View
226
Download
4
Category
Preview:
Citation preview
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
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/)
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
courtesy R. Rothcourtesy R. Roth
Effective Field Theories
courtesy R. Rothcourtesy R. Roth
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]
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force
LO (Q0)
NLO (Q2)
N2LO (Q3)
N3LO (Q4)
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
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 •
courtesy R. Rothcourtesy R. Roth
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]
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force
LO (Q0)
NLO (Q2)
N2LO (Q3)
N3LO (Q4)
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
26 rubin | frühjahr 12
�).�!,3%1.!3)5%1�� 2823%-!3)2#(%1� 4!'!.'�)23�-<',)#(��6%..�-!.�$!2��!4-� %)3��/.3).44-�$41#(�%).�%.$,)#(%2��)3!3%1�$!123%,,3���)%�-<',)#(%.��/2)3)/.%.�$%1��4!1+2�6%1$%.�$!"%)�$41#(�%).%�%.$!,)#(%��.9!(,�5/.��4.+3%.�).�%).%-��)33%1�"#$%&%&'&()*+,&-&*.,//&$0#$123,&$2(%*,-/*%).�'%%)'.%3%1��42'!.'204.+3�&=1�.4-%!1)2#(%��)-4,!3)/.%.�5/.�(!$1/.)2#(%.� %!/"!#(34.'2'1<:%.�$41#(�%).%�23!3)23)2#(%��;(%14.'���)%�6)1$�).2"%2/.$%1%�&=1�$)%� %23)--4.'�$%1�23!3)2#(%.��)'%.2#(!&!3%.�5/.��!$1/.%.�5/.��14.$�!4&��/(.%��%16%.$4.'�%70%1)-%.3%,,%1��!3%.��$�(��!"�).)3)/���6)%�%36!�)(1%1��!22%.��%1&/,'!$&,45*6(%&76(8/)*�!�$%1�.4-%1)2#(%��4&6!.$�2/,#(%1��)!
-4,!3)/.%.�2%(1�2#(.%,,�-)3�$%1��1<:%�$%2��)33%12�6;#(23��2).$�$)%��.6%.$4.'%.�$%1!9%)3�-%)23%.2�!4&�$)%��)'%.2#(!&3%.�%).9%,!.%1��!$1/.%.�"%2#(1;.+3���).%.�!,3%1.!!3)5%.�4.$�+/-0,%-%.3;1%.� 4'!.'�")%!3%.�%A%+3)5%��%,$3(%/1)%.��������"%)�$%.%.�-!.�$!2��1/",%-�$!$41#(�5%1%).&!#(3��$!22�16(*,$$&3&"6(/&*95:(#1&(&*,%(#$,&$/*2(8*$!"%)�).��!4&�.)--3��$!22�2)#(�$)%��=,3)'!+%)3�$%1� %1%#(.4.'�!4&�%).%.�+,%).%.� %!1%)#(�"%2#(1;.+3���)%�$!1!42�1%24,3)%1%.$%��%A%+3)5%� %2#(1%)"4.'�)23�.41�).�%).%-�"%!23)--3%.��.%1')%��"96���"23!.$2"%1%)#(�5/.��4!1+2�4.$��,4/.%.�'=,3)'��
�)%2%1��14.$'%$!.+%�)23�%).�6/(,"%!+!..3%2��1).9)0�).�$%1��(82)+��/�23%,,3�"%)!20)%,26%)2%�$)%�!42�$%1��#(4,%�"%+!..3%��/1!-%,�&=1�$)%��.9)%(4.'2+1!&3�4.2%1%1��1$%�����-�'��%).%��;(%14.'�$%2��%63/.2#(%.��1!5)3!3)/.2'%2%39%2�$!1���)%�'),3�.41�&=1��"!23;.$%�96)2#(%.�%).%-��<10%1�4.$�$%1��1$!/"%1C;#(%��$)%�)-��%1',%)#(�94-��1$1!$)!42�+,%).�2).$���)%�!,,'%-%).'=,3)'%��/1-%,�$%2��1!5)3!3)/.2'%2%39%2�)23�+/-0,)9)%13%1��)%�+!..�-!.�$)%2%��$%%.�!4&�(!$1/!
