Why is the 5 used in the axial Chiral current formulas?

In summary, the number 5 is used in the axial Chiral current formulas because it is the minimum number of chiral fermions needed to form a non-anomalous gauge theory. This number is significant because it ensures the existence of a gauge anomaly cancellation, which is necessary for a consistent quantum field theory. Additionally, the presence of a chiral symmetry breaking is also related to the number 5, as it is the minimum number of fermion flavors required for spontaneous chiral symmetry breaking. Therefore, the use of the number 5 in axial Chiral current formulas is crucial for the mathematical and theoretical validity of gauge theories and their predictions.
  • #1
Pet Scan
27
1
...the axial Chiral current formulas use the symbol J^5...i.e.,( "J" with a post script 5. ) What is the reason for the "5" ..??
 
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  • #2
I guess I put this in the wrong section, maybe.? since I got no answers... Axial anomaly is a QFT phenomena, but originated in Particle physics. However, chiral axial anomaly was recently "discovered" experimentally in condensed matter / solid state physics...in topological crystalline materials; see, for ex., http://phys.org/news/2015-09-long-sought-chiral-anomaly-crystalline-material.html
So where should I put this post, and how do I transfer it please...in order to get an answer?
Thx;
Pet Scan
 
  • #3
Becausee the five gamma's generate the 5-dimensional Clifford algebra.
 
  • #4
Thanks Neumaier...However, could you dumb it down for me;...real low.?... I know nothing of Clifford algebra...can you give it to me in terms of basic physics...
Thx
 
  • #6
More explicitly; we see in a phenomenological explanation, for example, given here, of the Chiral Magnetic Effect (which is an axial anomaly) .. which I find of great interest... https://www.researchgate.net/publication/47338061_Axial_anomaly_Dirac_sea_and_the_chiral_magnetic_effect ...
under the heading,...
2. The Chiral Magnetic Effect and Landau Levels of Dirac Fermions;
we see, in this derivation of parity violating current J, the difference in Fermi momenta of right and left handed spins (top of page 4), is given as u(R) - u(L) = 2u^5.
It is designated as u^5...(leading to an implied J^5 current density) ...why does he designate it as u^5 ...as opposed to u^4 or even u^6...IOW, from whence is the postscript derived.?

Sorry; I am lost when you speak of anything about Clifford algebra...even as I have tried to read about it o Wiki...I guess I need to learn it in order to get the understanding...
 
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  • #7
I don't know, what you are precisely referring to. An anomaly is defined as a symmetry valid in a classical theory, which implies the conservation of the corresponding current. For the axial U(1) symmetry the current is
$$j_5^{\mu}=\mathrm{i} \overline{\psi} \gamma^5 \gamma^{\mu} \psi,$$
and I guess the ##5## is indeed just due to the appearance of the ##\gamma^5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3## Dirac matrix in the current. It's just a name with not much deeper reason. Now it happens that when quantizing the theory, you cannot keep all symmetries.

For the axial U(1) anomaly it's intuitively understandable what happens. When calculating one-loop corrections to the three-point vertex function with one axial and two vector currents you have a linear divergence. Thus you have to regularize this loop integral. If you like to preserve the symmetry you should find a regulator that doesn't destroy the symmetry. You cannot find such a regulator for the axial U(1) symmetry. The usual suspects don't work: A momentum cutoff destroys the symmetry, because a scale enters, and chiral symmetry doesn't easily respect a scale. The same holds for Pauli-Villars: You have to introduce massive fermions, and these don't respect the chiral symmetry. Dimensional regularization fails, because you don't know what to do with the generically four-dimensional object ##\gamma^5##.

Indeed, a closer analysis shows that you can either keep the vector current conserved or the axial current or any linear combination thereof. You can however never preserve both. Physics tells you which current to conserve usually. E.g., if you have QED the vector current should be conserved, because otherweise you destroy electromagnetic gauge symmetry, because that's the current the em. field couples to. Thus you must break the conservation of the axial U(1).

I have a section on anomalies in my QFT manuscript (Sect. 7.6):

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf
 
  • #8
OK; I think you got it Vanhees...It comes from the Dirac matrix elements...another area I have not sufficient background in which to delve... thank you... I'm sure I will have more questions about the other details of your post later... very interesting.
 

1. What is the Chiral (axial) anomaly?

The Chiral (axial) anomaly is a quantum phenomenon that arises in certain gauge theories, such as quantum electrodynamics (QED) and quantum chromodynamics (QCD). It describes the violation of a symmetry called the chiral symmetry, which is related to the different handedness of particles (left-handed and right-handed).

2. How does the Chiral (axial) anomaly manifest in physics?

The Chiral (axial) anomaly is observed as a difference between the number of left-handed and right-handed particles produced in certain processes, such as the decay of a neutral pion into two photons. This difference is a direct consequence of the violation of chiral symmetry.

3. What is the significance of the Chiral (axial) anomaly in particle physics?

The Chiral (axial) anomaly plays a crucial role in our understanding of the Standard Model of particle physics. It is responsible for the mass generation of certain particles, such as the W and Z bosons, and has important implications for the study of the strong nuclear force.

4. How is the Chiral (axial) anomaly related to topological properties of space-time?

The Chiral (axial) anomaly is intimately connected to the topology of space-time. In particular, it is related to the presence of non-trivial topological structures, such as instantons, that can induce chiral symmetry breaking and therefore the anomaly.

5. What are some current research areas related to the Chiral (axial) anomaly?

Current research on the Chiral (axial) anomaly includes studying its effects in high-energy collisions, such as those at the Large Hadron Collider, and its role in condensed matter systems. There is also ongoing research on the connection between the Chiral (axial) anomaly and other areas of physics, such as gravity and cosmology.

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