Effective Field Theory Applied to Nuclei Evgeny Epelbaum, Jefferson Lab, USA PN12, 4 Nov 2004.
Isospin-dependence of nuclear forces Evgeny Epelbaum, Jefferson Lab ECT*, Trento, 16 June 2005.
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Transcript of Isospin-dependence of nuclear forces Evgeny Epelbaum, Jefferson Lab ECT*, Trento, 16 June 2005.
Isospin-dependence of nuclear forces
Evgeny Epelbaum, Jefferson Lab
ECT*, Trento, 16 June 2005
Isospin structure of the 2N and 3N forces
Isospin-breaking nuclear forces in chiral EFT:
Two nucleons
Three nucleons
Summary and outlook
Outline
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
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
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, …
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 ).
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
Class II Class III
Class IV (isospin mixing) Class II
Classes II, III
Isospin-violating 1π-exchange potentialIsospin-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.)
Isospin-violating 2π-exchange potential: order Q4
Class II
Trick(Friar & van Kolck ’99):
take isospin-symmetric potential, , and use and:
for pp and nn
for 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.
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)
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,
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)
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
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
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
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.
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.