Pions are Special Contents Chiral Symmetry and Its Breaking Symmetries and Conservation Laws Goldstone Theorem The Potential Linear Sigma Model Wigner

Transcription

1 Lecture 3 Pions as Goldstone Bosons of Chiral Symmetry Breaking Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora

2 Pions are Special Contents Chiral Symmetry and Its Breaking Symmetries and Conservation Laws Goldstone Theorem The Potential Linear Sigma Model Wigner-Weyl Weyl Mode Nambu-Goldstone Mode Explicit Chiral Symmetry Breaking Partially Conserved Axial Current Connecting with Experiment

3 Pions are Special Adapted from M.R. Pennington: Swimming with Quarks, Veracruz, Mexico, Light meson properties are tied to the dynamical breakdown of a symmetry called chiral symmetry. They are hadrons as well as Goldstone Bosons. What is all this about?

4 Pions are Special Pions were predicted by Yukawa to be the mediators of the strong nuclear force between protons and neutrons whose range is only meters. Pions are the lightest of hadrons. They do not have zero mass. Otherwise strong force will be a long range force. A typical meson like a ρ has a mass of 770 MeV while the nucleon has a mass of 940 MeV. This is consistent with a consituent u,d, mass of around 300 MeV. However, pions only weigh about 140 MeV, which is 1/5th of the mass of the ρ. This cannot be an accident.

5 Chiral Symmetry & Its Breaking Quarks have helicity +1/2 or -1/2. Helicity depends on the observer. For massless quarks, it becomes a good quantum number and can be identidied with chirality. Dirac equation for massless quarks is: Recall the γ 5 operator: It commutes with all the γ matrices. Thus:

6 Chiral Symmetry and Its Breaking Thus we can work with linear combinations of ψ and γ 5 ψ: L and R mean helicity -1/2 and +1/2 respectively. The massless Lagrangian: can also be written as which has enhanced symmetry: The mass term mixes the chiral partners:

7 Chiral Symmetry and Its Breaking Its spontaneous breaking leads to large effective quark masses and the existence of Goldstone bosons: pions. Nobel Prize 2008: for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics quark-anti-quark

8 Symmetry and Conservation Laws Noether s Theorem: Any continuous transformation which leaves the action invariant implies a conserved current: This implies a conserved charge: Other examples of conserved charges are the electric charge, the baryon number, etc.

9 Goldstone Theorem Explicit Symmetry Breaking: Example: For explicit chiral symmetry breaking in QCD, H 1 contains mass terms for light quarks. Spontaneous Symmetry Breaking: Example: Appearance of pions and large effective quark masses is due to SSB or dynamical chiral SB.

10 Goldstone Theorem Suppose A(x) and B(x) are some local operators which transform into each other under the symmetry charge Q. Suppose the SSB condition: This implies: and Q is a broken charge: Then for some state n which is a mass less excitation or a massless Goldstone boson.

11 Goldstone Theorem Note that despite the peculiarity of the ground state, U corresponding to Q is still the symmetry of H. Thus: This naturally implies:

12 Goldstone Theorem We now simplify the left hand side: Insert a unit operator built from the complete set of eigen-functions of the 4-momentum operator: Writing charge Q as current J and using translations:

13 Goldstone Theorem Operation of momentum operator on its eigen-states gives: We can now carry out space integration: The delta function indicates that only states with 3 vector contribute. This is represented by primed states: Hermiticity of the symmetry current operators gives:

14 Goldstone Theorem Hence:

15 Goldstone Theorem Example: Lagrangian with 2 scalar fields with μ 2 and λ both positive. This is invariant under the following continuous transformation Correspondingly, the conserved current and charge are:

16 Goldstone Theorem Start from equal time commutation relations: We can then show that: For SSB, one of these commutators has to be non-zero in vacuum. We make the following choice.

17 Goldstone Theorem Start from equal time commutation relations: Let be the state which corresponds to the Goldstone boson : with: Normalize: The above equation is satisfied if:

18 Lorentz covariance: Goldstone Theorem RHS: LHS: Therefore: Current conservation implies:

19 The Potential The original Lagrangian: Consider λ=0 for a moment. Then Corresponding equations of motion are (wrong sign): Correct equations for scalars of mass μ are:

20 The Potential The problem of wrong sign could be remedied by choosing μ 2 <0 which implies we start from massive σ and π. The potential then looks like: Oscillations in any of the two direction cost energy corresponding to these excitations being massive.

21 The Potential For SSB we consider μ 2 >0 and λ>0 in: The potential now looks like: There is not one ground state but an infinity along a circle defined by:

22 The Potential The condition of SSB is like choosing one of the ground states. Let us choose: Obviously, QFT should be developed with perturbations around the vacuum and not about origin. Thus we define: In terms of the new field, the Lagrangian is Rotational degree of freedom is masless and radial massive.

