Behavior of Current Divergences under SU 3 X SU 3 Published on July 22, 1968

1. Behavior of Current Divergences under SU3 X SU3 Published on July 22, 1968 Murry Gell-Mann – California Institute of Technology R.J. Oakes – Northwestern University B. Renner – California Institute of Technology Presented by Stephen Bello October 1st, 2013

2. Quick Introduction • The aim was to investigate the behavior of the hadron energy density 00 under SU3 X SU3 as well as the related question of how the divergences of the axial-vector currents and strangeness-shaping vectors currents transform. • They also wanted to explore the commutation relations between charges and current divergences. These relations arose from an energy density 00 where the SU3 X SU3violating part is of two terms: the first breaks the chiral symmetry of this but not SU3itself and the second breaks SU3(corresponding in a quark model to a mass-splitting between isotopic singlet and doublet states). • Remember that chirality in particle physics deals with left versus right handed spin. It is left handed if the spin and motion directions are opposite while right handed if the spin and motion point in the same direction. • The proposed behavior is simple. They are all (current divergences) belonging to a single representation of SU3 X SU3and parity. This theory contains just a simply universal parameter c that describes the strength of the SU3 symmetry breaking term relative to the chiral symmetry breaking term. It also determines the commutators of charges with divergences.

3. Left Versus Right Chirality

4. The Transformation Properties of the Current Divergences and the assumptions needed to get them. Some Applications of these properties while focusing on finding and examining the value of c, the relative scale between the two parts of H`

5. Transformation Properties of Current Divergences The algebraic behavior of the current divergence is related to the energy operator H and its properties under SU3 X SU3 through the following equation: Remember that 00 is the hadron energy density. The local generalization of the first equation turns out to be: Note that this local generalization relies heavily on the assumption that the tensor part in 00(x) must commute with all the charges. Only a Lorentz scalar part 00H`(x)(in some Lagrangian models is the mass term) is able to break the symmetry. Let this be our First Assumption.

6. If we decompose H`(x) into a multitude of terms that transform according to SU3x SU3 we can find the commutators of charges and current diverges for the system. • Second Assumption:. Let’s assume for the moment that our operators of isospin 2 are not admitted into the multiplet of the current divergences. This implies that there will be certain • commutators that will vanish: • These two equations were consistent with the experimental data available at the time. Our second assumption only allows components of H`(x) in the (3,3*) + (3*,3) and (1,8) + (8,1) representations of SU3x SU3. • These representations may look weird but we can simplify using our Third Assumption! Let’s assume the (1,8) + (8,1) representation of SU3 X SU3vanishes. This gives us the most simple theory for the behavior of H`(x) simply because it shows the least number of new operators to complete the current divergence multiplet.

7. Now with this new assumption we can derive the formula: H` = -u0 – cus where the value of c is known to be the relative scale between the two parts of H` and is uniquely defined by the transformation properties of the scalar & pseudoscalarnonets, ui & vi, in the (3,3*) + (3*,3) representation. • The value of i = (1,8) and j,k = (0,8). F05 (where i = 0) is not considered because there is no such operator that is partially conserved to anywhere near the same degree as the ones we do consider. What stops us entirely is that such an operator plays no known role in the hadron physical interactions! • Let us go back to the local generalization. The current divergences that follow from that are: • Wi(c) are defined for different values of i as:

8. The commutation of charges with the current divergences can now be read by simply combining the previous equations! • But wait…. • All of these assumptions were based on the fact that c was very, very small compared to the overall unity of the system. If we consider the analogy of –u0 - cuswhile using the mechanical masses of an isotopic doublet and singlet forming a triplet then the assumption about c suggests the doublet-singlet splitting is very small compared to the average mechanical mass. • If we take this in the limit of conservation for all 16 currents then the masses of the baryons go to zero. But if is the eight pseudoscalarmeson masses of the 0- octet that become zero then this is no longer reasonable. • The real world of hadrons seems to be not too far from eight masslesspseudoscalarmesons, SU3 degeneracy and conservation of all 16 currents. • It turns out we are in a world close to one with masslesspions and where SU2x SU2 is conserved.. In such a limit the value of c would be –Sqrt[2]. Since the pion is nearly massless we should expect the actual value of c to be close to the theoretical value.

9. Applications While the focus of my talk is on the value of c, there are other possibilities for applications of this including a discussion of the approximate equality of 2 and K2 decay constants, a small demonstration of the squared-mass formula, and some results for K3 decays. We need to make two new assumptions… 1. Pole dominance for axial current divergences through , K, and  mesons. 2. Application of approximate SU3 symmetry to vertices of certain operator octets involving multiplets with small mixing. We shall only apply this to form factors that are at points far enough away from important singularities that the differences in their distances due to SU3 violations can be neglected.

10. Finding c Consider the vertex of H`(0) between members of a pseudoscalar-meson octet in the low-energy limit: The limit shows the masses role in chiral symmetry breaking and their vanishing in the symmetry limit. Now, this gives us a way of breaking the effects of u0 away from the chirally invariant part in H. According to the paper this separation has yet to be achieved. Consider a more general scalar vertex with pseudoscalar mesons and apply SU3 symmetry to it. Neglecting (`) mixing: where α(0) and β(0) are related through the mass formula.

11. By taking the low-energy limit and ignoring the dependence of α and β on t: Using many different values for (i,j,k) we get the approximation (which states the equality of the two decay constants. Experimentally the two values differ by around 25% so take this with a grain of salt) This leads to: This equation gives us the value for c. This is similar to our theoretical value of –Sqrt[2].

12. Examining the value of c Because c≅ -1.25, this suggests the closeness of our theory to the symmetric limit of SU2x SU2with m = 0 and c = - Sqrt[2]. In fact, this allows us to replace the Second and Third assumptions with the requirements of SU2x SU2 symmetry. We now see SU3x SU3broken in two chains In low-lying multiplets, chiral symmetry appears to be realized via strong coupling to masslesspseudoscalar mesons rather than parity doubling. This is emphasized by Fubini (person) to be a quantitative matter in the real world

13. But wait a second…

14. Our Third Assumption said that the effects of the (1,8) + (8,1) representation of SU3x SU3on H` is so small it can be seen as zero. Let’s show why it is. Much like we did with the scalar and pseudoscalar members of the multiplet (ui and vi) earlier, create two new ones called gj(0) and hj(0). They will transform as follows: Here hk behaves opposite under the charge conjugation to the vk. If we do something similar to how we dealt with the (`) mixing for the contributions of gs in H`(0), we have: In the accurate low-energy limits, γ(0) ≈ 0 and <0|hk|Pk> ≈ 0. So any contribution that could come from this representation would be only from the order of corrections making it quite small.