# What Does Up-Down Asymmetry Mean?

There is strong experimental evidence (reported on in the linked paper), from more than one high energy physics experiment, that up-down asymmetry is present in the decays of certain charmed baryons. The abstract of the paper is follows (emphasis added):

We study the branching ratios and up-down asymmetries in the charmed baryon weak decays of Bc→BDM with Bc(D) with Bc(D) anti-triplet charmed (decuplet) baryon and M pseudo-scalar meson states based on the flavor symmetry of SU(3)F. We propose equal and physical-mass schemes for the hadronic states to deal with the large variations of the decuplet baryon momenta in the decays in order to fit with the current experimental data. We find that our fitting results of (Bc→BDM) are consistent with the current experimental data in both schemes, while the up-down asymmetries in all decays are found to be sizable, consistent with the current experimental data, but different from zero predicted in the literature. We also examine the the processes of Ξ0c→Σ′0KS/KL and derive the asymmetry between the KL/KS modes being a constant.

• Chao-Qiang Geng, Chia-Wei Liu, Tien-Hsueh Tsai, Yao Yu,"Charmed Baryon Weak Decays with Decuplet Baryon and SU(3) Flavor Symmetry" (April 25, 2019).

The experimental data are all consistent at the one sigma level with an up-down asymmetry of exactly 1. The experimental data are inconsistent with the hypothesis of zero up-down asymmetry (in experiments capable of distinguishing between the possibilities with a significance of at least two sigma) at from 2.9 sigma to more than 8 sigma of significance.

As I understand the matter, the existence or non-existence of up-down asymmetry in charmed baryon decays is not a trivial consequence of the physical laws that make up the Standard Model of particle physics (unlike, for example, lepton universality where recent experimentally results have appeared to show violations of this principle at lower statistical significances). Instead, as I understand it, it is an approximate symmetry only and is an "accidental symmetry" in that it isn't automatically and trivially apparent from the physical laws set forth in the Standard Model.

But, at least three studies in the theoretical literature using Standard Model physics, cited in the paper linked above, predicted that up-down asymmetry in charmed baryon decays should be indistinguishable from zero with the experimental precision found in the experiments showing this result. As explained at page 8 of the linked paper:

It is interesting to point out that the up-down asymmetries for all decays are expected to be zero by theoretical studies in Refs. [15, 16, 26] due to the vanishing D-wave amplitudes, which are different from our nonzero results and inconsistent with the current experimental result[.]

Those papers are (with links to the associated pre-prints where located, and otherwise to the abstract):

[15] Q. P. Xu and A. N. Kamal, Phys. Rev. D 46, 3836 (1992),

[16] J. G. Korner and M. Kramer, Z. Phys. C 55, 659 (1992), and

[26] K. K. Sharma and R. C. Verma, Phys. Rev. D 55, 7067 (1997).

I assume that the result has not made headlines because, while the experimental data is pretty much irrefutable since it exceeds the five sigma "discovery" threshold in the context of a very understood tried and true experimental set up used in many other contexts without apparent major flaws, the theoretical predictions are not emphatically certain and irrefutable, so it isn't entirely certain that this is really experimental evidence of "new physics."

Still, this could certainly be a very big deal, except for one thing. Even after reading this paper fairly carefully, I don't really understand what is meant by up-down asymmetry in this context, or what a particular numerical value of up-down asymmetry in the decay of a baryon means. The formula for the numerical value appears to be (9) in the linked paper, but I am at a loss to articulate what this means in words, or even to track back through the definitions of the variables and terms used in it.

The paper appears to be talking about a form of isospin violation described in Wikipedia as follows:

Isospin is the symmetry transformation of the weak interactions. The concept was first introduced by Werner Heisenberg in nuclear physics based on the observations that the masses of the neutron and the proton are almost identical and that the strength of the strong interaction between any pair of nucleons is the same, independent of whether they are protons or neutrons. This symmetry arises at a more fundamental level as a symmetry between up-type and down-type quarks. Isospin symmetry in the strong interactions can be considered as a subset of a larger flavor symmetry group, in which the strong interactions are invariant under interchange of different types of quarks. Including the strange quark in this scheme gives rise to the Eightfold Way scheme for classifying mesons and baryons.

Isospin is violated by the fact that the masses of the up and down quarks are different, as well as by their different electric charges. Because this violation is only a small effect in most processes that involve the strong interactions, isospin symmetry remains a useful calculational tool, and its violation introduces corrections to the isospin-symmetric results.

But, even after reading this, I still don't really understand what this means in a concrete nuts and bolts sense in this paper.

So my question is, in the context of the article linked above, what does up-down asymmetry mean and what do particular numerical values of it mean (e.g. is the numerical value a ratio of something to something else, and if so, what)?

There are some obvious followup questions the could be related to the answer to that question (e.g. what could be wrong with the predictions in the theoretical literature and what changes to the Standard Model could give rise to this if the theoretical predictions are correct), but I am limiting this question to just this very fundamental definitional issue and will save other question for a time at which I understand what the quantity that is measured to be anomalous means.

Why? Take the quark example of $$Σ$$ baryon ,
it has a $$Σ^+$$ going to $$uus$$ and a $$Σ^-$$ going to $$dds$$