* [30] Hiroshi Shimahara and Dierk Rainer.  Crossover from $\mathrm{V}$ortex $\mathrm{S}$tates to the $\mathrm{F}$ulde-$\mathrm{F}$errell- $\mathrm{L}$arkin-$\mathrm{O}$vchinnikov $\mathrm{S}$tate in $\mathrm{T}$wo-$\mathrm{D}$imensional $s$-and $d$-$\mathrm{W}$ave $\mathrm{S}$uperconductors.  Journal of the Physical Society of Japan, 66(11):3591–3599, 1997\.
  * [31] Hiroshi Shimahara.  Phase $\mathrm{F}$luctuations and $\mathrm{K}$osterlitz$-$$\mathrm{T}$houless $\mathrm{T}$ransition in $\mathrm{T}$wo-$\mathrm{D}$imensional $\mathrm{F}$ulde-$\mathrm{F}$errell- $\mathrm{L}$arkin-$\mathrm{O}$vchinnikov $\mathrm{S}$uperconductors.  Journal of the Physical Society of Japan, 67(6):1872–1875, 1998\.
  * [32] Yoji Ohashi.  On the $\mathrm{F}$ulde-$\mathrm{F}$errell $\mathrm{S}$tate in $\mathrm{S}$patially $\mathrm{I}$sotropic $\mathrm{S}$uperconductors.  Journal of the Physical Society of Japan, 71(11):2625–2628, 2002\.
  * [33] G. Sarma.  On the influence of a uniform exchange field acting on the spins of the conduction electrons in a superconductor.  Journal of Physics and Chemistry of Solids, 24(8):1029 – 1032, 1963\.
  * [34] W. Vincent Liu and Frank Wilczek.  Interior $\mathrm{G}$ap $\mathrm{S}$uperfluidity.  Phys. Rev. Lett., 90:047002, 2003.
  * [35] Elena Gubankova, W. Vincent Liu, and Frank Wilczek.  Breached $\mathrm{P}$airing $\mathrm{S}$uperfluidity: $\mathrm{P}$ossible $\mathrm{R}$ealization in $\mathrm{QCD}$.  Phys. Rev. Lett., 91:032001, 2003.
  * [36] H. Müther and A. Sedrakian.  Spontaneous $\mathrm{B}$reaking of $\mathrm{R}$otational $\mathrm{S}$ymmetry in $\mathrm{S}$uperconductors.  Phys. Rev. Lett., 88:252503, 2002.
  * [37] Armen Sedrakian, Jordi Mur-Petit, Artur Polls, and Herbert Müther.  Pairing in a two-component ultracold $\mathrm{F}$ermi gas: $\mathrm{P}$hases with broken-space symmetries.  Phys. Rev. A, 72:013613, 2005.
  * [38] Michael McNeil Forbes, Elena Gubankova, W. Vincent Liu, and Frank Wilczek.  Stability $\mathrm{C}$riteria for $\mathrm{B}$reached-$\mathrm{P}$air $\mathrm{S}$uperfluidity.  Phys. Rev. Lett., 94:017001, 2005.
  * [39] W. Vincent Liu, Frank Wilczek, and Peter Zoller.  Spin-dependent $\mathrm{H}$ubbard model and a quantum phase transition in cold atoms.  Phys. Rev. A, 70:033603, 2004.
  * [40] T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai, and Y. Onuki.  Unconventional heavy-fermion superconductor $\mathrm{CeCo}\mathrm{In}_{5}$: dc magnetization study at temperatures down to 50 mk.  Phys. Rev. B, 65:180504, 2002.
  * [41] C. Capan, A. Bianchi, R. Movshovich, A. D. Christianson, A. Malinowski, M. F. Hundley, A. Lacerda, P. G. Pagliuso, and J. L. Sarrao.  Anisotropy of thermal conductivity and possible signature of the $\mathrm{F}$ulde-$\mathrm{F}$errell- $\mathrm{L}$arkin-$\mathrm{O}$vchinnikov state in $\mathrm{CeCo}\mathrm{In}_{5}$.  Phys. Rev. B, 70:134513, 2004.
  * [42] C. Martin, C. C. Agosta, S. W. Tozer, H. A. Radovan, E. C. Palm, T. P. Murphy, and J. L. Sarrao.  Evidence for the $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov state in $\mathrm{CeCo}\mathrm{In}_{5}$ from penetration depth measurements.  Phys. Rev. B, 71:020503, 2005.
  * [43] G. Koutroulakis, V. F. Mitrovic, M. Horvatic, C. Berthier, G. Lapertot, and J. Flouquet.  Field $\mathrm{D}$ependence of the $\mathrm{G}$round $\mathrm{S}$tate in the $\mathrm{E}$xotic $\mathrm{S}$uperconductor $\mathrm{CeCo}\mathrm{In}_{5}$: $\mathrm{A}$ $\mathrm{N}$uclear $\mathrm{M}$agnetic $\mathrm{R}$esonance $\mathrm{I}$nvestigation.  Phys. Rev. Lett., 101:047004, 2008.
  * [44] K. Kumagai, M. Saitoh, T. Oyaizu, Y. Furukawa, S. Takashima, M. Nohara, H. Takagi, and Y. Matsuda.  Fulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov $\mathrm{S}$tate in a $\mathrm{P}$erpendicular $\mathrm{F}$ield of $\mathrm{Q}$uasi-$\mathrm{T}$wo-$\mathrm{D}$imensional $\mathrm{CeCo}\mathrm{In}_{5}$.  Phys. Rev. Lett., 97:227002, 2006.
  * [45] Simon Gerber, Marek Bartkowiak, Jorge L. Gavilano, Eric Ressouche, Nikola Egetenmeyer, Christof Niedermayer, Andrea D. Bianchi, Roman Movshovich, Eric D. Bauer, Joe D. Thompson, and Michel Kenzelmann.  Switching of magnetic domains reveals spatially inhomogeneous superconductivity.  Nature Physics, 10:126, 2013.
  * [46] Duk Y. Kim, Shi-Zeng Lin, Franziska Weickert, Michel Kenzelmann, Eric D. Bauer, Filip Ronning, J. D. Thompson, and Roman Movshovich.  Intertwined $\mathrm{O}$rders in $\mathrm{H}$eavy-$\mathrm{F}$ermion $\mathrm{S}$uperconductor $\mathrm{CeCo}\mathrm{In}_{5}$.  Phys. Rev. X, 6:041059, 2016.
  * [47] Shi-Zeng Lin, Duk Y. Kim, Eric D. Bauer, Filip Ronning, J. D. Thompson, and Roman Movshovich.  Interplay of the $\mathrm{S}$pin $\mathrm{D}$ensity $\mathrm{W}$ave and a $\mathrm{P}$ossible $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov $\mathrm{S}$tate in $\mathrm{CeCo}\mathrm{In}_{5}$ in $\mathrm{R}$otating $\mathrm{M}$agnetic $\mathrm{F}$ield.  Phys. Rev. Lett., 124:217001, 2020.
  * [48] J. A. Wright, E. Green, P. Kuhns, A. Reyes, J. Brooks, J. Schlueter, R. Kato, H. Yamamoto, M. Kobayashi, and S. E. Brown.  Zeeman-$\mathrm{D}$riven $\mathrm{P}$hase $\mathrm{T}$ransition within the $\mathrm{S}$uperconducting $\mathrm{S}$tate of $\kappa\mathrm{\text{$-$}}(\mathrm{BEDT}\mathrm{\text{$-$}}\mathrm{TTF}{)}_{2}\mathrm{Cu}(\mathrm{NCS}{)}_{2}$.  Phys. Rev. Lett., 107:087002, 2011.
  * [49] B. Bergk, A. Demuer, I. Sheikin, Y. Wang, J. Wosnitza, Y. Nakazawa, and R. Lortz.  Magnetic torque evidence for the $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov state in the layered organic superconductor $\kappa-(\mathrm{BEDT}-\mathrm{TTF}){}_{2}\mathrm{Cu}(\mathrm{NCS}){}_{2}$.  Phys. Rev. B, 83:064506, 2011.
  * [50] C. C. Agosta, Jing Jin, W. A. Coniglio, B. E. Smith, K. Cho, I. Stroe, C. Martin, S. W. Tozer, T. P. Murphy, E. C. Palm, J. A. Schlueter, and M. Kurmoo.  Experimental and semiempirical method to determine the $\mathrm{P}$auli-limiting field in quasi-two-dimensional superconductors as applied to $\kappa-(\mathrm{BEDT}-\mathrm{TTF}){}_{2}\mathrm{Cu}(\mathrm{NCS}){}_{2}$: $\mathrm{S}$trong evidence of a $\mathrm{FFLO}$ state.  Phys. Rev. B, 85:214514, 2012.
  * [51] H. Mayaffre, S. Kramer, M. HorvatiÄ, C. Berthier, K. Miyagawa, K. Kanoda, and V. F. Mitrovic.  Evidence of andreev bound states as a hallmark of the fflo phase in $\kappa-(\mathrm{BEDT}-\mathrm{TTF}){}_{2}\mathrm{Cu}(\mathrm{NCS}){}_{2}$.  Nature Physics, 10:928, 2014.
  * [52] K. Cho, H. Kim, M. A. Tanatar, Y. J. Song, Y. S. Kwon, W. A. Coniglio, C. C. Agosta, A. Gurevich, and R. Prozorov.  Anisotropic upper critical field and possible $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov state in the stoichiometric pnictide superconductor $\mathrm{LiFeAs}$.  Phys. Rev. B, 83:060502, 2011.
  * [53] Seunghyun Khim, Bumsung Lee, Jae Wook Kim, Eun Sang Choi, G. R. Stewart, and Kee Hoon Kim.  Pauli-limiting effects in the upper critical fields of a clean $\mathrm{LiFeAs}$ single crystal.  Phys. Rev. B, 84:104502, 2011.
  * [54] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin, A. Perali, P. Pieri, and G. C. Strinati.  Observation of pseudogap behaviour in a strongly interacting fermi gas.  Nat Phys, 6:569, 2010.
  * [55] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag, and R. Grimm.  Observation of the $\mathrm{P}$airing $\mathrm{G}$ap in a $\mathrm{S}$trongly $\mathrm{I}$nteracting $\mathrm{F}$ermi $\mathrm{G}$as.  Science, 305(5687):1128–1130, 2004.
  * [56] Alain Sewer, Xenophon Zotos, and Hans Beck.  Quantum $\mathrm{M}$onte $\mathrm{C}$arlo study of the three-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 66:140504, 2002.
