If leptons and quarks are not elementary particles but are composite, excited states of these particles can exist, called excited fermions. HERA, having electron and proton in the initial state, is a good place to search for them. They are sought via the decay into an ordinary fermion and a gauge boson, e.g. e* -> e gamma.
At HERA, excited electrons (e*) and excited quarks (q*) could be produced by t-channel exchange of a gamma or Z boson; similarly single excited neutrino (nu*) production could be achieved by t-channel exchange of a W boson :
/ e
/ e (nu) / nu (e) e /
e e* / e nu* / -------
------======= ------======= | / q
| \ | \ B | q* /
B | \ B (W) W | \ B (W) |=====
| | / \
--------------- -------------- q / \ C
Proton X Proton X /
-------------
B = gamma, Z B = gamma, Z Proton X
B = gamma, Z
C = gamma, Z, W, gluon
For e* production, the elastic contributions contributes to roughly
50% of the total. No elastic channel is possible for nu* production.
A phenomenological model (Hagiwara, Komamiya, Zeppenfeld) is commonly used in experimental searches. In this model, excited fermions are spin 1/2 states. Left-handed fermions f_L couple to right-handed excited fermions f*_R, (f*_L. f*_R) being a weak-isospin doublet. The coupling of this doublet to a gauge boson is proportionnal to (f_i / Lambda) * g_i, where g_i is the group gauge constant and the f_i are unknown parameters of the model. Lambda (GeV) is the compositeness scale. Usually f parameters are labelled f' for SU(2), f for U(1) and f_s for SU(3).
Hypothesis relating f, f' and f_s (e.g f=f'=f_s, or f=f' and f_s=0) allow to calculate branching ratios of the excited fermions, which are then independent on the model parameters. Under the same hypothesis, production cross-sections for f* only depend on one parameter, for example f/Lambda.
At LEP collider, excited leptons e*, mu*, tau* and nu* can be produced by pair up to the kinematical limit of sqrt(S)/2. Masses below sqrt(S)/2 have hence been excluded independly of f and f'.
Above the kinematical limit, single production of f* is possible. The production cross-section then depends on f/Lambda. It is higher for e* production than for mu* production for example (mu* production can only proceed via the splitting of an emitted photon in a mu mu_bar pair, while for e* production t-channel boson exchange is additionnaly possible). Hence, limits above sqrt(S)/2 can be derived in the plane f/Lambda versus M(f*), these limits being more stringent for e* than for other f*.