Bound states: Hadrons and the proton structure

Schroedinger equation for potentials: The hydrogen atom, fine structure and all that Positronium and quarkonia The proton and its structure: Parton distribution functions & experimental inputs

Bound states in quantum theory The Schroedinger equation Structure of the equation For the energy eigenfunctions the time-independent Schroedinger equation in a Coulomb potential reads:

Can rewrite in polar coordinates:

Bound states in quantum theory The Schroedinger equation Spherical symmetric solutions: Specialising on spherical symmetric solutions yields

Rescale r and E:

Ansatz for the solutions:

Bound states in quantum theory The Schroedinger equation Spherical symmetric solutions (cont'd): Choosing results in:

Solution by expanding g in a Taylor-series in ρ. Gives recursion relation for the coefficients:

a1 can be chosen for correct normalisation.

Bound states in quantum theory The Schroedinger equation Spherical symmetric solutions (cont'd): Potential problem: For

Solution: The recursion must terminate for some k=n:

 quantisation of energies.

Bound states in quantum theory The Schroedinger equation Spherical symmetric solutions (cont'd): Energy eigenvalues:

Energy eigenfunctions

Bound states in quantum theory The Schroedinger equation Adding in angular momentum Ansatz with spherical harmonics:

Spherical harmonics cancel out in Deq, thus:

Interpretation: Angular-momentum dependent potential acts as perturbation on original spherically symmetric solutions Bound states in quantum theory The Schroedinger equation Adding in angular momentum (cont'd) Again, solution through a Taylor-series

gives (again) recursion, this time:

must again terminate for some k=n, but: this time requirement that k and l different. More precisely:

Bound states in quantum theory The Schroedinger equation Visualisation of wave functions: Horizontally: s,p,d is “spectroscopic language”, refers to l=1,2,3. On the vertical: 1,2,3 are the nergy levels (principal quantum number n)

Bound states in quantum theory The Schroedinger equation Relativistic corrections – the fine structure Including relativistic effects:

Trick for evaluation

energy levels:

Bound states in quantum theory The Schroedinger equation Spin-orbit interactions – the fine structure Including the effect of the electron spin. The electron feels the magnetic field due to its own movement in the electric (Coulomb) potential. Extra term (spin-orbit coupling)

Bound states in quantum theory The Schroedinger equation Spin-orbit interactions – the fine structure But now neither L or s separately conserved, only J= L+s. Use

and

energy levels:

Bound states in quantum theory The Schroedinger equation Quantum corrections: There are further corrections: Lamb-shift and hyperfine structure. Lamb-shift is due to higher-order diagrams in QED Hyperfine splitting is due to spin-spin interaction of proton and electron

Bound states in quantum theory Positronium and quarkonia Similar reasoning can be applied to “exotic atoms”, e.g. bound states of an electron and a positron – called positronium – and its quark analogy, called quarkonium. The latter comes in a variety of realisations, summarily known under the name quarkonia. In both cases the trick is to go from the two-body problem to the equivalent one-body problem by introducing the reduced mass. For quarkonia this works only as long as the quarks move slowly in their potential.

Bound states in quantum theory Positronium The original Hamiltonian reads

where m = me/2 is the reduced mass with momentum p and distance r from the centre-of gravity. Therefore, energy levels etc. identical to hydrogen atom (up to the factor 2):

Bound states in quantum theory Positronium The relativistic corrections are evaluated as before (apart from some factors of 2 coming here and there):

Spin-orbit coupling is more intricate, due to the nucleus being non-stationary, unlike before. Therefore a new correction due to finite propagation time of the e.m. field. Altogether (ε j-dependent)

Bound states in quantum theory Positronium A new type of correction occurs: positronium may temporarily annihilate into an e+e--pair before going back to positronium

Like hydrogen, there is a spectrum with radiative transitions, but now the difference: positronium can annihilate into two or more photons.

Bound states in quantum theory Positronium The number of photons is connected with the charge conjugation property. For positronium, , while for n photons . Therefore, invariance under charge conjugation enforces the selection rule: This decays happen only, when electron and positron “meet”  l=0. Naively, s of the positronium is given by adding two spin-1/2 systems, giving a singlet (spin-0) and a triplet (spin-1). For the spin-0 ground state:

Bound states in quantum theory Quarkonia In principle very similar, but caveat: Strong interactions are different, have Coulomb potential at short distances plus linear term at large distances (confinement). Also, have to add factors due to colour. for heavy quarks (c,b), this works, for light quarks, the potential is so strong that quarks move very fast  huge relativistic terms, no corrections !