23 Linear Sigma Model Consider the following Lagrangian which has meson-nucleon nucleon interactions: [Schwinger , Polkinghorne , Gell- Mann and Levy ] SU(2) L x SU(2) R - Model N=(p,n p,n) is an iso-doublet doublet. π = (π 1,π 2,π 3 ) is an iso-triplet and σ is the iso-scalar scalar partner of the pion. As usual:

24 This Lagrangian is invariant under the following SU(2) V vector transformations: Conserved quantities: Linear Sigma Model Tha Lagrangian is also invariant under the SU(2) A axial transformations: Conserved quantities:

25 The symmetry charges of the two transformations satisfy the commutation relations: Linear Sigma Model These are SU(2) L x SU(2) R relations which can be checked by defining: We can now confirm that:

26 Linear Sigma Model We can work either with the symmetry transformations SU(2) V X SU(2) A or SU(2) L X SU(2) R. As the nucleons are massless, the SU(2) L X SU(2) R transformations correspond to left and right handed nucleons respectively. The symmetry will be broken spontaneously for μ 2 & λ>0. As usual, the circle of minima lies at: We can choose: The symmetry of the Lagrangian is broken spontaneously from SU(2) L X SU(2) R to SU(2) V.

27 Linear Sigma Model Using the easily derivable commutation relation: it is readily seen that: It implies that the axial vector charges no longer annihilate the vacuum: However, the vector charges continue to do so:

28 Linear Sigma Model Make the change of variables: The Lagrangian acquires the following form: Thus through the SU(2) L X SU(2) R symmetry breaking of the linear sigma model, three massless goldstone bosons (pions pions) emerge. Nucleons and the sigma acquire mass:

29 Wigner-WeylWeyl Mode Symmetry realized in Wigner-Weyl Weyl mode: The standard realization of a symmetry in the energy spectrum is called Wigner-Weyl Weyl mode. For example the rotational symmetry of a three dimensional system leads to a (2l+ l+1)-fold degeneracy of the spectrum. This is an example of the Wigner-Weyl Weyl mode of symmetry. The mass terms have the correct sign. SU(2) V X SU(2) A or SU(2) L X SU(2) R symmetry is realized.

30 Wigner-WeylWeyl Mode Note that the three pions alone do not form an irreducible representation of the SU(2) group. Together with a σ, they form (1/2,1/2) representation. When chiral symmetry is fully realized in the Wigner-Weyl Weyl mode, there ought to be a scalar partner of the pions with the same mass as the pions. Otherwise Lagrangian loses the symmetry. Recall that the variation of the π field is proportional to the σ field in the SU(2) A transformations. And vice versa: variation of the σ field involves the π field. Thus chiral symmetry requires the existence of parity partners of the same mass.

31 Nambu-Goldstone Mode It was in the late 50 s Nambu and Goldstone discovered a new way through which a symmetry of a system can manifest itself: spontaneous breaking of the symmetry. This realization of a symmetry is called Nambu-Goldstone mode. In this mode, the symmetry is not realized in the particle mass/energy spectrum. Parity partners are non-degenerate in masses.

32 Explicit Symmetry Breaking Let us look at the potential: Minimization conditions: Thus in the presence of c-term, minimization condition is no longer:

33 Explicit Symmetry Breaking Minimization conditions are now: This condition automatically picks out a vacuum. We thus make the change of variables again: Let us see how the new potential looks. Concentrate only on the quadratic terms to deduce the masses.

34 Explicit Symmetry Breaking We thus get: Mass spectrum Choose c negative c -> -c

35 Partially Conserved Axial Current (PCAC) We can readily check that: What is c? We thus have (PCAC): In the limit when c=0, m π =0 and we recuperate axial vector current conservation:

36 Partially Conserved Axial Current (PCAC) Recall from general discussion on the Goldstone theorem: 1. Why did we use f π instead of v? 2. What is f π? It is called pion decay constant. Why? 3. The above is the matrix element for the pion decay, i.e., one pion state to the vacuum through the axial current. As pion is a negative parity particle, it can be connected to the vacuum only through the axial current % of the time, a charged pion decays to a muon and its neutrino.

37 Partially Conserved Axial Current (PCAC) Let us calculate the decay rate for the process: We find: Note that f π appears as the number characterizing the decay of the pion. Hence pion leptonic decay constant. The experimental value is: After including radiative corrections and comparing with the experiment, we deduce:

38 Connecting with Experiment Show: At classical level: Compare with: Thus: Check: This fits to within 30%!!! Can we understand it all starting from the fundamental dynamics of quarks?

39 What Next? How do the quarks interact with each other? How can we construct a Lagrangian describing interactions between these fundamental degrees of freedom? If the interactions are described by a gauge theory, what and how many are the gauge bosons? What are the similarities and differences with, e.g., QED, the gauge theory we already know.