error of ±0.02 fm, thismodel dependence can be tested experimentally [18].

034325-7

N"N"

N"

2"

N"N"

N"

2"

QCD-based description of nuclear structure is within reach Complementary experimental approaches will provide stringent test for actual theories

Phase diagram of QCD accessible by different methods @ finite density: from the Lab to the star On the wedge of turning qualitative insight into quantitative understanding

SCALES of NM

PHASES of NM

nuclear physics and cosmology …. from outer space to … to nuclear astrophysics …. why modern...

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