1. Nuclear Forces - Lecture 3 -
12th CNS International Summer School (CNSSS13)
28 August to 03 September CNS Wako Campus
Nuclear Forces - Lecture 3 -
QCD and Nuclear Forces;
Effective Field Theory (EFT) for Low-Energy QCD
R. Machleidt
University of Idaho
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

2. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Prologue
In Lecture 2, we have seen how beautifully the meson theory of nuclear forces works. This may suggest that we are done with the theory of nuclear forces.
Well, as it turned out, the fundamental theory of the strong interaction is QCD (and not meson theory). Thus, if we want to understand the nuclear force on the most fundamental level, then we have to base it upon QCD.
So, we have to start all-over again; this time from QCD.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

3. Lecture 3: QCD and Nuclear Forces; EFT for Low-Energy QCD
The nuclear force in the light of QCD
QCD-based models (“quark models”)
The symmetries of low-energy QCD
An EFT for low-energy QCD: Why?
An EFT for low-energy QCD: How?
Power counting
The Lagrangians
The Diagrams: Hierarchy of nuclear forces
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

4. The nuclear force in the light of QCD
If nucleons do not overlap, …. ?
then we have two separated colorless objects. In lowest order, they do not interact. This is analogous to the interaction between two neutral atoms. Like the Van der Waals force, the nuclear force is a residual interaction. Such interactions are typically weak as compared to the “pure” version of the force (Coulomb force between electron and proton, strong force between two quarks, respectively).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

5. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The analogy
Van der Waals force
Nuclear force + - + -
Residual force between two electric charge-neutral atoms
Residual force between two color charge-neutral nucleons
Note that residual forces are typically weak as compared to the “pure” version of the force (order of magnitude weaker).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

6. No! No! How far does the analogy go?
What is the mechanism for each residual force?
Van der Waals force + -
Residual force between two electric charge-neutral atoms
Nuclear force
Residual force between two color charge-neutral nucleons
No!
Because it would create a force of infinite range (gluons are massless), but the nuclear force is of finite range.
Two-photon exchange (=dipole-dipole interaction)
No!
The perfect analogy would be a two-gluon exchange.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

7. Non-overlapping nucleons, cont’d
But they could interact by quark/anti-quark exchange:
This is meson exchange!
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

8. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
When nucleons DO overlap, we have a six-quark problem with non-perturbative interactions between the quarks (non-perturbative gluon-exchanges). A formidable problem! ?
Attempts to calculate this by lattice QCD are under way; see e.g., T. Hatsuda, Prog. Part. Nucl. Phys. 67, 122 (2012); M.J. Savage, ibid. 67, 140 (2012).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

9. QCD-based models (“quark models”)
To make 2- and 3-quark (hadron spectrum) or 5- or 6-quark (hadron-hadron interaction) calculations feasible, simple assumptions on the quark-quark interactions are made. For example, the quark-quark interaction is assumed to be just one-gluon-exchange or even meson-exchange.
This is, of course, not real QCD. It is a model; a “QCD-inspired” model.
The positive thing one can say about these models is that they make a connection between the hadron spectrum and the hadron-hadron interaction.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

10. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Conclusion
Quark models are not QCD. So, they are not the solution.
Lattice QCD is QCD, but too elaborate for every-day nuclear physics.
What now?
Let’s take another look at QCD.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

11. The symmetries of low-energy QCD
The QCD Lagrangian
Symmetries of the QCD Lagrangian
Broken symmetries
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

12. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The QCD Lagrangian
with
where r=red, g=green, and b=blue,
and the gauge-covariant derivative
and the gluon field tensor
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

13. Symmetries of the QCD Lagrangian
Quark masses
Assuming
the QCD Lagrangian for just up and down quarks reads
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

14. Symmetries of the QCD Lagrangian, cont’d
Introduce projection operators
“Right-handed”
“Left-handed”
Properties
Complete
Idempotent
Orthogonal
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

15. Symmetries of the QCD Lagrangian, cont’d
Consider Dirac spinor for a particle of large energy or zero mass:
Thus, for mass-less particles, helicity/chirality eigenstates exist
and the projection operators project them out
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

16. = “Chiral Symmetry” Symmetries of the QCD Lagrangian, cont’d
For zero-mass quarks,
quark field of right-handed chirality
quark field of left-handed chirality:
the QCD Lagrangian can be written
= “Chiral Symmetry”
(Because the mass term is absent; mass term destroys chiral symmetry.)
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

17. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Interim summary
(approximate) Chiral Symmetry
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

18. Chiral symmetry and Noether currents
Noether’s Theorem
Six conserved currents
3 left-handed currents
3 right-handed currents
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

19. Note: “vector” is Isospin!
Alternatively, …
3 vector currents
3 axial-vector currents
6 conserved “charges”
Note: “vector” is Isospin!
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

20. Explicit symmetry breaking due to non-zero quark masses
The mass term that we neglected breaks chiral symmetry explicitly:
with
Both terms break chiral symmetry,
but the 1st term is invariant under
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

21. Spontaneous symmetry breaking
From the chiral symmetry of the QCD Lagrangian, one would expect that “parity doublets” exist; because:
Let
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

