U(1)'

U(1)'
Name	U(1)'
Type	Lie group (compact, abelian)
Typical representation	complex phase rotations
Applications	gauge symmetry in extensions of the Standard Model, condensed matter, string compactifications

U(1).

Introduction

U(1)' appears as a one‑dimensional compact abelian Lie symmetry used in extensions of the Standard Model, in string theory constructions such as heterotic string compactifications, and in effective field theories employed at colliders like the Large Hadron Collider and in deep underground experiments such as Super-Kamiokande. It provides an extra conserved charge beyond existing charges associated with electromagnetism, flavor symmetries explored at CERN, or global symmetries discussed in contexts like CP violation studies. Historically, additional U(1) factors have arisen in grand unified theories like SO(10) and E6, in early phenomenological proposals examined at SLAC National Accelerator Laboratory and Fermilab, and in model building influenced by results from experiments such as LEP and Tevatron.

Mathematical definition and properties

Mathematically, the group is isomorphic to the circle group S^1 familiar from studies at institutions such as Princeton University and Oxford University in the context of Lie theory. Representations are one‑dimensional complex phases classified by integer or rational charges that occur in constructions by authors associated with Institute for Advanced Study and research programs at Max Planck Institute for Physics. As an abelian Lie group, its Lie algebra is one‑dimensional and commutes with nonabelian algebras like SU(2) and SU(3), a structure exploited in combined gauge algebras found in grand unified theories proposed by research groups at CERN Theory Group and Brookhaven National Laboratory. Topological properties such as fundamental group and first Chern class are standard topics in seminars at California Institute of Technology and Massachusetts Institute of Technology when discussing fiber bundles and characteristic classes.

Role in particle physics and gauge theories

In particle physics, an extra U(1)' gauge symmetry provides a mediator often called Z' in phenomenological literature produced by collaborations at ATLAS and CMS. Model builders linked to Perimeter Institute and groups at University of Tokyo embed U(1)' as a factor commuting with SU(3)×SU(2)×U(1) of the Standard Model or as a remnant of symmetry breaking in SO(10) and E6 chains investigated at CERN. It furnishes new gauge bosons that couple to charges carried by fermions such as quarks studied at Brookhaven and leptons studied at SLAC, with coupling strengths constrained by precision measurements from experiments at LEP and flavor facilities like Belle II and LHCb. The symmetry can be gauged or global in contexts explored by researchers at Perimeter Institute and appears in dark sector portals examined by teams at Fermilab and JLab.

Symmetry breaking and Higgs mechanism for U(1)'

Spontaneous breaking of the extra U(1)' is commonly implemented by scalar fields acquiring vacuum expectation values, an approach developed in theoretical work at University of Cambridge and Stanford University. The associated Higgs mechanism yields a massive Z' boson, similar to mass generation for the W and Z bosons studied at CERN and described in textbooks influenced by curricula at Harvard University and Yale University. Models often introduce singlet scalars or extended Higgs sectors appearing in supersymmetric constructions from groups at University of California, Berkeley and Imperial College London. The pattern of breaking can connect to cosmological histories explored by cosmology groups at Princeton University and Kavli Institute for Cosmological Physics, affecting processes like baryogenesis addressed by collaborations at Fermilab.

Phenomenology and experimental searches

Experimental searches for Z' bosons and associated signatures have been central to programs at ATLAS and CMS, with complementary limits set by fixed‑target experiments at Jefferson Lab and beam‑dump searches coordinated by teams at SLAC. Collider signatures include dilepton resonances, dijet excesses, and deviations in electroweak precision observables measured at LEP and reanalyzed by theorists at CERN Theory Group. Indirect probes arise from flavor experiments such as Belle II and LHCb, neutrino scattering facilities like MINERvA, and cosmic probes discussed by groups at Planck (spacecraft) and Fermi Gamma-ray Space Telescope. Dedicated dark photon and dark Z' searches have been pursued by collaborations at BaBar and low‑energy facilities at TRIUMF.

Anomalies and anomaly cancellation

Gauge anomalies associated with U(1)' gauge currents require cancellation to preserve consistency, a requirement emphasized in anomaly cancellation studies originating from work at Princeton University and Harvard University. Cancellation mechanisms include appropriate charge assignments across fermion families as in models influenced by Glashow–Iliopoulos–Maiani (GIM) mechanism thinking from CERN and the introduction of new fermions such as vector‑like states proposed by groups at University of California, Santa Barbara. String theory constructions at Caltech and Institute for Advanced Study often supply Green–Schwarz mechanisms or Stueckelberg couplings to render anomalous U(1) factors consistent, techniques also used in heterotic and Type II model building studied at Rutgers University.

Extensions and models involving U(1)'

Extensions incorporating U(1)' include Z' models inspired by E6 breaking patterns, B‑L models linked to Baryon number and Lepton number considerations explored at CERN, supersymmetric realizations like the UMSSM studied at DESY and University of Chicago, and hidden‑sector portals motivated by dark matter programs at Fermi National Accelerator Laboratory. Embeddings in string theory and compactifications producing multiple U(1) factors have been developed by researchers at Perimeter Institute and Max Planck Institute for Physics. Phenomenological variants connect to neutrino mass models worked on at IPMU and cosmological inflation scenarios examined at Kavli Institute for Cosmology.

Category:Gauge symmetries