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nuclear lattice effective field theory ( nleft ) is a first - principles approach , in which chiral eft for nucleons is combined with numerical auxiliary - field quantum monte carlo ( afqmc ) lattice calculations . nleft differs from other _ ab initio _ methods  @xcite in that it is an unconstrained monte carlo calculation , which does not rely on truncated basis expansions or many - body perturbation theory , nor on prior information about the structure of the nuclear wave function .    as in chiral eft , our calculations are organized in powers of a generic soft scale @xmath5 associated with factors of momenta and the pion mass  @xcite . we denote @xmath6 as leading order ( lo ) , @xmath7 as next - to - leading order ( nlo ) , and @xmath8 as next - to - next - to - leading order ( nnlo ) contributions . the present calculations are performed up to nnlo . we define @xmath9 as the lo lattice hamiltonian , and @xmath10 as the equivalent hamiltonian with the pion - nucleon coupling @xmath11 and contact interactions that respect wigner s su(4 ) symmetry .    in our nleft calculations , @xmath9 is treated non - perturbatively ( see ref . @xcite for a review ) . the nlo contribution to the two - nucleon force ( 2nf ) , the electromagnetic and strong isospin - breaking contributions ( emib ) , and the three - nucleon force ( 3nf ) which first enters at nnlo , are all treated as perturbations . it should be noted that our `` lo '' calculations use smeared short - range interactions that capture much of the corrections usually treated at nlo and higher orders  @xcite . at nnlo , the 3nf overbinds nuclei with @xmath12 due to a clustering instability which involves four nucleons on the same lattice site . a long - term objective of nleft is to remedy this problem by decreasing the lattice spacing and including the next - to - next - to - next - to - leading order ( n3lo ) corrections in chiral eft . in the mean time , the overbinding problem has been rectified by means of a 4n contact interaction , tuned to the empirical binding energy of either @xmath13he or @xmath14be  @xcite . while this provides a good description of the alpha nuclei up to @xmath15 including the hoyle state  @xcite , the overbinding is found to increase more rapidly for @xmath16 . in ref . @xcite , a non - local 4n interaction which accounts for all possible configurations of four nucleons on adjacent lattice sites was introduced , and adjusted to the empirical binding energy of @xmath3 mg . a detailed study of the spectrum and electromagnetic properties of @xmath1o ( with inclusion of the effective 4n interaction ) has been reported in ref . the hoyle state is a resonance with spin - parity quantum numbers @xmath17 in the spectrum of @xmath0c , which plays an important role in resonantly enhancing the reaction rate for the so - called triple - alpha process , which is responsible for the production of carbon in massive stars that have reached the red giant stage in their evolution . this reaction represents a significant bottleneck in the stellar nucleosynthesis , as @xmath14be is an unstable ( though relatively long - lived ) resonance . for @xmath0c to form , a third alpha particle must combine with the @xmath14be resonance to form the hoyle state , which subsequently decays electromagnetically to the ground state of @xmath0c . this reaction may then proceed further ( non - resonantly ) to form @xmath1o through addition of a fourth alpha particle . however , the temperature of the stellar plasma at which the triple - alpha process takes place depends exponentially on the energy @xmath18 of the hoyle state above the triple - alpha threshold , which is experimentally known to be @xmath19  kev . stellar model calculations  @xcite have shown that only a narrow window of @xmath20  kev exists in @xmath18 where sizable amounts of carbon and oxygen can be produced simultaneously .    .lattice results at leading order ( lo ) and available experimental values for the root - mean - square charge radii and quadrupole moments of the @xmath0c states . [ hoyle1 ] [ cols="^,^,^",options="header " , ]     in table  [ oxygen2 ] , we note that the lo charge radius @xmath21 of the ground state of @xmath1o is smaller than the empirical value @xmath22 . this leads to a systematic deviation , which arises from the overall size of the second moment of the charge distribution . to compensate for this overall scaling mismatch , we have also calculated `` rescaled '' quantities multiplied by powers of the ratio @xmath23 , according to the length dimension of each observable . with such a scaling factor included , we find that the nleft predictions for the @xmath24 and @xmath25 transitions are in good agreement with available experimental values . we have presented an overview of the central nleft results for the low - lying even - parity spectra of @xmath0c and @xmath1o . this includes the hoyle state of @xmath0c which plays a central role in the stellar nucleosynthesis of life - essential elements . we have also shown that the electromagnetic properties and transition rates of @xmath0c and @xmath1o are in agreement with available experimental data . while the long - term objectives of nleft are to decrease the lattice spacing and include higher orders in the eft expansion , we also find that the missing physics up to @xmath4si can be approximated by an `` effective '' 4n interaction . these results represent an important step towards more comprehensive nleft calculations of medium - mass nuclei in the near future . we are grateful for the help in automated data collection by thomas luu . partial financial support from the deutsche forschungsgemeinschaft ( sino - german crc 110 ) , the helmholtz association ( contract no . vh - vi-417 ) , bmbf ( grant no . 05p12pdfte ) , and the u.s . department of energy ( de - fg02 - 03er41260 ) is acknowledged . this work was further supported by the eu hadronphysics3 project , and funds provided by the erc project no . 259218 nucleareft . the computational resources were provided by the jlich supercomputing centre at the forschungszentrum jlich and by rwth aachen . b.  borasoy , e.  epelbaum , h.  krebs , d.  lee , and ulf - g . meiner , eur . j.  a * 31 * , 105 ( 2007 ) . e.  epelbaum , h.  krebs , d.  lee , and ulf - g . meiner , phys . lett .   * 104 * , 142501 ( 2010 ) ; _ ibid . _ , eur . j.  a * 45 * , 335 ( 2010 ) . e.  epelbaum , h.  krebs , d.  lee , and ulf .- g . meiner , phys . lett .   * 106 * , 192501 ( 2011 ) . e.  epelbaum , h.  krebs , t.  a.  lhde , d.  lee , and ulf - g . meiner , phys . * 109 * , 252501 ( 2012 ) ; _ ibid . _ , lett .  * 110 * , 112502 ( 2013 ) ; _ ibid . _ , eur . phys .  j.  a * 49 * , 82 ( 2013 ) . t.  a.  lhde , e.  epelbaum , h.  krebs , d.  lee , ulf - g . meiner , and g.  rupak , arxiv:1311.0477 [ nucl - th ] . e.  epelbaum , h.  krebs , t.  a.  lhde , d.  lee , ulf - g . meiner , and g.  rupak , _ to appear in physical review letters _ , arxiv:1312.7703 [ nucl - th ]
we review the calculation of the hoyle state of @xmath0c in nuclear lattice effective field theory ( nleft ) and its anthropic implications for the nucleosynthesis of @xmath0c and @xmath1o in red giant stars . 
 we also review the extension of nleft to the regime of medium - mass nuclei , with emphasis on the determination of the ground - state energies of the alpha nuclei @xmath1o , @xmath2ne , @xmath3 mg and @xmath4si by means of euclidean time projection . finally , we review recent nleft results for the spectrum , electromagnetic properties , and alpha - cluster structure of @xmath1o .