  * [57] Thereza Paiva, Richard Scalettar, Mohit Randeria, and Nandini Trivedi.  Fermions in 2$\mathrm{D}$ $\mathrm{O}$ptical $\mathrm{L}$attices: $\mathrm{T}$emperature and $\mathrm{E}$ntropy $\mathrm{S}$cales for $\mathrm{O}$bserving $\mathrm{A}$ntiferromagnetism and $\mathrm{S}$uperfluidity.  Phys. Rev. Lett., 104:066406, 2010.
  * [58] M. G. Ries, A. N. Wenz, G. Zürn, L. Bayha, I. Boettcher, D. Kedar, P. A. Murthy, M. Neidig, T. Lompe, and S. Jochim.  Observation of $\mathrm{P}$air $\mathrm{C}$ondensation in the $\mathrm{Q}$uasi-2$\mathrm{D}$ $\mathrm{BEC}-\mathrm{BCS}$ $\mathrm{C}$rossover.  Phys. Rev. Lett., 114:230401, 2015.
  * [59] C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle.  Pairing $\mathrm{W}$ithout $\mathrm{S}$uperfluidity: $\mathrm{T}$he $\mathrm{G}$round $\mathrm{S}$tate of an $\mathrm{I}$mbalanced $\mathrm{F}$ermi $\mathrm{M}$ixture.  Science, 316:867–870, 2007.
  * [60] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle.  Observation of $\mathrm{P}$hase $\mathrm{S}$eparation in a $\mathrm{S}$trongly $\mathrm{I}$nteracting $\mathrm{I}$mbalanced $\mathrm{F}$ermi $\mathrm{G}$as.  Phys. Rev. Lett., 97:030401, 2006.
  * [61] Yean-an Liao, Ann Sophie C. Rittner, Tobias Paprotta, Wenhui Li, Guthrie B. Partridge, Randall G. Hulet, Stefan K. Baur, and Erich J. Mueller.  Spin-imbalance in a one-dimensional $\mathrm{F}$ermi gas.  Nature, 467:567, 2010.
  * [62] M. Taglieber, A.-C. Voigt, T. Aoki, T. W. Hänsch, and K. Dieckmann.  Quantum $\mathrm{D}$egenerate $\mathrm{T}$wo-$\mathrm{S}$pecies $\mathrm{F}$ermi-$\mathrm{F}$ermi $\mathrm{M}$ixture $\mathrm{C}$oexisting with a $\mathrm{B}$ose-$\mathrm{E}$instein $\mathrm{C}$ondensate.  Phys. Rev. Lett., 100:010401, 2008.
  * [63] D. Naik, A. Trenkwalder, C. Kohstall, F. M. Spiegelhalder, M. Zaccanti, G. Hendl, F. Schreck, R. Grimm, T. M. Hanna, and P. S. Julienne.  Feshbach resonances in the ${}^{6}\mathrm{Li}$$-$${}^{40}\mathrm{K}$ $\mathrm{F}$ermi- $\mathrm{F}$ermi mixture: elastic versus inelastic interactions.  The European Physical Journal D, 65(1):55–65, 2011.
  * [64] E. Wille, F. M. Spiegelhalder, G. Kerner, D. Naik, A. Trenkwalder, G. Hendl, F. Schreck, R. Grimm, T. G. Tiecke, J. T. M. Walraven, S. J. J. M. F. Kokkelmans, E. Tiesinga, and P. S. Julienne.  Exploring an $\mathrm{U}$ltracold $\mathrm{F}$ermi-$\mathrm{F}$ermi $\mathrm{M}$ixture: $\mathrm{I}$nterspecies $\mathrm{F}$eshbach $\mathrm{R}$esonances and $\mathrm{S}$cattering $\mathrm{P}$roperties of ${}^{6}\mathrm{Li}$ and ${}^{40}\mathrm{K}$.  Phys. Rev. Lett., 100:053201, 2008.
  * [65] L. Costa, J. Brachmann, A.-C. Voigt, C. Hahn, M. Taglieber, T. W. Hänsch, and K. Dieckmann.  $s$-$\mathrm{W}$ave $\mathrm{I}$nteraction in a $\mathrm{T}$wo-$\mathrm{S}$pecies $\mathrm{F}$ermi- $\mathrm{F}$ermi $\mathrm{M}$ixture at a $\mathrm{N}$arrow $\mathrm{F}$eshbach $\mathrm{R}$esonance.  Phys. Rev. Lett., 105:123201, 2010.
  * [66] A.-C. Voigt, M. Taglieber, L. Costa, T. Aoki, W. Wieser, T. W. Hänsch, and K. Dieckmann.  Ultracold $\mathrm{H}$eteronuclear $\mathrm{F}$ermi-$\mathrm{F}$ermi $\mathrm{M}$olecules.  Phys. Rev. Lett., 102:020405, 2009.
  * [67] Mingwu Lu, Nathaniel Q. Burdick, and Benjamin L. Lev.  Quantum $\mathrm{D}$egenerate $\mathrm{D}$ipolar $\mathrm{F}$ermi $\mathrm{G}$as.  Phys. Rev. Lett., 108:215301, 2012.
  * [68] Albert Frisch, Kiyotaka Aikawa, Michael Mark, Francesca Ferlaino, Ekaterina Berseneva, and Svetlana Kotochigova.  Hyperfine structure of laser-cooling transitions in fermionic $\mathrm{E}$rbium-167.  Phys. Rev. A, 88:032508, 2013.
  * [69] F. M. Spiegelhalder, A. Trenkwalder, D. Naik, G. Kerner, E. Wille, G. Hendl, F. Schreck, and R. Grimm.  All-optical production of a degenerate mixture of ${}^{6}\mathrm{Li}$ and ${}^{40}\mathrm{K}$ and creation of heteronuclear molecules.  Phys. Rev. A, 81:043637, 2010.
  * [70] P. Pieri, A. Perali, G. C. Strinati, S. Riedl, M. J. Wright, A. Altmeyer, C. Kohstall, E. R. Sánchez Guajardo, J. Hecker Denschlag, and R. Grimm.  Pairing-gap, pseudogap, and no-gap phases in the radio-frequency spectra of a trapped unitary ${}^{6}\mathrm{Li}$ gas.  Phys. Rev. A, 84:011608, 2011.
  * [71] S. Nascimbasne, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon.  Exploring the thermodynamics of a universal $\mathrm{F}$ermi gas.  Nature, 463:1057–1060, 2010.
  * [72] Leo Radzihovsky.  Fluctuations and phase transitions in $\mathrm{L}$arkin$-$$\mathrm{O}$vchinnikov liquid$-$crystal states of a population$-$imbalanced resonant $\mathrm{F}$ermi gas.  Phys. Rev. A, 84:023611, 2011.
  * [73] V. Gurarie and L. Radzihovsky.  Resonantly paired fermionic superfluids.  Annals of Physics, 322(1):2 – 119, 2007.  January Special Issue 2007.
  * [74] Roberto Casalbuoni and Giuseppe Nardulli.  Inhomogeneous superconductivity in condensed matter and $\mathrm{QCD}$.  Rev. Mod. Phys., 76:263–320, 2004.
  * [75] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger.  Many-body physics with ultracold gases.  Rev. Mod. Phys., 80:885–964, 2008.
  * [76] W. Kohn.  Analytic $\mathrm{P}$roperties of $\mathrm{B}$loch $\mathrm{W}$aves and $\mathrm{W}$annier $\mathrm{F}$unctions.  Phys. Rev., 115:809–821, 1959.
  * [77] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller.  Cold bosonic atoms in optical lattices.  Phys. Rev. Lett., 81:3108–3111, 1998.
  * [78] Gerald D. Mahan.  Many particle physics, $\mathrm{S}$pringer-$\mathrm{V}$erlag $\mathrm{N}$ew $\mathrm{Y}$ork $\mathrm{I}$nc., $\mathrm{N}$ew york, 1990.
  * [79] John Dirk Walecka Alexander L. Fetter.  Quantum $\mathrm{T}$heory of $\mathrm{M}$any-$\mathrm{P}$article $\mathrm{S}$ystems, $\mathrm{M}$c$\mathrm{G}$raw-$\mathrm{H}$ill, $\mathrm{S}$an $\mathrm{F}$ransisco, 1971.
  * [80] Nobukatsu Yoshida and S.-K. Yip.  Larkin-$\mathrm{O}$vchinnikov state in resonant $\mathrm{F}$ermi gas.  Phys. Rev. A, 75:063601, 2007.
  * [81] G. G. Batrouni, M. H. Huntley, V. G. Rousseau, and R. T. Scalettar.  Exact $\mathrm{N}$umerical $\mathrm{S}$tudy of $\mathrm{P}$air $\mathrm{F}$ormation with $\mathrm{I}$mbalanced $\mathrm{F}$ermion $\mathrm{P}$opulations.  Phys. Rev. Lett., 100:116405, 2008.
  * [82] J. E. Baarsma and P. Törmä.  Larkin-$\mathrm{O}$vchinnikov phases in two-dimensional square lattices.  Journal of Modern Optics, 63(18):1795–1804, 2016.
  * [83] Orso G Feiguin A E, Heidrich-Meisner F and Zwerger W.  $\mathrm{BCS}$$–$$\mathrm{BEC}$ $\mathrm{C}$rossover and $\mathrm{U}$nconventional $\mathrm{S}$uperfluid $\mathrm{O}$rder in $\mathrm{O}$ne $\mathrm{D}$imension ($\mathrm{B}$erlin: $\mathrm{S}$pringer), 2012.
  * [84] M. M. Parish.  The $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover Quantum Gas Experiments $\mathrm{C}$h 9 (London: Imperial College Press), 2014.
  * [85] Mohit Randeria and Edward Taylor.  Crossover from $\mathrm{B}$ardeen-$\mathrm{C}$ooper-$\mathrm{S}$chrieffer to $\mathrm{B}$ose- $\mathrm{E}$instein $\mathrm{C}$ondensation and the $\mathrm{U}$nitary $\mathrm{F}$ermi $\mathrm{G}$as.  Annual Review of Condensed Matter Physics, 5(1):209–232, 2014.
  * [86] A. Sanpera M. Lewenstein and V. Ahufinger.  Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems (Oxford: Oxford University Press), 2012.
  * [87] A. J. Leggett.  Modern Trends in the Theory of Condensed Matter, Springer-Verlag, Berlin, 1979.
  * [88] D. M. Eagles.  Possible $\mathrm{P}$airing without $\mathrm{S}$uperconductivity at $\mathrm{L}$ow $\mathrm{C}$arrier $\mathrm{C}$oncentrations in $\mathrm{B}$ulk and $\mathrm{T}$hin-$\mathrm{F}$ilm $\mathrm{S}$uperconducting $\mathrm{S}$emiconductors.  Phys. Rev., 186:456–463, 1969.
  * [89] Daniel Rohe and Walter Metzner.  Pair-fluctuation-induced pseudogap in the normal phase of the two-dimensional attractive hubbard model at weak coupling.  Phys. Rev. B, 63:224509, 2001.
  * [90] R. Micnas, J. Ranninger, and S. Robaszkiewicz.  Superconductivity in narrow-band systems with local nonretarded attractive interactions.  Rev. Mod. Phys., 62:113–171, 1990.