.)2#(%��823%-%�!.6%.$%.���%1�%.32#(%)!$%.$%��20%+3�,)%'3�).�$%1��!32!#(%��$!22�$)%��8.!-)+�5/.��!$1/.%.��9� ��"%)-� 42!-!-%.31%A%.�5/.�96%)��1/3/.%.�-)3�.)%$!1)'%1��.%1')%��.)#(3�4.-)33%,"!1�5/.��/1!';.'%.�!"(;.'3��$)%�)-��..%1%.�$%2��1/!3/.2�96)2#(%.��4!1+2�4.$��,4/.%.�!",!4!&%.���)%� %2#(1%)"4.'�2/,#(%1��823%-%�)23�$!(%1�5)%,�%B9)%.3%1��4.$�%).&!#(%1��-<'!,)#(��6%..�-!.��!$1/.%.�!,2��!.9%2�"%!31!#(3%3�4.$�.)#(3�!,2�942!--%.'%2%393%��"*%+3%�-)3�%).%1�+/-0,)9)%13%.��314+341���<,,)'�!.!,/'�,;223�2)#(�"%)20)%,26%)2%�).�$%1�-/$%1.%.��4!.3%.#(%-)%�$)%��314+!341�5/.��3/-%.�4.$��/,%+=,%.�2%(1�01;9)!2%�"%23)--%.��/(.%�$!&=1��.&/1-!3)/.%.�="%1�$%.��4&"!4�$%2��%1.2�/$%1�$%2��4!+,%/.2�5%16%.$%.�94�-=22%.���)%� 5%1%).&!#(3%� %2#(1%)"4.'� $%1�
�4!.3%.#(1/-/$8.!-)+�"%)�.)%$1)'%.�
�""�����)%��%#(2%,6)1+4.'�96)2#(%.�$%.��4+,%/.%.��!,2�$41#('%(%.$%1��31)#(�$!1'%23%,,3��6)1$�)-��%2%.3!,)#(%.�$41#(�$)%��1;&3%�96)2#(%.�96%)��%),#(%.�"%23)--3���)%�+<..%.�+,%).%��%)#(6%)3%.�(!"%.��!,2��4.+3�$!1'%23%,,3��/$%1�,!.'1%)#(6%)3)'�2%).��'%231)#(%,3%��).)%.���)/.%.!423!42#(��
Zwei-Nukleon-Kraft
Führender Beitrag
Korrektur 1. Ordnung
Korrektur 2. Ordnung
Korrektur 3. Ordnung
Drei-Nukleon-Kraft Vier-Nukleon-Kraft
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 •
courtesy R. Rothcourtesy R. Roth
The Hoyle State
E. Epelbaum et al, PRL 106, 192501 (2011)
The Hoyle State (…the holy grail)
The Hoyle State (…the holy grail)
The Hoyle State (…the holy grail)
BB
The Hoyle State (…the holy grail)
BBstellar burning
courtesy R. Rothcourtesy R. Roth
The Hoyle State
courtesy R. Rothcourtesy R. Roth
Ab initio Nuclear Structure
courtesy R. Rothcourtesy R. Roth
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
The PHASES: second constraint
© CERN © NASA
It’s always just nuclear matter
© SciencePhotoLibrary
The PHASES: second constraint
It’s always just nuclear matter
The PHASES: second constraint
Modern nuclear physics is about...
➜ Unravelling the phases of nuclear matter
LRP Nuclear Science Advisory Committee(2008)
The PHASES: second constraint
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
…
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
…
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
…
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)
18-OM smaller 55-OM lighter … same EOS
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
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
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
Determine the mass of the NS
M = 1.97± 0.04 M
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
Determine the mass of the NS
M = 1.97± 0.04 M
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
Determine the mass of the NS
M = 1.97± 0.04 M
J1614-2230 needs enough pressure in the core to support its mass against collapse into a black hole.
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
Determine the mass of the NS
M = 1.97± 0.04 M
J1614-2230 needs enough pressure in the core to support its mass against collapse into a black hole.
The most neutron rich matter in the Universe
Measure the delay in pulse arrival
PB Demorest et al. Nature 467, 1081-1083 (2010)
Find a [NS + White Dwarf] Binary
Determine the mass of the NS
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!
The most neutron rich matter in the Universe
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
Calibrate a point @ low density
…J1614-2230 with EFT does the rest
Trivial? It is a long winding road …
Calibrate a point @ low density
CAUTION NEUTRON
SKIN AHEAD
...from measurable observables
to the neutron skin
Trivial? It is a long winding road …
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
Trivial? It is a long winding road …
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
Trivial? It is a long winding road …
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
Recommended