Contents

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

Hadron supersymmetry (Miyazawa 66, Miyazawa 68) is an approximate (dynamically broken) supersymmetry among the experimentally observed spectra of hadrons, hence an approximate symmetry of quantum hadrodynamics, relating the masses of mesons (which are bosons) to baryons (which are fermions) and further to tetraquarks (which are again bosons).

This experimentally observed hadron flavor supersymmetry is in contrast to the more popular but also more speculative idea of color charge supersymmetry (super QCD, MSSM), which hypothesizes that quarks and gluons have superpartners. While these (squarks and gluinos) have not been observed, even single quarks have never been observed, due to their confinement inside hadrons.

In terms of geometric engineering of QFT via intersecting D-brane models, hadron supersymmetry means that the flavor brane is supersymmetric, while the color brane may not be.

## Theory

From the constituent quark model of hadrons in quantum chromodynamics, the phenomenon of hadron supersymmetry follows (Catto-Gürsey 85, Catto-Gürsey 88) from the fact that

1. both single antiquarks $\bar q$ as well as diquarks $q\!q$ carry the same color charge $\overline{\mathbf{3}} \in Rep_\mathbb{C}(SU(3)_c)$,

2. color charge predominantly determines the properties of bound states in QCD,

3. the exchange of anti-quarks with diquarks inside the bound state with yet another quark exchanges mesons with baryons,

this gives an approximate symmetry of confined QCD as shown on the left here:

$\array{ \text{antiquark} & & \text{diquark} \\ \phantom{q} \bar q & \leftrightarrow & \phantom{q} q\!q \\ q \bar q & \leftrightarrow & q q\!q & \leftrightarrow & \bar q\!\!\bar{q} q\!q \\ \text{meson} & &\text{baryon} & & \text{tetraquark} }$

Dually, both quarks as well as anti-diquarks carry color charge $\mathbf{3} \in Rep_\mathbb{C}(SU(3)_c)$, which leads to a further such transformation exchanging baryons with tetraquarks, as shown on the right.

These transformations organize into a supersymmetry operation, successively relating mesons, baryons and tetraquarks:

In the formulation of confined QCD as holographic light front QCD, this hadron supersymmetry is brought out explicitly by the fact that the light cone quantum mechanics of the system refines to supersymmetric quantum mechanics (dTDB 14, see e.g. Nielsen-Brodsky 18).

This supersymmetry would be an exact symmetry of quantum hadrodynamics if antiquarks and diquarks had not only the same color charge but also the same mass (and other properties), for instance if the quarks involved all had vanishing mass, as assumed, to lowest order, in chiral perturbation theory. Any discrepancy causes breaking of the symmetry, hence here: supersymmetry breaking.

## Phenomenology

A direct consequence of hadron supersymmetry is the equality of slopes of the Regge trajectories of mesons and hadrons. This is indeed observed, with high accuracy:

## References

### Review

Review of the original work Miyazawa 66, Miyazawa 68, Catto-Gürsey 85, Catto-Gürsey 88:

Review with emphasis on the formulation via holographic light front QCD:

### Original articles

The phenomenon of hadron supersymmetry was first noted and formalized (together with early classification of supersymmetry super Lie algebras) in:

The argument that hadron supersymmetry is in fact implied by the constituent quark model in quantum chromodynamics:

### Via holographic light front QCD

In view of tetraquarks:

### Via octonions

Formulation via octonions:

• Sultan Catto, Yasemin Gürcan, Amish Khalfan and Levent Kurt, Algebraic formulation of hadronic supersymmetry based on octonions: new mass formulas and further applications, Journal of Physics: Conference Series, Volume 563, XXII International Conference on Integrable Systems and Quantum Symmetries (ISQS-22) 23–29 June 2014, Prague, Czech Republic (doi:10.1088/1742-6596/563/1/012006/meta)