  * [91] N. Dupuis.  Berezinskii-$\mathrm{K}$osterlitz-$\mathrm{T}$houless transition and $\mathrm{BCS}$-$\mathrm{B}$ose crossover in the two-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 70:134502, 2004.
  * [92] K. Borejsza and N. Dupuis.  Antiferromagnetism and single-particle properties in the two-dimensional half-filled $\mathrm{H}$ubbard model: $\mathrm{A}$ nonlinear sigma model approach.  Phys. Rev. B, 69:085119, 2004.
  * [93] P. Nozières and S. Schmitt-Rink.  Bose condensation in an attractive fermion gas: $\mathrm{F}$rom weak to strong coupling superconductivity.  Journal of Low Temperature Physics, 59(3):195–211, 1985.
  * [94] J. J. Deisz, D. W. Hess, and J. W. Serene.  Phase diagram for the attractive hubbard model in two dimensions in a conserving approximation.  Phys. Rev. B, 66:014539, 2002.
  * [95] H. Tamaki, Y. Ohashi, and K. Miyake.  $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover and effects of density fluctuations in a two-component $\mathrm{F}$ermi gas described by the three-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. A, 77:063616, 2008.
  * [96] B. Kyung, S. Allen, and A.-M. S. Tremblay.  Pairing fluctuations and pseudogaps in the attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 64:075116, 2001.
  * [97] T. K. Kopeć.  Superconducting phase coherence and pairing gap in the three-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 65:054509, 2002.
  * [98] R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto.  Phase diagram of the two-dimensional negative-${U}$ $\mathrm{H}$ubbard model.  Phys. Rev. Lett., 62:1407–1410, 1989.
  * [99] A. Moreo and D. J. Scalapino.  Two-dimensional negative-${U}$ $\mathrm{H}$ubbard model.  Phys. Rev. Lett., 66:946–948, 1991.
  * [100] Adriana Moreo, Douglas J. Scalapino, and Steven R. White.  Quasiparticle gap in a two-dimensional $\mathrm{K}$osterlitz-$\mathrm{T}$houless superconductor.  Phys. Rev. B, 45:7544–7546, 1992.
  * [101] Mohit Randeria, Nandini Trivedi, Adriana Moreo, and Richard T. Scalettar.  Pairing and spin gap in the normal state of short coherence length superconductors.  Phys. Rev. Lett., 69:2001–2004, 1992.
  * [102] Nandini Trivedi and Mohit Randeria.  Deviations from $\mathrm{F}$ermi-$\mathrm{L}$iquid $\mathrm{B}$ehavior above ${T}_{c}$ in 2$\mathrm{D}$ $\mathrm{S}$hort $\mathrm{C}$oherence $\mathrm{L}$ength $\mathrm{S}$uperconductors.  Phys. Rev. Lett., 75:312–315, 1995.
  * [103] S. Allen, H. Touchette, S. Moukouri, Y. M. Vilk, and A.-M. S. Tremblay.  Role of $\mathrm{S}$ymmetry and $\mathrm{D}$imension in $\mathrm{P}$seudogap $\mathrm{P}$henomena.  Phys. Rev. Lett., 83:4128–4131, 1999.
  * [104] Singer, J. M., Schneider, T., and Meier, P. F.  Spectral properties of the attractive $\mathrm{H}$ubbard model.  Eur. Phys. J. B, 7(1):37–51, 1999.
  * [105] Thereza Paiva, Raimundo R. dos Santos, R. T. Scalettar, and P. J. H. Denteneer.  Critical temperature for the two-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 69:184501, 2004.
  * [106] M. Keller, W. Metzner, and U. Schollwöck.  Dynamical $\mathrm{M}$ean-$\mathrm{F}$ield $\mathrm{T}$heory for $\mathrm{P}$airing and $\mathrm{S}$pin $\mathrm{G}$ap in the $\mathrm{A}$ttractive $\mathrm{H}$ubbard $\mathrm{M}$odel.  Phys. Rev. Lett., 86:4612–4615, 2001.
  * [107] M. Capone, C. Castellani, and M. Grilli.  First-$\mathrm{O}$rder $\mathrm{P}$airing $\mathrm{T}$ransition and $\mathrm{S}$ingle-$\mathrm{P}$article $\mathrm{S}$pectral $\mathrm{F}$unction in the $\mathrm{A}$ttractive $\mathrm{H}$ubbard $\mathrm{M}$odel.  Phys. Rev. Lett., 88:126403, 2002.
  * [108] Arti Garg, H. R. Krishnamurthy, and Mohit Randeria.  $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover at $\mathrm{T}=0$: $\mathrm{A}$ dynamical mean-field theory approach.  Phys. Rev. B, 72:024517, 2005.
  * [109] J. Bauer and A. C. Hewson.  Quasiparticle excitations and dynamic susceptibilities in the BCS-BEC crossover.  Euro Phys. Lett., 85(2):27001, 2009.
  * [110] J. Bauer, A. C. Hewson, and N. Dupuis.  Dynamical mean-field theory and numerical renormalization group study of superconductivity in the attractive $\mathrm{H}$ubbard model.  Phys. Rev. B, 79:214518, 2009.
  * [111] Akihisa Koga and Philipp Werner.  Low-temperature properties of the infinite-dimensional attractive $\mathrm{H}$ubbard model.  Phys. Rev. A, 84:023638, 2011.
  * [112] S. Tarat and P. Majumdar.  A real space auxiliary field approach to the $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover.  Eur. Phys. Jour. B, 88:68, 2015.
  * [113] A. Sumiyama, R. Onuki, Y. Oda, H. Shishido, R. Settai, and Y. Ånuki.  Point-contact study of the heavy-fermion superconductor $\mathrm{CeCo}$$\mathrm{In}_{5}$.  Journal of Physics and Chemistry of Solids, 69(12):3018 – 3021, 2008.
  * [114] Y. Fasano, P. SzabÄ, J. KaÄmarÄk, Z. PribulovÄ, P. Pedrazzini, P. Samuely, and V.F. Correa.  Unconventional superconductivity in the strong-coupling limit for the heavy fermion system $\mathrm{CeCo}\mathrm{In}_{5}$.  Physica B: Condensed Matter, 536:798, 2018.
  * [115] Jochen Wosnitza.  Superconductivity in $\mathrm{L}$ayered $\mathrm{O}$rganic $\mathrm{M}$etals.  Crystals, 2(2):248–265, 2012.
  * [116] M. Abdel-Hafiez, S. Aswartham, S. Wurmehl, V. Grinenko, C. Hess, S.-L. Drechsler, S. Johnston, A. U. B. Wolter, B. Büchner, H. Rosner, and L. Boeri.  Specific heat and upper critical fields in $\mathrm{K}$$\mathrm{Fe}_{2}$$\mathrm{As}_{2}$ single crystals.  Phys. Rev. B, 85:134533, 2012.
  * [117] J. Hubbard.  Calculation of $\mathrm{P}$artition $\mathrm{F}$unctions.  Phys. Rev. Lett., 3:77–78, 1959.
  * [118] H. J. Schulz.  Effective action for strongly correlated fermions from functional integrals.  Phys. Rev. Lett., 65:2462–2465, 1990.
  * [119] S. Q. Wang, W. E. Evenson, and J. R. Schrieffer.  Theory of $\mathrm{I}$tinerant $\mathrm{F}$erromagnets $\mathrm{E}$xhibiting $\mathrm{L}$ocalized- $\mathrm{M}$oment $\mathrm{B}$ehavior above the $\mathrm{C}$urie $\mathrm{P}$oint.  Phys. Rev. Lett., 23:92–95, 1969.
  * [120] W. E. Evenson, J. R. Schrieffer, and S. Q. Wang.  New $\mathrm{A}$pproach to the $\mathrm{T}$heory of $\mathrm{I}$tinerant $\mathrm{E}$lectron $\mathrm{F}$erromagnets with $\mathrm{L}$ocal$-$$\mathrm{M}$oment $\mathrm{C}$haracteristics.  Journal of Applied Physics, 41(3):1199–1204, 1970.
  * [121] Yonatan Dubi, Yigal Meir, and Yshai Avishai.  Nature of the superconductor–insulator transition in disordered superconductors.  Nature, 449:876–880, 2007.
  * [122] S. Kumar and P. Majumdar.  A travelling cluster approximation for lattice fermions strongly coupled to classical degrees of freedom.  Eur. Phys. J. B, 50:571–579, 2006.
  * [123] Madhuparna Karmakar and Pinaki Majumdar.  Population-imbalanced lattice fermions near the $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover: $\mathrm{T}$hermal physics of the breached pair and $\mathrm{F}$ulde-$\mathrm{F}$errell- $\mathrm{L}$arkin-$\mathrm{O}$vchinnikov phases.  Phys. Rev. A, 93:053609, 2016.
  * [124] Madhuparna Karmakar.  Thermal transitions, pseudogap behavior, and $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover in $\mathrm{F}$ermi-$\mathrm{F}$ermi mixtures.  Phys. Rev. A, 97:033617, 2018.
  * [125] Madhuparna Karmakar and Pinaki Majumdar.  Anomalous pseudogap in population imbalanced $\mathrm{F}$ermi superfluids.  The European Physical Journal D, 70(10):220, Oct 2016.
  * [126] Madhuparna Karmakar.  Pauli limited d-wave superconductors: quantum breached pair phase and thermal transitions.  Journal of Physics: Condensed Matter, 32:405604, 2020.
  * [127] D.-H. Kim, J. J. Kinnunen, J.-P. Martikainen, and P. Törmä.  Exotic $\mathrm{S}$uperfluid $\mathrm{S}$tates of $\mathrm{L}$attice $\mathrm{F}$ermions in $\mathrm{E}$longated $\mathrm{T}$raps.  Phys. Rev. Lett., 106:095301, 2011.
