Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics
Abstract
:1. Introduction
2. Gauge Theory and Composite Dynamics
2.1. Fields, Symmetries, and Observables
2.2. Perturbative Considerations
2.3. Low-Energy EFT
Hidden Local Symmetry
2.4. Phenomenological Applications
2.4.1. Composite Higgs
2.4.2. Top Partial Compositeness
2.4.3. Composite Dark Matter
- Gauge theories with group, coupled to families of fundamental matter, might play a central role in SIMP model building, and it is hence a priority to study them on the lattice, both in the minimal realisation and its extensions.
- In dark matter models, the mass of the lightest spin-1 composite state lies between that of the PNGBs, and about twice of it. This is to be contrasted with the CHM context, where addressing the little hierarchy problem requires a scale separation between PNGBs and heavier states. This is diametrically opposite to TC, where gauge invariance forbids fermion masses. For lattice practitioners, this observation makes the quenched calculations into a reasonable approximation of the true dynamics in the phenomenologically relevant region of parameter space.
- Many variations of the mechanism yielding the SIMP dark matter relic abundance exist, including SIMP adaptations [162] of the freeze-in mechanism [330,331,332,333,334], and more are likely to be proposed in the near future. This observation suggests to carry out broad, unprejudiced explorations of the whole parameter space. As high precision measurements are not yet a priority, feasible investigation strategies for these explorations make reasonable use of available computing resources.
- The (percolation) temperature (or Hubble parameter ) at which the phase transition ends. The phase transition starts at the critical temperature .
- The inverse duration of the transition, measured by the bubble nucleation rate computed at , defined in terms of , the 3-dimensional action of the system:
- The parameter , determining the strength of the transition, depends on , the jump at the transition in trace of the stress-energy tensor , and the enthalpy in the high-T phase:
- The bubble wall speed —the efficiency parameter (the ratio of bulk kinetic energy to vacuum energy) depends on and [335].
- The number of degrees of freedom after the phase transition.
3. Lattice Gauge Theories
3.1. Lattice Action
3.2. Simulation Strategies
3.3. Scale Setting and Topology
3.4. Measurements: Two-Point Functions, Masses, and Decay Constants
3.5. Bulk Phase Structure and Finite Volume Effects
4. Numerical Investigation I: Pure
4.1. Glueballs and String Tension
4.2. Quenched Mesons
4.3. Topology
5. Numerical Investigations II: Dynamical Fermions in Sp(4)
5.1. Fundamental Fermions
5.2. Antisymmetric and Multiple Representation Dynamical Fermions
6. Summary and Outlook
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
(as) | 2-index antisymmetric (representation) |
(A)T | (Axial-)Tensor (operator, particle) |
(A)V | (Axial-)Vector (operator, particle) |
BZ | Banks-Zaks |
CB | Chimera Baryon |
CDM | Cold Dark Matter |
CERN | European Organisation for Nuclear Research |
CHM | Composite Higgs Model |
ChRMT | Chiral Random Matrix Theory |
CoDM | Composite Dark Matter |
(E) | Euclidean (space-time) |
EFT | Effective Field Theory |
ETC | Extended Technicolor |
EW(SB) | ElectroWeak (Symmetry Breaking) |
(f) | fundamental (representation) |
FCNC | Flavor Changing Neutral Current |
FV | Finite Volume |
GIM | Glashow-Iliopoulos-Maiani (mechanism) |
GMOR | Gell-Mann-Oakes-Renner |
GW | Gravitational Wave |
HB | Heat Bath |
(R)HMC | (Rational) Hybrid Monte Carlo |
IR | Infra-Red |
KSRF | Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (relation) |
HLS | Hidden Local Symmetry |
HPC | High Performance Computing |
LHC | Large Hadron Collider |
LISA | Laser Interferometer Space Antenna |
LLR | Logarithmic Linear Relaxation |
(M) | Minkowski (space-time) |
MC | Monte Carlo |
MD | Molecular Dynamics |
NDA | Naive Dimensional Analysis |
NLO | Next-to-Leading Order |
OR | Over-Relaxation |
PNGB | Pseudo-Nambu–Goldstone Boson |
PS | Pseudoscalar (operator, particle) |
QCD | Quantum Chromodynamics |
RG(E) | Renormalisation Group (Equation) |
S | Scalar (operator, particle) |
SIMP | Strongly Interacting Massive Particle |
SM | Standard Model (of particle physics) |
(W)TC | (Walking) Technicolor |
TPC | Top Partial Compositeness |
URL | Uniform Resource Locator |
UV | Ultra-Violet |
VEV | Vacuum Expectation Value |
WIMP | Weakly Interacting Massive Particle |
WZW | Wess–Zumino–Witten (interaction term) |
(W)PT | (Wilson) Chiral Perturbation Theory |
Appendix A. Groups, Algebras and Technical Details
Appendix B. Data and Analysis Code
- A potential reader might be interested in learning how to apply one of the techniques that we have used in our work to their own research. Some technical detail might have been omitted from the published paper for presentation reasons (length or readability constraints). The reader will benefit from direct inspection of the complete procedure we followed, which can be found in the associated code release.
- A reader, who seeks to independently replicate one of our findings, might discover some tension between the results of our and their own implementation of the analysis. Direct inspection of the software we used would enable this reader to identify at what point the divergence between the two processes occurs, avoiding protracted arguments on reproducibility—see, e.g., the case described in Ref. [383].
- Lattice studies frequently generate more data than what can be feasible to fully exploit for a single group of researchers. The interested reader may perform their own, additional analysis on our data, with alternative methodologies, without the need to regenerate the data from scratch (which might require a significant investment of computer time). For example, more advanced fitting algorithms may give more detailed or precise results, or gain access to additional observables.
