isospin-dependence of nuclear forces
DESCRIPTION
ECT*, Trento, 16 June 2005. Isospin-dependence of nuclear forces. Evgeny Epelbaum, Jefferson Lab. Isospin structure of the 2N and 3N forces Isospin-breaking nuclear forces in chiral EFT: Two nucleons Three nucleons Summary and outlook. Outline. Class II (charge independence breaking):. - PowerPoint PPT PresentationTRANSCRIPT
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Isospin-dependence of nuclear forces
Evgeny Epelbaum, Jefferson Lab
ECT*, Trento, 16 June 2005
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Isospin structure of the 2N and 3N forcesIsospin-breaking nuclear forces in chiral EFT:
Two nucleonsThree nucleons
Summary and outlook
Outline
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Class I (isospin invariant forces):
Class III (charge symmetry breaking, no isospin mixing):
Class IV (charge symmetry breaking and isospin mixing):
(Henley & Miller 1979)Isospin structure of the 2N force
Class II (charge independence breaking):
charge reflection
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Conservation of is not suitable for generalization to since, in general: but
Class I (isospin invariant forces):
Class II (charge symmetry conserving):
Class III (charge symmetry breaking):
Generalization to 3 nucleons
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Chiral EFT à la Weinberg
N of loopsN of nucleons N of vertices of type i
N of nucleon fieldsN of powers of the small scale
Unified expansion:
isospin invariant
Vertices:
isospin breaking
van Kolck ’93, ‘95Friar et al. ’03, ’04, …
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Q0
Q1
Q3
Q4
Class I Class II Class III Class IV
Q2
Q5
Hierarchy of the two-nucleon forces
+ pure electromagnetic interactions (V1γ, V2γ, …)
Class I > Class II > Class III > Class IV van Kolck ’93, ’95
(This hierarchy is valid for the specified power counting rules and assuming ).
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Long-range electromagnetic forces
Dominated by the Coulomb interaction, vacuum polarization and the magnetic moment interaction (Ueling ’35, Durand III ’57, Stoks & de Swart ’90). Contribute to Classes I, II, III, IV.Big effects in low-energy scattering due to long range.
πγ - exchange
Worked out by van Kolck et al., ‘98. Contributes to Class II NN force at order Q4 .Numerically small (α/π-times weaker than the isospin-invariant V1π).
Isospin-violating contact terms
Up to order Q5 contribute to 1S0 and P-waves (Classes III, IV):1S0
P-waves, spin & isospin mixing
P-waves, CSB
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Class II Class III
Class IV (isospin mixing) Class II
Classes II, III
Isospin-violating 1π-exchange potential
Charge-dependent πNN coupling constant:
Q4Q3Q2
van Kolck ’93, ‘95; van Kolck, Friar & Goldman ’96; Friar et al. ’04; E.E. & Meißner ‘05
[largely unknown…]
Class IV potential:
where(the NN Hamiltonian is still
Galilean invariant, see Friar et al. ’04.)
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Isospin-violating 2π-exchange potential: order Q4
Class II
Trick(Friar & van Kolck ’99):
take isospin-symmetric potential, , and use and:
for pp and nnfor np, T=1
for np, T=0
Class III
CSB potential (non-polynomial pieces):
where and
Niskanen ’02; Friar et al. ’03, ’04; E.E. & Meißner ‘05.
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Class II
The CIB potential can be obtained using the above trick
Isospin-violating 2π-exchange potential: order Q5
Class III
CSB potential
where
and
(E.E. & Meißner ’05)
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CSB 2π-exchange potential: size estimation
Subleading 2π-exchange potential is proportional to LECs c1, c3 and c4 which are large expect large contribution to the potential at order Q5
r [fm]
In the numerical estimation we use:
GL ’82:
charge independent πN coupling, i.e.: .
dimensional regularization,
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Q3
Q4
Class I Class II Class III
Q5
Hierarchy of the three-nucleon forces
work in progress…
Notice that formally: Class I > Class III > Class II
(in an energy-independent formulation)
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3N force: order Q4
All 3NFs at Q4 are charge-symmetry breaking!
(E.E., Meißner & Palomar ’04; Friar, Payne & van Kolck ‘04)
Class II
Class III
Class III
Feynman graphs = iteration of the NN potential (in an E-
independent formulation)
1/m suppressed
yield nonvanishing 3NF proportional to
yields nonvanishing 3NF proportional to
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Other diagrams lead to vanishing 3NF contributions:
3N force: order Q5
Class II
3N force: order Q5 (E.E., Meißner & Palomar ’04)
Classes II, III
Lead to nonvanishing 3NFs proportional to
Leads to nonvanishing 3NF proportional to ,
Feynman graphs = iteration of the NN potential
1/m suppressed
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Size estimation (very rough)
The strength of the Class III 3NFs:
The strength of the Class II 3NFs: (!)
The formally subleading Class II 3NF is strong due to large values of ’s
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Q4
Q4
Q5
Q5
Role of the Δ
Δ-less EFT EFT with explicit Δ
EFT with explicit Δ’s would probably lead to the nuclear force contributions of a more natural size, since the big portion of the terms is shifted to lower orders.
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Summary
Isospin breaking nuclear forces have been studied up to order Q5.2N force
Outlook Numerical calculations in few-nucleon systems should be performed in
order to see how large the effects actually are.
First contribute at order Q2. Up to Q5, is given by 1γ-, 2γ-, πγ-, 1π-, 2π-exchange & contact terms. Subleading (i.e. order- Q5) 2π-exchange numerically large!The only unknown LECs in the long-range part are the charge dependent πNN coupling constants. They can [in principle] be fixed in PWA.
3N forceFirst contribute at order Q4. Depends on and the unknown LEC .Numerically large CS-conserving force.