  * [128] Miikka O. J. Heikkinen, Dong-Hee Kim, and Päivi Törmä.  Finite-temperature stability and dimensional crossover of exotic superfluidity in lattices.  Phys. Rev. B, 87:224513, 2013.
  * [129] M. O. J. Heikkinen, D.-H. Kim, M. Troyer, and P. Törmä.  Nonlocal $\mathrm{Q}$uantum $\mathrm{F}$luctuations and $\mathrm{F}$ermionic $\mathrm{S}$uperfluidity in the $\mathrm{I}$mbalanced $\mathrm{A}$ttractive $\mathrm{H}$ubbard $\mathrm{M}$odel.  Phys. Rev. Lett., 113:185301, 2014.
  * [130] Jan Gukelberger, Sebastian Lienert, Evgeny Kozik, Lode Pollet, and Matthias Troyer.  Fulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov pairing as leading instability on the square lattice.  Phys. Rev. B, 94:075157, 2016.
  * [131] M. J. Wolak, B. Grémaud, R. T. Scalettar, and G. G. Batrouni.  Pairing in a two-dimensional $\mathrm{F}$ermi gas with population imbalance.  Phys. Rev. A, 86:023630, 2012.
  * [132] V.L. Berezinsky.  Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. $\mathrm{C}$lassical systems.  Sov. Phys. JETP, 32:493–500, 1971.
  * [133] V. L. Berezinsky.  Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. $\mathrm{Q}$uantum systems.  Soviet Journal of Experimental and Theoretical Physics, 34:610, 1972\.
  * [134] J M Kosterlitz and D J Thouless.  Ordering, metastability and phase transitions in two-dimensional systems.  Journal of Physics C: Solid State Physics, 6:1181–1203, 1973.
  * [135] T K Koponen, T Paananen, J-P Martikainen, M R Bakhtiari, and P Törmä.  FFLO state in 1-, 2- and 3-dimensional optical lattices combined with a non-uniform background potential.  New Journal of Physics, 10:045014, 2008.
  * [136] T. K. Koponen, T. Paananen, J.-P. Martikainen, and P. Törmä.  Finite-$\mathrm{T}$emperature $\mathrm{P}$hase $\mathrm{D}$iagram of a $\mathrm{P}$olarized $\mathrm{F}$ermi $\mathrm{G}$as in an $\mathrm{O}$ptical $\mathrm{L}$attice.  Phys. Rev. Lett., 99:120403, 2007.
  * [137] Yen Lee Loh and Nandini Trivedi.  Detecting the $\mathrm{E}$lusive $\mathrm{L}$arkin-$\mathrm{)}$vchinnikov $\mathrm{M}$odulated $\mathrm{S}$uperfluid $\mathrm{P}$hases for $\mathrm{I}$mbalanced $\mathrm{F}$ermi $\mathrm{G}$ases in $\mathrm{O}$ptical $\mathrm{L}$attices.  Phys. Rev. Lett., 104:165302, 2010.
  * [138] Simone Chiesa and Shiwei Zhang.  Phases of attractive spin-imbalanced fermions in square lattices.  Phys. Rev. A, 88:043624, 2013.
  * [139] Peter Rosenberg, Simone Chiesa, and Shiwei Zhang.  FFLO order in ultra-cold atoms in three-dimensional optical lattices.  Journal of Physics: Condensed Matter, 27:225601, 2015.
  * [140] Daniel E. Sheehy and Leo Radzihovsky.  $\mathrm{BEC}$$-$$\mathrm{BCS}$ crossover, phase transitions and phase separation in polarized resonantly-paired superfluids.  Annals of Physics, 322(8):1790 – 1924, 2007.
  * [141] Yong Xu, Chunlei Qu, Ming Gong, and Chuanwei Zhang.  Competing superfluid orders in spin-orbit-coupled fermionic cold-atom optical lattices.  Phys. Rev. A, 89:013607, 2014.
  * [142] Kuei Sun and C. J. Bolech.  Pair tunneling, phase separation, and dimensional crossover in imbalanced fermionic superfluids in a coupled array of tubes.  Phys. Rev. A, 87:053622, 2013.
  * [143] Meera M. Parish, Stefan K. Baur, Erich J. Mueller, and David A. Huse.  Quasi-$\mathrm{O}$ne-$\mathrm{D}$imensional $\mathrm{P}$olarized $\mathrm{F}$ermi $\mathrm{S}$uperfluids.  Phys. Rev. Lett., 99:250403, 2007.
  * [144] Erhai Zhao and W. Vincent Liu.  Theory of quasi-one-dimensional imbalanced $\mathrm{F}$ermi gases.  Phys. Rev. A, 78:063605, 2008.
  * [145] A. B. Vorontsov, J. A. Sauls, and M. J. Graf.  Phase diagram and spectroscopy of $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov states of two-dimensional $d$-wave superconductors.  Phys. Rev. B, 72:184501, 2005.
  * [146] Anton B. Vorontsov and Matthias J. Graf.  Fermi-liquid effects in the $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov state of two-dimensional $d$-wave superconductors.  Phys. Rev. B, 74:172504, 2006.
  * [147] Robert Beaird, Anton B. Vorontsov, and Ilya Vekhter.  Pauli-limited superconductivity with classical magnetic fluctuations.  Phys. Rev. B, 81:224501, 2010.
  * [148] A. B. Vorontsov, I. Vekhter, and M. J. Graf.  Pauli-limited upper critical field in dirty $d$-wave superconductors.  Phys. Rev. B, 78:180505, 2008.
  * [149] Kun Yang and S. L. Sondhi.  Response of a ${d}_{{x}^{2}-{y}^{2}}$ superconductor to a $\mathrm{Z}$eeman magnetic field.  Phys. Rev. B, 57:8566–8570, 1998.
  * [150] Tao Zhou and C. S. Ting.  Phase diagram and local tunneling spectroscopy of the $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov states of a two-dimensional square-lattice $d$-wave superconductor.  Phys. Rev. B, 80:224515, 2009.
  * [151] Yuhki Hatakeyama and Ryusuke Ikeda.  Strong-$\mathrm{C}$oupling $\mathrm{A}$pproach to $\mathrm{A}$ntiferromagnetic $\mathrm{O}$rdering $\mathrm{D}$riven by $\mathrm{P}$aramagnetic $\mathrm{P}$air-$\mathrm{B}$reaking in $d$-$\mathrm{W}$ave $\mathrm{S}$uperconducting $\mathrm{P}$hase.  Journal of the Physical Society of Japan, 83(2):024713, 2014.
  * [152] Yuhki Hatakeyama and Ryusuke Ikeda.  Antiferromagnetic order oriented by $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov superconducting order.  Phys. Rev. B, 91:094504, 2015.
  * [153] Ken-ichi Hosoya and Ryusuke Ikeda.  Angular dependence of antiferromagnetic order induced by paramagnetism in a $d$-wave superconductor.  Phys. Rev. B, 88:094513, 2013.
  * [154] Yuhki Hatakeyama and Ryusuke Ikeda.  Emergent antiferromagnetism in a $d$-wave superconductor with strong paramagnetic pair-breaking.  Phys. Rev. B, 83:224518, 2011.
  * [155] Ken-ichi Hosoya and Ryusuke Ikeda.  Possible triplet superconducting order in a magnetic superconducting phase induced by paramagnetic pair breaking.  Phys. Rev. B, 95:224513, 2017.
  * [156] Kyosuke Adachi and Ryusuke Ikeda.  Possible $\mathrm{F}$ield$-$$\mathrm{T}$emperature $\mathrm{P}$hase $\mathrm{D}$iagrams of $\mathrm{T}$wo-$\mathrm{B}$ and $\mathrm{S}$uperconductors with $\mathrm{P}$aramagnetic $\mathrm{P}$air- $\mathrm{B}$reaking.  Journal of the Physical Society of Japan, 84(6):064712, 2015.
  * [157] Nayuta Takemori and Akihisa Koga.  Low-$\mathrm{T}$emperature $\mathrm{P}$roperties of the $\mathrm{F}$ermionic $\mathrm{M}$ixtures with $\mathrm{M}$ass $\mathrm{I}$mbalance in $\mathrm{O}$ptical $\mathrm{L}$attice.  Journal of the Physical Society of Japan, 81(6):063002, 2012.
  * [158] Ryo Hanai, Takashi Kashimura, Ryota Watanabe, Daisuke Inotani, and Yoji Ohashi.  Excitation properties and effects of mass imbalance in the $\mathrm{BCS}$$-$$\mathrm{BEC}$ crossover regime of an ultracold $\mathrm{F}$ermi gas.  Phys. Rev. A, 88:053621, 2013.
  * [159] Dietrich Roscher, Jens Braun, and Joaquin E. Drut.  Phase structure of mass- and spin-imbalanced unitary $\mathrm{F}$ermi gases.  Phys. Rev. A, 91:053611, 2015.
  * [160] Jens Braun, Joaquin E. Drut, Thomas Jahn, Martin Pospiech, and Dietrich Roscher.  Phases of spin- and mass-imbalanced ultracold $\mathrm{F}$ermi gases in harmonic traps.  Phys. Rev. A, 89:053613, 2014.
  * [161] Ryo Hanai and Yoji Ohashi.  Heteropairing and component-dependent pseudogap phenomena in an ultracold $\mathrm{F}$ermi gas with different species with different masses.  Phys. Rev. A, 90:043622, 2014.
  * [162] Hao Guo, Chih-Chun Chien, Qijin Chen, Yan He, and K. Levin.  Finite-temperature behavior of an interspecies fermionic superfluid with population imbalance.  Phys. Rev. A, 80:011601, 2009.
  * [163] K. B. Gubbels, J. E. Baarsma, and H. T. C. Stoof.  Lifshitz $\mathrm{P}$oint in the $\mathrm{P}$hase $\mathrm{D}$iagram of $\mathrm{R}$esonantly $\mathrm{I}$nteracting ${}^{6}\mathrm{Li}$$-$${}^{40}\mathrm{K}$ $\mathrm{M}$ixtures.  Phys. Rev. Lett., 103:195301, 2009.
