Twisted mass fermion

{{Short description|Lattice fermion discretisation}}

{{Technical|date=November 2023}}

In lattice field theory, twisted mass fermions are a fermion discretization that extends Wilson fermions for two mass-degenerate fermions.{{cite journal

|title=A local formulation of lattice QCD without unphysical fermion zero modes

|volume=83–84

|issn=0920-5632

|url=http://dx.doi.org/10.1016/S0920-5632(00)91852-8

|doi=10.1016/s0920-5632(00)91852-8

|journal=Nuclear Physics B - Proceedings Supplements

|author=Frezzotti, Roberto; Grassi, Pietro Antonio; Sint, Stefan; Weisz, Peter

|year=2000

|pages=941–946|arxiv=hep-lat/9909003

|bibcode=2000NuPhS..83..941F

|s2cid=17757436

}}

They are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD.{{cite book

|author=FLAG Working Group; Aoki, S.

|display-authors=etal

|title = Review of Lattice Results Concerning Low-Energy Particle Physics

|arxiv=1310.8555

|doi = 10.1140/epjc/s10052-014-2890-7

|series = Eur. Phys. J. C

|volume = 74

|pages = 116–117

|date = 2014

|chapter=A.1 Lattice actions|issue=9

|pmid=25972762

|pmc=4410391

}}

The original motivation for the use of twisted mass fermions in lattice QCD simulations was the observation that the two lightest quarks (up and down) have very similar mass and can therefore be approximated with the same (degenerate) mass. They form a so-called isospin doublet and are both represented by Wilson fermions in the twisted mass formalism. The name-giving twisted mass is used as a numerical trick, assigned to the two quarks with opposite signs. It acts as an infrared regulator, that is it allows to avoid unphysical configurations at low energies. In addition, at vanishing physical mass $m=0$ (maximal or full twist) it allows $\backslash mathcal\{O\}(a)$ improvement, getting rid of leading order lattice artifacts linear in the lattice spacing $a$.{{cite book|last1=Gattringer|first1=C.|last2=Lang|first2=C.B.|date=2009|title=Quantum Chromodynamics on the Lattice: An Introductory Presentation|series=Lecture Notes in Physics 788|url=|doi=10.1007/978-3-642-01850-3|location=|publisher=Springer|chapter=10 More about lattice fermions|pages=253–260|isbn=978-3642018497}}

The twisted mass Dirac operator is constructed from the (massive) Wilson Dirac operator $D\_W$ and reads{{cite journal

|title=An introduction to chiral symmetry on the lattice

|volume=53

|issn=0146-6410

|url=http://dx.doi.org/10.1016/j.ppnp.2004.05.003

|doi=10.1016/j.ppnp.2004.05.003

|number=2

|journal=Progress in Particle and Nuclear Physics

|publisher=Elsevier BV

|author=Chandrasekharan, S.

|year=2004

|pages=373–418 |arxiv=hep-lat/0405024

|bibcode=2004PrPNP..53..373C

|s2cid=17473067

}}{{cite journal

|title = Going chiral: twisted mass versus overlap fermions

|journal = Computer Physics Communications

|volume = 169

|number = 1

|pages = 362–364

|year = 2005

|issn = 0010-4655

|doi = 10.1016/j.cpc.2005.03.080

|url = https://www.sciencedirect.com/science/article/pii/S0010465505001773

|author = Karl Jansen|bibcode = 2005CoPhC.169..362J

|url-access = subscription

}}

:$$

D_\text{tw} = D_W + i\mu\gamma_5\sigma_3

where $\backslash mu$ is the twisted mass and acts as an infrared regulator (all eigenvalues $\backslash lambda$ of $D\_\backslash text\{tw\}$ obey $\backslash lambda\backslash ge\backslash mu^2>0$). $\backslash sigma\_3$ is the third Pauli matrix acting in the flavour space spanned by the two fermions. In the continuum limit $a\backslash rightarrow0$ the twisted mass becomes irrelevant in the physical sector and only appears in the doubler sectors which decouple due to the use of Wilson fermions.