- Phenomenologists and other researchers who look to build on the numerical results of lattice computations may import the data from our work directly into their computational environment, without the need to resort to copying and pasting from published tables (or reading numbers off published plots). By doing so, one reduces the risk of introducing additional uncertainties, and avoids one source of potential human error.
Appendix B.1. Data Release
- Raw data, such as correlation functions and gradient flow histories, are released in their native formats as generated by the HiRep code [345,347], in accordance with the principle of “keeping raw data raw” [385]. By doing so, we reduce the chances of human error in transcription of data formats, while increasing the opportunity to detect such type of errors in a subsequent validation process.
- Reformatted raw data, obtained by taking the output files of raw data, condense the salient information in tables stored in HDF5 format [386]. Commonly available library functions can read the data in this format, so that one does not need to write a parser to interpret the bespoke formats generated by HiRep. Currently this information is generated from the raw log files as part of the analysis process.
- Metadata are collections of parameter values that identify the analysis performed. They include physical parameters, such as the lattice coupling , algorithmic ones, such as the number of trajectories between successive configurations, and analysis ones, such as the start and end of plateaux in effective mass plots. The metadata we publish are primarily those which enable the analysis.
- Final results, also presented in tabular form in the corresponding publications, are released in CSV format; they are typically compact enough that using a denser format such as HDF5 would not yield a significant benefit (in file size, for example), and the use of CSV files makes the data accessible without specialist software tools.
Appendix B.2. Analysis Code Release
- 1.
- Dependencies between steps are automatically managed. The ordering of steps is automatically decided, rather than requiring the user’s input.
- 2.
- Steps can automatically be run in parallel, with Make ensuring that no step runs before its prerequisites are complete. This allows the analysis process to scale with the available compute capacity.
- 3.
- The workflow can be interrupted partway and resumed subsequently, without the need to re-run previously completed steps.
- 4.
- Make is able to re-run only the steps of the analysis that depend on specific files, if data are updated, thereby expediting the debugging cycle.
Appendix B.3. Closing Remarks
1 | We borrow the terminology and nomenclature associated with mesons and baryons from QCD, when referring to the analogous composite states in new strongly coupled gauge theories. |
2 | In the presence of fermions transforming in different representations of the gauge group, the triangle anomaly gives mass to only one linear combination of the PNGBs associated with the breaking of the chiral symmetries acting on the different flavour species. The phenomenological implications are discussed for example in Ref. [47]. |
3 | |
4 | We denote the set of PNGBs of as , for consistency with the conventional notation of , as there are three inequivalent representations with 20 degrees of freedom, usually denoted as 20, , and [266]. |
5 | Although only integer values of are physically meaningful, is treated as a continuous variable. An alternative argument could be made by taking the large- (Veneziano) limit while holding fixed the continuous ratio . |
6 | |
7 | This critical condition should agree with , yet it gives rise to different results at finite order in the expansion. This critical condition reproduces the value of the critical coupling obtained from the Schwinger–Dyson analysis in the ladder approximation [296], and furthermore has a square-root singularity with respect to when the IR and UV fixed point merge [295]. |
8 | By preserving the whole , the model preserves custodial symmetry, suppressing new physics contributions to the T parameter [306]. |
9 | The important difference between CHMs and TC is that , so that can naturally be larger than the TeV scale. |
10 | In this work we use “reproduce” to mean “perform the same analysis on the same data and obtain the same result”, and “replicate” to mean “repeat the same or a similar analysis on freshly-obtained data and obtain compatible results”, as suggested by the Turing Way [382]. |
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Label M | Interpolating Operator | Mesons in QCD | |||
---|---|---|---|---|---|
PS | 5 | 1 | |||
S | 5 | 1 | |||
V | 10 | 1 | |||
T | 10 | 1 | |||
AV | 5 | 1 | |||
AT | 10 | 1 | |||
ps | 1 | ||||
s | 1 | ||||
v | 1 | 15 | |||
t | 1 | 15 | |||
av | 1 | ||||
at | 1 | 15 |
Label | Interpolating Operator | ||
---|---|---|---|
5 | 6 | ||
5 | 6 | ||
1 | 2 | 3 | 4 | ∞ | |
---|---|---|---|---|---|
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Bennett, E.; Holligan, J.; Hong, D.K.; Hsiao, H.; Lee, J.-W.; Lin, C.-J.D.; Lucini, B.; Mesiti, M.; Piai, M.; Vadacchino, D. Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics. Universe 2023, 9, 236. https://doi.org/10.3390/universe9050236
Bennett E, Holligan J, Hong DK, Hsiao H, Lee J-W, Lin C-JD, Lucini B, Mesiti M, Piai M, Vadacchino D. Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics. Universe. 2023; 9(5):236. https://doi.org/10.3390/universe9050236
Chicago/Turabian StyleBennett, Ed, Jack Holligan, Deog Ki Hong, Ho Hsiao, Jong-Wan Lee, C.-J. David Lin, Biagio Lucini, Michele Mesiti, Maurizio Piai, and Davide Vadacchino. 2023. "Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics" Universe 9, no. 5: 236. https://doi.org/10.3390/universe9050236
APA StyleBennett, E., Holligan, J., Hong, D. K., Hsiao, H., Lee, J. -W., Lin, C. -J. D., Lucini, B., Mesiti, M., Piai, M., & Vadacchino, D. (2023). Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics. Universe, 9(5), 236. https://doi.org/10.3390/universe9050236