  * [164] B. Wang, Han-Dong Chen, and S. Das Sarma.  Quantum phase diagram of fermion mixtures with population imbalance in one-dimensional optical lattices.  Phys. Rev. A, 79:051604, 2009.
  * [165] M. Dalmonte, K. Dieckmann, T. Roscilde, C. Hartl, A. E. Feiguin, U. Schollwöck, and F. Heidrich-Meisner.  Dimer, trimer, and $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov liquids in mass- and spin-imbalanced trapped binary mixtures in one dimension.  Phys. Rev. A, 85:063608, 2012.
  * [166] Shanna Pahl and Zlatko Koinov.  Phase $\mathrm{D}$iagram of a ${}^{6}\mathrm{Li}$$-$${}^{40}\mathrm{K}$ $\mathrm{M}$ixture in a $\mathrm{S}$quare $\mathrm{L}$attice.  Journal of Low Temperature Physics, 176:113, 2014.
  * [167] Jens Braun, Joaquín E. Drut, and Dietrich Roscher.  Zero-$\mathrm{T}$emperature $\mathrm{E}$quation of $\mathrm{S}$tate of $\mathrm{M}$ass-$\mathrm{I}$mbalanced $\mathrm{R}$esonant $\mathrm{F}$ermi $\mathrm{G}$ases.  Phys. Rev. Lett., 114:050404, 2015.
  * [168] Victor Galitski and Ian B. Spielman.  Spin–orbit coupling in quantum gases.  Nature, 494:49, 2013.
  * [169] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas, and Patrik Öhberg.  Colloquium: $\mathrm{A}$rtificial gauge potentials for neutral atoms.  Rev. Mod. Phys., 83:1523–1543, 2011.
  * [170] Jami J Kinnunen, Jildou E Baarsma, Jani-Petri Martikainen, and Päivi Törmä.  The fulde–ferrell–larkin–ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review.  Reports on Progress in Physics, 81:046401, 2018.
  * [171] J. Higbie and D. M. Stamper-Kurn.  Periodically dressed $\mathrm{B}$ose–$\mathrm{E}$instein condensate: A superfluid with an anisotropic and variable critical velocity.  Phys. Rev. Lett., 88:090401, 2002.
  * [172] Y. J. Lin, K. Jiménez-García, and I. B. Spielman.  Spin–orbit coupled $\mathrm{B}$ose–$\mathrm{E}$instein condensates.  Nature, 471:83, 2011.
  * [173] Pengjun Wang, Zeng-Qiang Yu, Zhengkun Fu, Jiao Miao, Lianghui Huang, Shijie Chai, Hui Zhai, and Jing Zhang.  Spin–orbit coupled degenerate $\mathrm{F}$ermi gases.  Phys. Rev. Lett., 109:095301, 2012.
  * [174] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadzibabic, Tarik Yefsah, Waseem S. Bakr, and Martin W. Zwierlein.  Spin–injection spectroscopy of a spin–orbit coupled $\mathrm{F}$ermi gas.  Phys. Rev. Lett., 109:095302, 2012.
  * [175] J. Ruseckas, G. Juzeliūnas, P. Öhberg, and M. Fleischhauer.  Non-$\mathrm{A}$belian $\mathrm{G}$auge $\mathrm{P}$otentials for $\mathrm{U}$ltracold $\mathrm{A}$toms with $\mathrm{D}$egenerate $\mathrm{D}$ark $\mathrm{S}$tates.  Phys. Rev. Lett., 95:010404, 2005.
  * [176] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein.  Cold $\mathrm{A}$toms in $\mathrm{N}$on-$\mathrm{A}$belian $\mathrm{G}$auge $\mathrm{P}$otentials: $\mathrm{F}$rom the $\mathrm{H}$ofstadter $"\mathrm{M}oth"$ to $\mathrm{L}$attice $\mathrm{G}$auge $\mathrm{T}$heory.  Phys. Rev. Lett., 95:010403, 2005.
  * [177] D. L. Campbell, G. Juzeliūnas, and I. B. Spielman.  Realistic $\mathrm{R}$ashba and $\mathrm{D}$resselhaus spin-orbit coupling for neutral atoms.  Phys. Rev. A, 84:025602, 2011.
  * [178] Zhen Zheng, Ming Gong, Xubo Zou, Chuanwei Zhang, and Guangcan Guo.  Route to observable $\mathrm{F}$ulde–$\mathrm{F}$errell–$\mathrm{L}$arkin–$\mathrm{O}$vchinnikov phases in three-dimensional spin-orbit-coupled degenerate $\mathrm{F}$ermi gases.  Phys. Rev. A, 87:031602, 2013.
  * [179] Yong Xu, Chunlei Qu, Ming Gong, and Chuanwei Zhang.  Competing superfluid orders in spin-orbit-coupled fermionic cold-atom optical lattices.  Phys. Rev. A, 89:013607, 2014.
  * [180] M. Iskin.  Trapped $\mathrm{F}$ermi gases with $\mathrm{R}$ashba spin-orbit coupling in two dimensions.  Phys. Rev. A, 86:065601, 2012.
  * [181] M. Iskin and A. L. Subaş ı.  Topological superfluid phases of an atomic $\mathrm{F}$ermi gas with in-and out-of-plane $\mathrm{Z}$eeman fields and equal $\mathrm{R}$ashba-$\mathrm{D}$resselhaus spin-orbit coupling.  Phys. Rev. A, 87:063627, 2013.
  * [182] Kangjun Seo, Chuanwei Zhang, and Sumanta Tewari.  Topological uniform superfluid and $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin- $\mathrm{O}$vchinnikov phases in three-dimensional to one-dimensional crossover of spin-orbit-coupled $\mathrm{F}$ermi gases.  Phys. Rev. A, 88:063601, 2013.
  * [183] Hui Hu and Xia-Ji Liu.  Fulde$–$$\mathrm{F}$errell superfluidity in ultracold $\mathrm{F}$ermi gases with $\mathrm{R}$ashba spin–orbit coupling.  New Journal of Physics, 15:093037, 2013.
  * [184] Xia-Ji Liu and Hui Hu.  Inhomogeneous $\mathrm{F}$ulde-$\mathrm{F}$errell superfluidity in spin-orbit-coupled atomic $\mathrm{F}$ermi gases.  Phys. Rev. A, 87:051608, 2013.
  * [185] Xia-Ji Liu.  Fulde-$\mathrm{F}$errell pairing instability of a $\mathrm{R}$ashba spin-orbit-coupled $\mathrm{F}$ermi gas.  Phys. Rev. A, 88:043607, 2013.
  * [186] Lin Dong, Lei Jiang, and Han Pu.  Fulde$-$$\mathrm{F}$errell pairing instability in spin–orbit coupled $\mathrm{F}$ermi gas.  New Journal of Physics, 15(7):075014, 2013.
  * [187] Xiang-Fa Zhou, Guang-Can Guo, Wei Zhang, and Wei Yi.  Exotic pairing states in a $\mathrm{F}$ermi gas with three-dimensional spin-orbit coupling.  Phys. Rev. A, 87:063606, 2013.
  * [188] M. Iskin.  Spin-orbit-coupling-induced $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov- like $\mathrm{C}$ooper pairing and skyrmion-like polarization textures in optical lattices.  Phys. Rev. A, 88:013631, 2013.
  * [189] Sumanta Tewari, Tudor D Stanescu, Jay D Sau, and S Das Sarma.  Topologically non-trivial superconductivity in spin–orbit-coupled systems: bulk phases and quantum phase transitions.  New Journal of Physics, 13:065004, 2011.
  * [190] Wei Zhang and Wei Yi.  Topological $\mathrm{F}$ulde–$\mathrm{F}$errell–$\mathrm{L}$arkin–$\mathrm{O}$vchinnikov states in spin–orbit-coupled $\mathrm{F}$ermi gases.  Nature Communications, 4:2711, 2013.
  * [191] Chunlei Qu, Zhen Zheng, Ming Gong, Yong Xu, Li Mao, Xubo Zou, Guangcan Guo, and Chuanwei Zhang.  Topological superfluids with finite-momentum pairing and $\mathrm{M}$ajorana fermions.  Nature Communications, 4:2710, 2013.
  * [192] Ye Cao, Shu-Hao Zou, Xia-Ji Liu, Su Yi, Gui-Lu Long, and Hui Hu.  Gapless $\mathrm{T}$opological $\mathrm{F}$ulde-$\mathrm{F}$errell $\mathrm{S}$uperfluidity in $\mathrm{S}$pin-$\mathrm{O}$rbit $\mathrm{C}$oupled $\mathrm{F}$ermi $\mathrm{G}$ases.  Phys. Rev. Lett., 113:115302, 2014.
  * [193] Xia-Ji Liu and Hui Hu.  Topological superfluid in one-dimensional spin-orbit-coupled atomic $\mathrm{F}$ermi gases.  Phys. Rev. A, 85:033622, 2012.
  * [194] Yong Xu and Chuanwei Zhang.  Berezinskii-$\mathrm{K}$osterlitz-$\mathrm{T}$houless $\mathrm{P}$hase $\mathrm{T}$ransition in 2$\mathrm{D}$$\mathrm{S}$pin-$\mathrm{O}$rbit-coupled $\mathrm{F}$ulde-$\mathrm{F}$errell $\mathrm{S}$uperfluids.  Phys. Rev. Lett., 114:110401, 2015.
  * [195] Ye Cao, Xia-Ji Liu, Lianyi He, Gui-Lu Long, and Hui Hu.  Superfluid density and $\mathrm{B}$erezinskii-$\mathrm{K}$osterlitz-$\mathrm{T}$houless transition of a spin-orbit-coupled $\mathrm{F}$ulde-$\mathrm{F}$errell superfluid.  Phys. Rev. A, 91:023609, 2015.
  * [196] Jibiao Wang, Yanming Che, Leifeng Zhang, and Qijin Chen.  Instability of $\mathrm{F}$ulde-$\mathrm{F}$errell-$\mathrm{L}$arkin-$\mathrm{O}$vchinnikov states in atomic $\mathrm{F}$ermi gases in three and two dimensions.  Phys. Rev. B, 97:134513, 2018.
  * [197] Peter Rosenberg, Hao Shi, and Shiwei Zhang.  Ultracold $\mathrm{A}$toms in a $\mathrm{S}$quare $\mathrm{L}$attice with $\mathrm{S}$pin-$\mathrm{O}$rbit $\mathrm{C}$oupling: $\mathrm{C}$harge $\mathrm{O}$rder, $\mathrm{S}$uperfluidity, and $\mathrm{T}$opological $\mathrm{S}$ignatures.  Phys. Rev. Lett., 119:265301, 2017.