22. Spontaneous symmetry breaking, cont’d
What would parity doublets look like?
Nucleons of positive parity: p(1/2+,938.3), n(1/2+,939.6), I=1/2;
nucleons of negative parity: N(1/2-,1535), I=1/2.
But, the masses are very different: NOT a parity doublet!
A meson of negative parity:
the “same” with positive parity:
But again, the masses are very different: NOT a parity doublet!
Conclusion: Parity doublets are not observed in the low-energy hadron spectrum.
Chiral symmetry is spontaneously broken.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

23. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Definition:
When the ground state does not have the same symmetries as the Langrangian, then one speaks of “spontaneous symmetry breaking”.
Classical examples
Lagrangian has rot. symmetry;
groundstate has not.
Symmetry is spontaneously broken.
Lagrangian and ground state have both rotational symmetry.
No symmetry breaking.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

24. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Goldstone’s Theorem
When a continuous symmetry is broken, then there exists a (massless) boson with the quantum numbers of the broken generator.
Here: The 3 axial generators are
broken, therefore, 3 pseudoscalar bosons
exist: the 3 pions
This explains the small mass of the pion. The pion mass is not exactly zero, because the u and d quark masses are no exactly zero either.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

25. More about Goldstone Bosons
The breaking of the axial symmetry in the QCD ground state (QCD vacuum) implies that the ground state is not invariant under axial transformations, i.e.
Thus, a physical state must be generated by the axial charge,
Goldstone bosons interact weakly.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

26. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Summary
QCD in the u/d sector has approximate chiral symmetry; but this symmetry is broken in two ways:
explicitly broken,
because the u and d quark masses are not exactly zero;
spontaneously broken:
i.e., axial symmetry is broken,
while isospin symmetry is intact.
There exist 3 Goldstone bosons: the pions
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

27. Knowing the symmetries, we can now apply …
Weinberg’s
“Folk Theorem”
Weinberg
R. Machleidt
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13) 27

28. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Weinberg’s Folk Theorem is the stepping stone to what has become known as an Effective Field Theory (EFT). Here, in short, the reasoning:
QCD at low energy is non-perturbative and, therefore, not solvable analytically in terms of the “fundamental” degrees of freedom (dof); quarks and gluons are ineffective dof at low energy.
Below the chiral symmetry breaking scale pions and nucleons are the appropriate dof.
Moreover, pions are Goldstone bosons which typically interact weakly. Thus, there is hope that a perturbative approach might work.
To ensure the connection with QCD, we must observe the same symmetries as low-energy QCD, particularly, spontaneously broken chiral symmetry, such that our chiral EFT can be indentified with low-energy QCD.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

29. How to do this EFT program? Take the following steps:
Write down the most general Lagrangian including all terms consistent with the assumed symmetries, particularly, spontaneously broken chiral symmetry. (Note: There will be infinitely many terms.)
Calculate Feynman diagrams (Note: There will be infinitely many diagrams.)
Find a scheme for assessing the importance of the various diagrams (because we cannot calculate infinitely many diagrams).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

30. The organizational scheme
“Power Counting”
Chiral Perturbation Theory (ChPT)
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

31. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The Lagrangian
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

32. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The pi-pi Lagrangian
At order two (leading order)
Derivative part
with (any number of pion fields).
Goldstone bosons can only interact when they carry momentum → derivative coupling Lorentz invariance → even number of derivatives.
Plus symmetry breaking mass term:
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

33. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The pi-N Lagrangian with one derivative (=lowest order or “leading” order)
Non-linear realization of chiral symmetry; Callan, Coleman, Wess, and Zumino, PR 177, 2247 (1969).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

34. pi-N Lagrangian, cont’d
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

35. pi-N Lagrangian with two derivatives (“next-to-leading” order)
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

36. The N-N Langrangian (“contact terms”)
At order zero (leading order):
At order two:
At order four:
R. Machleidt
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13) 36

37. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
Why contact terms?
Two reasons:
Renormalization:
Absorb infinities from loop integrals.
A physics reason (see next slide).
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

38. + + + … What is the physics of contact terms?
Consider the contribution from the exchange of a heavy meson + +
+ …
Contact terms take care of the short range structures without resolving them.
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

39. Remember the homework we have to do✔
Write down the most general Lagrangian including all terms consistent with the assumed symmetries, particularly, spontaneously broken chiral symmetry. (Note: There will be infinitely many terms.)
Calculate Feynman diagrams (Note: There will be infinitely many diagrams.)
Find a scheme for assessing the importance of the various diagrams (because we cannot calculate infinitely many diagrams). ✔
R. Machleidt
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13) 39

40. Nuclear Forces - Lecture 3 QCD & Nucl. Forces (CNSSS13)
The Diagrams
Power counting for Feynman diagrams
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)

41. The Hierarchy of Nuclear Forces
2N forces
3N forces
4N forces
Leading Order
The Hierarchy of Nuclear Forces
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order

42. In Lecture 4, we will discuss the diagrams in more detail
This finishes Lecture 3.
In Lecture 4, we will discuss the diagrams in more detail
A pedagogical review article about all this:
R. Machleidt and D.R. Entem,
Chiral Effective Field Theory and Nuclear Forces,
Physics Reports 503, 1 (2011).
R. Machleidt
R. Machleidt
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13)
Nuclear Forces - Lecture QCD & Nucl. Forces (CNSSS13) 42