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 ofis 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 loops N of nucleons N of vertices of type i N of nucleon fields N of powers of the small scale Unified expansion: isospin invariant Vertices: isospin breaking van Kolck ’93, ‘95 Friar et al. ’03, ’04, …

Q0Q0 Q1Q1 Q3Q3 Q4Q4 Class I Class IIClass III Class IV Q2Q2 Q5Q5 Hierarchy of the two-nucleon forces + pure electromagnetic interactions ( V 1γ, V 2γ, …) 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 Q 4. Numerically small ( α/π -times weaker than the isospin-invariant V 1π ). Isospin-violating contact terms Up to order Q 5 contribute to 1 S 0 and P-waves ( Classes III, IV ): 1S01S0 P-waves, spin & isospin mixing P-waves, CSB

Class IIClass III Class IV (isospin mixing) Class II Classes II, III Isospin-violating 1π -exchange potential Charge-dependent πNN coupling constant: Q4Q4 Q3Q3 Q2Q2 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 Q 4 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 Q 5 Class III CSB potential where and (E.E. & Meißner ’05)

CSB 2π -exchange potential: size estimation Subleading 2π -exchange potential is proportional to LECs c 1, c 3 and c 4 which are large expect large contribution to the potential at order Q 5 r [fm] In the numerical estimation we use: GL ’82: charge independent πN coupling, i.e.:. dimensional regularization,

Q3Q3 Q4Q4 Class I Class IIClass III Q5Q5 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 Q 4 All 3NFs at Q 4 are charge-symmetry breaking! (E.E., Meißner & Palomar ’04; Friar, Payne & van Kolck ‘04) Class II 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 Q 5 Class II 3N force: order Q 5 (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

Q4Q4 Q4Q4 Q5Q5 Q5Q5 Role of the Δ Δ-less EFTEFT 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 Q 5. 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 Q 2. Up to Q 5, is given by 1γ-, 2γ-, πγ-, 1π-, 2π -exchange & contact terms. Subleading (i.e. order- Q 5 ) 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 force First contribute at order Q 4. Depends on and the unknown LEC. Numerically large CS-conserving force.