# Study of $J/\psi\to\eta\phi\pi^{+}\pi^{-}$ at BESIII

M. Ablikim1, M. N. Achasov8,a, X. C. Ai1, O. Albayrak4, M. Albrecht3, D. J.
Ambrose43, A. Amoroso47A,47C, F. F. An1, Q. An44, J. Z. Bai1, R. Baldini
Ferroli19A, Y. Ban30, D. W. Bennett18, J. V. Bennett4, M. Bertani19A, D.
Bettoni20A, J. M. Bian42, F. Bianchi47A,47C, E. Boger22,h, O. Bondarenko24, I.
Boyko22, R. A. Briere4, H. Cai49, X. Cai1, O. Cakir39A,b, A. Calcaterra19A, G.
F. Cao1, S. A. Cetin39B, J. F. Chang1, G. Chelkov22,c, G. Chen1, H. S. Chen1,
H. Y. Chen2, J. C. Chen1, M. L. Chen1, S. J. Chen28, X. Chen1, X. R. Chen25,
Y. B. Chen1, H. P. Cheng16, X. K. Chu30, G. Cibinetto20A, D. Cronin-
Hennessy42, H. L. Dai1, J. P. Dai33, A. Dbeyssi13, D. Dedovich22, Z. Y. Deng1,
A. Denig21, I. Denysenko22, M. Destefanis47A,47C, F. De Mori47A,47C, Y.
Ding26, C. Dong29, J. Dong1, L. Y. Dong1, M. Y. Dong1, S. X. Du51, P. F.
Duan1, J. Z. Fan38, J. Fang1, S. S. Fang1, X. Fang44, Y. Fang1, L.
Fava47B,47C, F. Feldbauer21, G. Felici19A, C. Q. Feng44, E. Fioravanti20A, M.
Fritsch13,21, C. D. Fu1, Q. Gao1, Y. Gao38, I. Garzia20A, K. Goetzen9, W. X.
Gong1, W. Gradl21, M. Greco47A,47C, M. H. Gu1, Y. T. Gu11, Y. H. Guan1, A. Q.
Guo1, L. B. Guo27, T. Guo27, Y. Guo1, Y. P. Guo21, Z. Haddadi24, A. Hafner21,
S. Han49, Y. L. Han1, F. A. Harris41, K. L. He1, Z. Y. He29, T. Held3, Y. K.
Heng1, Z. L. Hou1, C. Hu27, H. M. Hu1, J. F. Hu47A, T. Hu1, Y. Hu1, G. M.
Huang5, G. S. Huang44, H. P. Huang49, J. S. Huang14, X. T. Huang32, Y.
Huang28, T. Hussain46, Q. Ji1, Q. P. Ji29, X. B. Ji1, X. L. Ji1, L. L. Jiang1,
L. W. Jiang49, X. S. Jiang1, J. B. Jiao32, Z. Jiao16, D. P. Jin1, S. Jin1, T.
Johansson48, A. Julin42, N. Kalantar-Nayestanaki24, X. L. Kang1, X. S. Kang29,
M. Kavatsyuk24, B. C. Ke4, R. Kliemt13, B. Kloss21, O. B. Kolcu39B,d, B.
Kopf3, M. Kornicer41, W. Kuehn23, A. Kupsc48, W. Lai1, J. S. Lange23, M.
Lara18, P. Larin13, C. H. Li1, Cheng Li44, D. M. Li51, F. Li1, G. Li1, H. B.
Li1, J. C. Li1, Jin Li31, K. Li12, K. Li32, P. R. Li40, T. Li32, W. D. Li1, W.
G. Li1, X. L. Li32, X. M. Li11, X. N. Li1, X. Q. Li29, Z. B. Li37, H. Liang44,
Y. F. Liang35, Y. T. Liang23, G. R. Liao10, D. X. Lin13, B. J. Liu1, C. L.
Liu4, C. X. Liu1, F. H. Liu34, Fang Liu1, Feng Liu5, H. B. Liu11, H. H. Liu1,
H. H. Liu15, H. M. Liu1, J. Liu1, J. P. Liu49, J. Y. Liu1, K. Liu38, K. Y.
Liu26, L. D. Liu30, P. L. Liu1, Q. Liu40, S. B. Liu44, X. Liu25, X. X. Liu40,
Y. B. Liu29, Z. A. Liu1, Zhiqiang Liu1, Zhiqing Liu21, H. Loehner24, X. C.
Lou1,e, H. J. Lu16, J. G. Lu1, R. Q. Lu17, Y. Lu1, Y. P. Lu1, C. L. Luo27, M.
X. Luo50, T. Luo41, X. L. Luo1, M. Lv1, X. R. Lyu40, F. C. Ma26, H. L. Ma1, L.
L. Ma32, Q. M. Ma1, S. Ma1, T. Ma1, X. N. Ma29, X. Y. Ma1, F. E. Maas13, M.
Maggiora47A,47C, Q. A. Malik46, Y. J. Mao30, Z. P. Mao1, S. Marcello47A,47C,
J. G. Messchendorp24, J. Min1, T. J. Min1, R. E. Mitchell18, X. H. Mo1, Y. J.
Mo5, C. Morales Morales13, K. Moriya18, N. Yu. Muchnoi8,a, H. Muramatsu42, Y.
Nefedov22, F. Nerling13, I. B. Nikolaev8,a, Z. Ning1, S. Nisar7, S. L. Niu1,
X. Y. Niu1, S. L. Olsen31, Q. Ouyang1, S. Pacetti19B, P. Patteri19A, M.
Pelizaeus3, H. P. Peng44, K. Peters9, J. L. Ping27, R. G. Ping1, R. Poling42,
Y. N. Pu17, M. Qi28, S. Qian1, C. F. Qiao40, L. Q. Qin32, N. Qin49, X. S.
Qin1, Y. Qin30, Z. H. Qin1, J. F. Qiu1, K. H. Rashid46, C. F. Redmer21, H. L.
Ren17, M. Ripka21, G. Rong1, X. D. Ruan11, V. Santoro20A, A. Sarantsev22,f, M.
Savrié20B, K. Schoenning48, S. Schumann21, W. Shan30, M. Shao44, C. P. Shen2,
P. X. Shen29, X. Y. Shen1, H. Y. Sheng1, M. R. Shepherd18, W. M. Song1, X. Y.
Song1, S. Sosio47A,47C, S. Spataro47A,47C, B. Spruck23, G. X. Sun1, J. F.
Sun14, S. S. Sun1, Y. J. Sun44, Y. Z. Sun1, Z. J. Sun1, Z. T. Sun18, C. J.
Tang35, X. Tang1, I. Tapan39C, E. H. Thorndike43, M. Tiemens24, D. Toth42, M.
Ullrich23, I. Uman39B, G. S. Varner41, B. Wang29, B. L. Wang40, D. Wang30, D.
Y. Wang30, K. Wang1, L. L. Wang1, L. S. Wang1, M. Wang32, P. Wang1, P. L.
Wang1, Q. J. Wang1, S. G. Wang30, W. Wang1, X. F. Wang38, Y. D. Wang19A, Y. F.
Wang1, Y. Q. Wang21, Z. Wang1, Z. G. Wang1, Z. H. Wang44, Z. Y. Wang1, T.
Weber21, D. H. Wei10, J. B. Wei30, P. Weidenkaff21, S. P. Wen1, U. Wiedner3,
M. Wolke48, L. H. Wu1, Z. Wu1, L. G. Xia38, Y. Xia17, D. Xiao1, Z. J. Xiao27,
Y. G. Xie1, G. F. Xu1, L. Xu1, Q. J. Xu12, Q. N. Xu40, X. P. Xu36, L. Yan44,
W. B. Yan44, W. C. Yan44, Y. H. Yan17, H. X. Yang1, L. Yang49, Y. Yang5, Y. X.
Yang10, H. Ye1, M. Ye1, M. H. Ye6, J. H. Yin1, B. X. Yu1, C. X. Yu29, H. W.
Yu30, J. S. Yu25, C. Z. Yuan1, W. L. Yuan28, Y. Yuan1, A. Yuncu39B,g, A. A.
Zafar46, A. Zallo19A, Y. Zeng17, B. X. Zhang1, B. Y. Zhang1, C. Zhang28, C. C.
Zhang1, D. H. Zhang1, H. H. Zhang37, H. Y. Zhang1, J. J. Zhang1, J. L. Zhang1,
J. Q. Zhang1, J. W. Zhang1, J. Y. Zhang1, J. Z. Zhang1, K. Zhang1, L. Zhang1,
S. H. Zhang1, X. J. Zhang1, X. Y. Zhang32, Y. Zhang1, Y. H. Zhang1, Z. H.
Zhang5, Z. P. Zhang44, Z. Y. Zhang49, G. Zhao1, J. W. Zhao1, J. Y. Zhao1, J.
Z. Zhao1, Lei Zhao44, Ling Zhao1, M. G. Zhao29, Q. Zhao1, Q. W. Zhao1, S. J.
Zhao51, T. C. Zhao1, Y. B. Zhao1, Z. G. Zhao44, A. Zhemchugov22,h, B. Zheng45,
J. P. Zheng1, W. J. Zheng32, Y. H. Zheng40, B. Zhong27, L. Zhou1, Li Zhou29,
X. Zhou49, X. K. Zhou44, X. R. Zhou44, X. Y. Zhou1, K. Zhu1, K. J. Zhu1, S.
Zhu1, X. L. Zhu38, Y. C. Zhu44, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, B. S.
Zou1, J. H. Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
5 Central China Normal University, Wuhan 430079, People’s Republic of China
6 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
7 COMSATS Institute of Information Technology, Lahore, Defence Road, Off
Raiwind Road, 54000 Lahore, Pakistan
8 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
9 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
10 Guangxi Normal University, Guilin 541004, People’s Republic of China
11 GuangXi University, Nanning 530004, People’s Republic of China
12 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
13 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
14 Henan Normal University, Xinxiang 453007, People’s Republic of China
15 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
16 Huangshan College, Huangshan 245000, People’s Republic of China
17 Hunan University, Changsha 410082, People’s Republic of China
18 Indiana University, Bloomington, Indiana 47405, USA
19 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
20 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
21 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
22 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
23 Justus Liebig University Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
24 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
25 Lanzhou University, Lanzhou 730000, People’s Republic of China
26 Liaoning University, Shenyang 110036, People’s Republic of China
27 Nanjing Normal University, Nanjing 210023, People’s Republic of China
28 Nanjing University, Nanjing 210093, People’s Republic of China
29 Nankai University, Tianjin 300071, People’s Republic of China
30 Peking University, Beijing 100871, People’s Republic of China
31 Seoul National University, Seoul, 151-747 Korea
32 Shandong University, Jinan 250100, People’s Republic of China
33 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
34 Shanxi University, Taiyuan 030006, People’s Republic of China
35 Sichuan University, Chengdu 610064, People’s Republic of China
36 Soochow University, Suzhou 215006, People’s Republic of China
37 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
38 Tsinghua University, Beijing 100084, People’s Republic of China
39 (A)Istanbul Aydin University, 34295 Sefakoy, Istanbul, Turkey; (B)Dogus
University, 34722 Istanbul, Turkey; (C)Uludag University, 16059 Bursa, Turkey
40 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
41 University of Hawaii, Honolulu, Hawaii 96822, USA
42 University of Minnesota, Minneapolis, Minnesota 55455, USA
43 University of Rochester, Rochester, New York 14627, USA
44 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
45 University of South China, Hengyang 421001, People’s Republic of China
46 University of the Punjab, Lahore-54590, Pakistan
47 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
48 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
49 Wuhan University, Wuhan 430072, People’s Republic of China
50 Zhejiang University, Hangzhou 310027, People’s Republic of China
51 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
b Also at Ankara University, 06100 Tandogan, Ankara, Turkey
c Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia and at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
d Currently at Istanbul Arel University, Kucukcekmece, Istanbul, Turkey
e Also at University of Texas at Dallas, Richardson, Texas 75083, USA
f Also at the PNPI, Gatchina 188300, Russia
g Also at Bogazici University, 34342 Istanbul, Turkey
h Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia

###### Abstract

Based on a sample of $2.25\times 10^{8}J/\psi$ events taken with the BESIII
detector at the BEPCII collider, we present the results of a study of the
decay $J/\psi\to\eta\phi\pi^{+}\pi^{-}$. The $Y(2175)$ resonance is observed
in the invariant mass spectrum of $\phi f_{0}(980)$ with a statistical
significance of greater than $10\sigma$. The corresponding mass and width are
determined to be $M=2200\pm 6\mathrm{(stat.)}\pm
5\mathrm{(syst.)}~{}\mathrm{MeV}/c^{2}$ and $\Gamma=104\pm
15\mathrm{(stat.)}\pm 15\mathrm{(syst.)}$ MeV, respectively, and the product
branching fraction is measured to be $\mathcal{B}(J/\psi\to\eta Y(2175)$,
$Y(2175)\to\phi f_{0}(980)$, $f_{0}(980)\to\pi^{+}\pi^{-})=(1.20\pm
0.14\mathrm{(stat.)}\pm 0.37\mathrm{(syst.)})\times 10^{-4}$. The results are
consistent within errors with those of previous experiments. We also measure
the branching fraction of $J/\psi\to\phi f_{1}(1285)$ with
$f_{1}(1285)\to\eta\pi^{+}\pi^{-}$ and set upper limits on the branching
fractions for $J/\psi\to\phi\eta(1405)$/$\phi X(1835)$/$\phi X(1870)$ with
$\eta(1405)$/$X(1835)$/$X(1870)\to\eta\pi^{+}\pi^{-}$ at the 90% confidence
level.

###### pacs:

12.39.Mk, 13.25.Gv, 14.40.Be, 14.40.Rt

##  I Introduction

The $Y(2175)$, also referred to as the $\phi(2170)$ by the Particle Data Group
(PDG 2014) PDG , was first observed by the BABAR experiment aaaa_babar  in the
$e^{+}e^{-}\rightarrow\gamma_{ISR}\phi f_{0}(980)$ initial-state-radiation
(ISR) process. It was later confirmed by the BESII experiment in
$J/\psi\rightarrow\eta\phi f_{0}(980)$ decays aaab_wanx  and via the same ISR
process by the BELLE aaba_belle  and BABAR experiments babar_y2175  with
increased statistics. Since the $Y(2175)$ resonance is produced via ISR in
$e^{+}e^{-}$ collisions, it is known to have $J^{PC}=1^{--}$. This observation
stimulated the speculation that the $Y(2175)$ may be an $s$-quark counterpart
to the $Y(4260)$ babay4260 ; bn978 , since both are produced in $e^{+}e^{-}$
annihilation and exhibit similar decay patterns. Like for the $Y(4260)$, a
number of different interpretations have been proposed for the $Y(2175)$ with
predicted masses that are consistent, within errors, with the experimental
measurements. These include: an $s\overline{s}$-gluon hybrid Yml1 ; an excited
$\phi$ state Yml2 ; a tetraquark state tetraq ; a $\Lambda\overline{\Lambda}$
bound state EK15 ; CFQ ; or an ordinary $\phi f_{0}(980)$ resonance produced
by interactions between the final state particles 27mev .

A recent review zhusl  discusses the basic problem of the large expected decay
widths into two mesons, which contradicts experimental observations. Around
the mass of the $Y(2175)$, there are two conventional $1^{--}$ $s\bar{s}$
states in the quark model, $2^{3}D_{1}$ and $3^{3}S_{1}$. According to Ref.
barnes10 , the width of the $3^{3}S_{1}$ $s\bar{s}$ state is expected to be
about 380 MeV. The total width of the $2^{3}D_{1}$ state from both
${}^{3}P_{0}$ and flux tube model is expected to be around $(150\sim 250)$ MeV
Yml2 . However, the predictions from these strong decay models sometimes
deviate from the experimentally found width by a factor of two or three. For
comparison, the widths of the $3^{3}S_{1}$ and $2^{3}D_{1}$ charmonium are
less than 110 MeV wmy . Fortunately, the characteristic decay modes of Y(2175)
as either a hybrid or $s\bar{s}$ state are quite different, which may be used
to distinguish the hybrid and $s\bar{s}$ schemes. The possibility of $Y(2175)$
arising from $S$-wave threshold effects is not excluded. As of now, none of
these interpretations have been either established or ruled out by experiment.
The confirmation and study of the $Y(2175)$ in
$J/\psi\to\eta\phi\pi^{+}\pi^{-}$ with a large data sample is necessary for
clarifying its nature.

The $J/\psi\rightarrow\eta\phi\pi^{+}\pi^{-}$ decay also offers a unique
opportunity to investigate the properties of the $f_{1}(1285)$, the
$\eta(1295)$, and the $\eta(1405)/\eta(1475)$ resonances. The $f_{1}(1285)$ is
usually considered to be a member of the axial vector meson nonet, but the
interpretation of the $\eta(1295)$ is less clear. Both the $f_{1}(1285)$ and
the $\eta(1295)$ were seen in fixed target experiments, but the $\eta(1295)$
was not evident in central production, in $\gamma\gamma$ collisions, or in
$J/\psi$ decays. Therefore it has been speculated that either the
$f_{1}(1285)$, at least in some cases, contains an $\eta(1295)$ component both
, or that the $\eta(1295)$ does not exist. The $\eta(1405)/\eta(1475)$
pseudoscalar was once regarded as a glueball candidate since it is copiously
produced in $J/\psi$ radiative decays jpsirad  and there was only an upper
limit from $\gamma\gamma$ collisions ggc1 . But this viewpoint changed when
the $\eta(1405)/\eta(1475)$ was also observed in untagged $\gamma\gamma$
collisions L3  and in $J/\psi$ hadronic decays.

In addition, two interesting resonances, the $X(1835)$ and the $X(1870)$, were
observed in $J/\psi\rightarrow\gamma\pi^{+}\pi^{-}\eta^{\prime}$
X1835_pipietap1 ; X1835_pipietap2  and
$J/\psi\rightarrow\omega\pi^{+}\pi^{-}\eta$ liuk , respectively. The
$X(1835)$, in particular, inspired many possible theoretical interpretations,
including a $p\overline{p}$ bound state ppbs1 ; ppbs2 , a glueball X1835g1 ;
X1835g2 ; X1835g3 , and final state interactions (FSI) between a proton and
antiproton fsi1 ; fsi2 ; fsi3 . To better understand the properties of these
two resonances, one needs to further study their production in different
$J/\psi$ decay modes. For example, the search for them in the
$\eta\pi^{+}\pi^{-}$ mass spectrum recoiling against the $\phi$ in $J/\psi$
decays would be rather interesting for clarifying their nature.

In this paper, we present a study of the decay
$J/\psi\rightarrow\eta\phi\pi^{+}\pi^{-}$ with $\eta\to\gamma\gamma$ and
$\phi\to K^{+}K^{-}$ decay modes using a sample of $2.25\times 10^{8}$
$J/\psi$ events collected with the Beijing Spectrometer (BESIII) located at
the Beijing Electron-Positron Collider (BEPCII) BEPCII . The mass and width of
the $Y(2175)$, as well as its production rate, are measured. In addition, the
production rates of the $f_{1}(1285)$, the $\eta(1405)/\eta(1475)$, the
$X(1835)$, and the $X(1870)$ in $J/\psi$ hadronic decays associated with a
$\phi$ meson are investigated.

##  II Detector and Monte Carlo simulation

The BESIII detector is a magnetic spectrometer BEPCII  located at BEPCII,
which is a double-ring $e^{+}e^{-}$ collider with a design peak luminosity of
$10^{33}$ cm-2 s-1 at a center-of-mass energy of 3.773 GeV. The cylindrical
core of the BESIII detector consists of a helium-based main drift chamber
(MDC), a plastic scintillator time-of-flight system (TOF), and a CsI (Tl)
electromagnetic calorimeter (EMC), which are all enclosed in a superconducting
solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported
by an octagonal flux-return yoke with modules of resistive plate muon counters
interleaved with steel. The acceptance for charged particles and photons is
93% of the full 4$\pi$ solid angle. The momentum resolution for a charged
particle at 1 GeV/$c$ is 0.5%, and the ionization energy loss per unit path-
length ($dE/dx$) resolution is 6%. The EMC measures photon energies with a
resolution of 2.5% (5%) at 1 GeV in the barrel (end-caps). The time resolution
for the TOF is 80 ps in the barrel and 110 ps in the end-caps.

The GEANT-based simulation software BOOST MC1  is used to simulate the desired
Monte Carlo (MC) samples. An inclusive $J/\psi$ MC sample is used to estimate
the backgrounds. The production of the $J/\psi$ resonance is simulated by the
MC event generator KKMC KKMC1 ; KKMC2 , while the decays are generated by
BesEvtGen besevtgen1 ; besevtgen2 ; evtgen  for known decay modes with
branching fractions set at the PDG PDG  world average values, and by the Lund-
Charm model lundcharm  for the remaining unknown decays.

In this analysis, a signal MC sample for the process $J/\psi\to\eta Y(2175)$,
$Y(2175)\to\phi f_{0}(980)$ and $f_{0}(980)\to\pi^{+}\pi^{-}$, is generated to
optimize the selection criteria and determine the detection efficiency. Since
the $J^{PC}$ of the $Y(2175)$ is $1^{--}$, a $P$-wave orbital angular momentum
is used for the $\eta-Y(2175)$ system, while $S$-wave is used for the $\phi-
f_{0}(980)$ and $\pi^{+}-\pi^{-}$ systems. The shape of the $f_{0}(980)$ is
parameterized with the $\rm Flatt\acute{e}$ formula Flatte , and the
corresponding parameters are taken from the measurement of BESII phipipi . For
the signal MC sample of $J/\psi\to\phi\eta(1405)/f_{1}(1285)$, the angular
distributions are also considered in the simulation.

##  III Event selection

To select candidate events of the process $J/\psi\to\eta\phi\pi^{+}\pi^{-}$
with $\phi\to K^{+}K^{-}$ and $\eta\to\gamma\gamma$, the following criteria
are imposed on the data and MC samples.

We select charged tracks in the MDC within the polar angle range
$|\cos\theta|<0.93$ and require that the points of closest approach to the
beam line be within $\pm 20$ cm of the interaction point in the beam direction
and within $2$ cm in the plane perpendicular to the beam. The TOF and $dE/dx$
information are combined to form particle identification (PID) confidence
levels for the $\pi$, $K$, $p$ hypotheses, and each track is assigned to the
particle type corresponding to the hypothesis with the highest confidence
level. Two kaon and two pion particles with opposite charges are required.