Bound states in quantum theory Quarkonia QCD potential and quarkonia spectrum:

Bound states in quantum theory Quarkonia The OZI rule The OZI rule states that decays are disfavoured, where quark lines “pop out of the vacuum”. For the example of the ϕ this implies that annihilation into kaons is largely preferred over pions, in spite of the large phase space of the pions.

The proton as a bound state Description of extended objects Formfactors and charge radius Extended objects have a matter density:

Its Fourier transform is called a form factor:

Naive modification of cross sections:

The proton as a bound state Experiments: Elastic scattering The Rosenbluth formula Simple test of the proton structure: elastic ep scattering (exchange of a photon). Rosenbluth-formula (E and E' are energy of electron before and after scattering, q2 is the – spacelike – momentum transfer, and θ is the scattering angle):

Compare with Rutherford scattering: The proton as a bound state Experiments: Elastic scattering Form factor parametrization (dipole form) Differences due to relativistic kinematics plus recoil of the protons (in Rutherford scattering, the nuclei stay at rest). Also: inner structure there are two form

factors K1,2 that remain in the cross section formula after using symmetry arguments. They are related to the electric and magnetic form factors (see next slide) Low energy parametrization:

The proton as a bound state Experiments: Elastic scattering Formfactors and the proton charge radius Alternative form of Rosenbluth formula:

Connection to charge radius (F1 = F2)

 the charge radius The proton as a bound state Experiments: Inelastic scattering Structure functions What happens if scattering inelastic (i.e. if proton does not remain intact) ? Formally speaking: form factors are replaced by structure functions (do depend on two kinematic parameters rather than one). Equation for cross section ( )

The proton as a bound state Experiments: Inelastic scattering Kinematics of Deep-Inelastic Scattering (DIS) (in white: Lorentz-invariant, in red: proton rest frame)

The proton as a bound state Experiments: Inelastic scattering Partons and Bjorken scaling Around 1968 J.D.Bjorken postulated that there is no characteristic size, and the structure functions depend on one parameter only (Bjorken scaling)

Bjorken scaling implies that there is no special scale

present in the dynamics of this scattering process.

The proton as a bound state Experiments: Inelastic scattering The Callan-Gross relation In 1969 Callan and Gross suggested that Bjorken's scaling functions are related:

This reflects the fact that the quarks are spin-1/2 particles (spin-0 for example would lead to )

The proton as a bound state Experiments: Inelastic scattering Derivation of the Callan-Gross relation Basic idea: Compare cross section for scattering of an electron with a free quark

with electron-proton cross section:

The proton as a bound state Experiments: Inelastic scattering Partons and Bjorken scaling This implies that the proton consists of point-like objects, named “partons” by R.P.Feynman. He postulated that the partons share the proton's momentum: If is the probability to find a parton with energy fraction between x and x+dx,

Also sum rules:

The proton as a bound state Experiments: Inelastic scattering Scaling and scaling violations

at small x: scaling violated (dependence on Q)

at low Q2 and moderate x: scaling quite well satisfied

The proton as a bound state The QCD-improved parton model Partons and the content of the proton To test any ideas, consider first how the structure function(s) are related to the parton content of the proton:

Three extreme ideas for the proton: three quarks, no interactions three bound quarks quarks + other “stuff2

The proton as a bound state The QCD-improved parton model Partons and the content of the proton:

The proton as a bound state The QCD-improved parton model Partons and the content of the proton: Are quarks the full story? Let's put pieces together.

“Valence” quarks carry only half the proton's momentum there must be a large “sea”

The proton as a bound state The QCD-improved parton model The partons as Russian dolls Explanation: As the proton is hit harder and harder (larger Q2), the virtual photon starts resolving gluons and quark-antiquark fluctuations (partons in partons!). The scale Q2 plays the role of “resolution parameter”. Described by the DGLAP equations. Basic structure:

(quark emits gluon)

(gluon emits quark pair)

The proton as a bound state The QCD-improved parton model Parton distribution functions:

The proton as a bound state The QCD-improved parton model Cross section calculations When calculating cross sections with incoming hadrons (e.g. protons), must integrate over possible x- values of the incident partons:

Here connects parton and hadron-level kinematics. Note: scale Q2 has been omitted. x-values may be constrained by rapidity:

The proton as a bound state Collider consequences: Tevatron Typical cross sections for some SM processes Total cross sections for W and Z production:

The proton as a bound state Collider consequences: Tevatron Typical cross sections for some SM processes Differential cross sections in Z production:

The proton as a bound state Collider consequences: Tevatron Typical cross sections for some SM processes Differential cross sections in W production:

The proton as a bound state Collider consequences: LHC Typical cross sections: Signals and SM backgrounds More to come here.