Isospin-dependence of nuclear forces

Evgeny Epelbaum, Jefferson Lab

ECT*, Trento, 16 June 2005

Isospin structure of the 2N and 3N forcesIsospin-breaking nuclear forces in chiral EFT:

Two nucleonsThree 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 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 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.

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.