Photon candidates are reconstructed by clustering signals in EMC crystals. The
energy deposited in nearby TOF counters is included to improve the photon
reconstruction efficiency and the photon energy resolution. At least two
photon candidates are selected, the minimum energy of which are required to be
$25$ MeV for barrel showers ($|\cos\theta|<0.80$) and $50$ MeV for endcap
showers ($0.86<|\cos\theta|<0.92$). To exclude showers due to the
bremsstrahlung of charged particles, the angle between the nearest charged
track and the shower must be greater than $10^{\circ}$. EMC cluster timing
requirements are applied to suppress electronic noise and energy deposits
unrelated to the event.

A four-constraint kinematic fit using energy-momentum conservation is
performed to the $J/\psi\to K^{+}K^{-}\pi^{+}\pi^{-}\gamma\gamma$ hypothesis.
All combinations of two photons are tried and the one with the smallest
$\chi^{2}_{4C}$ value is retained. To further suppress background,
$\chi^{2}_{4C}<200$ is required.

After the above selection process, a scatter plot of the invariant mass of the
$\gamma\gamma$ system ($M(\gamma\gamma)$) versus the invariant mass of the
$K^{+}K^{-}$ system ($M(K^{+}K^{-})$) in data is shown in Fig.1(a), where the
events concentrated in the region indicated by the dotted-line box correspond
to the $J/\psi\to\eta\phi\pi^{+}\pi^{-}$ signal. The $\phi$ and $\eta$ signal
regions are defined as $|M(K^{+}K^{-})-M_{\phi}|<0.013$ GeV$/c^{2}$ and
$|M(\gamma\gamma)-M_{\eta}|<0.019$ GeV$/c^{2}$, where $M_{\phi}$ and
$M_{\eta}$ are world average values of the $\phi$ and $\eta$ masses,
respectively. Fig.1(b) and (c) show the $\gamma\gamma$ and $K^{+}K^{-}$
invariant mass distributions for events with a $K^{+}K^{-}$ invariant mass
within the $\phi$ signal region and a $\gamma\gamma$ invariant mass within the
$\eta$ signal region, respectively. Both $\eta$ and $\phi$ signals are clearly
seen with very low background levels.

(a)(b)(c)

Figure 1:  (a) Scatter plot of $M(\gamma\gamma)$ versus $M(K^{+}K^{-})$. The
boxes with the dotted and solid lines show the $\eta$ and $\phi$ signal and
sidebands regions, respectively. (b) The $\gamma\gamma$ invariant mass
spectrum for events with the $K^{+}K^{-}$ invariant mass in the $\phi$ signal
region. (c) The $K^{+}K^{-}$ invariant mass spectrum for events with the
$\gamma\gamma$ invariant mass in the $\eta$ signal region. In plots (b) and
(c), the dotted arrows show the signal regions and the solid lines show the
sideband regions, which are described in the text.

##  IV Measurement of $J/\psi\to\eta Y(2175)$ with $Y(2175)\to\phi f_{0}(980)$
and $f_{0}(980)\to\pi^{+}\pi^{-}$

With the above requirements on the $\eta$ and $\phi$ candidate masses, the
$\pi^{+}\pi^{-}$ invariant mass distribution is shown in Fig. 2(a). A clear
$f_{0}(980)$ signal is visible. The non-$\phi$ and/or non-$\eta$ backgrounds
are estimated with the events in the $\eta-\phi$ sideband regions, shown as
the shaded histogram in Fig. 2(a). The $\eta$ sideband is defined by $0.480$
GeV$/c^{2}<M(\gamma\gamma)<0.499$ GeV$/c^{2}$ or $0.577$
GeV/$c^{2}<M(\gamma\gamma)<0.596$ GeV$/c^{2}$, and the $\phi$ sideband is
defined by $1.070$ GeV$/c^{2}<M(K^{+}K^{-})<1.096$ GeV$/c^{2}$. Using a mass
requirement of $0.90$ GeV$/c^{2}<M(\pi^{+}\pi^{-})<1.05$ GeV$/c^{2}$ to select
the $f_{0}(980)$ signal, the invariant mass distribution of $\phi f_{0}(980)$
is shown in Fig. 2(d), where a broad structure around $2.2$ GeV$/c^{2}$ is
evident. Figure 2(c) shows a two-dimensional histogram of
$M(\phi\pi^{+}\pi^{-})$ versus $M(\pi^{+}\pi^{-})$. A cluster of events
populating the $Y(2175)$ and $f_{0}(980)$ signal regions is observed, which
corresponds to the decay of $Y(2175)\to\phi f_{0}(980)$ with
$f_{0}(980)\to\pi^{+}\pi^{-}$.

Since the contribution from non-$\eta$ background events in the $f_{0}(980)$
mass region is small and can be neglected, the two-dimensional
$\phi$-$f_{0}(980)$ sidebands are used to estimate the background events in
this analysis. With the $\eta$ mass requirement applied, the non-$\phi$ and/or
non-$f_{0}(980)$ events are estimated by the weighted sums of horizontal and
vertical sidebands, with the entries in the diagonal side bands subtracted to
compensate for the double counting of background components. The definition of
the two-dimensional side bands is illustrated in Fig. 2(b). The weighting
factors for the events in the horizontal, vertical and the diagonal side bands
are measured to be 0, and 0.66, -0.085 respectively, which are determined from
the results of a two-dimensional fit to the mass spectrum of $M(K^{+}K^{-})$
versus $M(\pi^{+}\pi^{-})$. No signal of $f_{0}(980)$ is evident in non-$\phi$
processes as shown in the scatter plot of $M(\pi^{+}\pi^{-})$ versus
$M(K^{+}K^{-})$. Hence, the weighting factor for the events in the horizontal
side band is zero, and the non-$\phi$ events in the horizontal side band are
not used in the background estimation. The two-dimensional Probability Density
Functions (PDFs) for $J/\psi\to\eta\phi f_{0}(980)$, $\phi$ but
non-$f_{0}(980)$, non-$\phi$ and non-$f_{0}(980)$ processes are constructed by
the product of one-dimensional functions, where the resonant peaks are
parameterized by Breit-Wigner functions (for $\phi$) and a shape taken from
simulation (for $f_{0}(980)$), and the non-resonant parts are described by
polynomials with coefficients left free in the fit. To account for the
difference of the background shape between the signal region and side bands
due to the varying phase space, the obtained background mass distribution is
multiplied by a correction curve determined from an MC sample of 1 million
events of the phase space processes $J/\psi\to\eta\phi\pi^{+}\pi^{-}$. The
estimated $K^{+}K^{-}\pi^{+}\pi^{-}$ invariant mass distribution for the total
non-$\phi$ or non-$f_{0}(980)$ components is shown by the shaded histogram in
Fig. 2(d). No evident $Y(2175)$ signal is observed.

(a)(b)

(c)(d)

Figure 2:  (a) The $\pi^{+}\pi^{-}$ invariant mass spectrum. The shaded
histogram shows the non-$\eta$ background estimated with $\eta$ sideband
region; the dotted and solid arrows denote the $f_{0}(980)$ signal and
sideband regions, respectively. (b) The scatter plot of $M(\pi^{+}\pi^{-})$
versus $M(K^{+}K^{-})$. The solid box shows the signal region, and the dotted
boxes show the sideband regions of $\phi$ and $f_{0}$. (c) The scatter plot of
$M(\phi\pi^{+}\pi^{-})$ versus $M(\pi^{+}\pi^{-})$. (d) The
$\phi\pi^{+}\pi^{-}$ invariant mass distribution after imposing the
$f_{0}(980)$ signal mass window requirement. The shaded histogram shows the
background distribution estimated with the sideband method described in the
text.

To extract the yield of $Y(2175)$, an unbinned maximum likelihood fit to the
$\phi f_{0}(980)$ invariant mass is performed. The $Y(2175)$ signal, the
direct three-body decay of $J/\psi\to\eta\phi f_{0}(980)$, and the background
from the above estimation shown as the shaded histogram in Fig. 2(b) are
included in the fit. With the assumption of no interference between the
$Y(2175)$ signal and the direct three-body decay of $J/\psi\to\eta\phi
f_{0}(980)$, the probability density function (PDF) can be written as

$\epsilon(m)\times(G\otimes|A(m)|^{2})+A(J/\psi\to\eta\phi f_{0})+\rm BKG,$
(1)

where $A(m)=\frac{P_{J\to\eta Y}^{l_{1}}P_{Y\to\phi
f_{0}}^{l_{2}}}{m^{2}-M_{0}^{2}+iM_{0}\Gamma_{0}}$ is a Breit-Wigner function
representing the $Y(2175)$ signal shape, taking into account the phase space
factor of a two-body decay. $M_{0}$ and $\Gamma_{0}$ are left free in the fit.
$P_{J\to\eta Y}$ and $P_{Y\to\phi f_{0}}$ denote the momentum of the $\eta$ in
the rest frame of the $J/\psi$ and that of the $\phi$ in the rest frame of the
$Y(2175)$, respectively. $l_{1}$ and $l_{2}$, which label the relative orbital
angular momenta of the $\eta-Y(2175)$ and $\phi-f_{0}(980)$ systems, are set
to be $1$ and $0$ in the fit, respectively. $G$ is a Gaussian function
representing the mass resolution, and the corresponding parameters are taken
from MC simulation. $\epsilon(m)$, the detection efficiency as a function of
the $\phi f_{0}(980)$ invariant mass, is also obtained from MC simulation.
$A(J/\psi\to\eta\phi f_{0})$ represents the component of the direct decay of
$J/\psi\to\eta\phi f_{0}(980)$ with the shape derived from the phase space MC
sample. Finally, $\rm BKG$ refers to the background component estimated from
the two-dimensional weighted sideband method.

Figure 3 shows the results of the fit, where the circular dots with error bars
show the distribution for the signal and the triangular dots with error bars
are for the backgrounds estimated by the sidebands. The solid curve is the
overall fit projection, the dotted curve the fit for the backgrounds, and the
dashed curve for the sum of the direct decay of $J/\psi\to\eta\phi f_{0}$ and
backgrounds. The mass and width of the $Y(2175)$ are determined to be
$M=2200\pm 6$ MeV/c2 and $\Gamma=104\pm 15$ MeV, respectively. The fit yields
$471\pm 54$ $Y(2175)$ events with a statistical significance of greater than
10$\sigma$, which is determined by the change of the log-likelihood value and
the number of degree of freedom in the fit with and without the $Y(2175)$
signal. Taking into account the detection efficiency, $(9.10\pm 0.01)\%$,
obtained from MC simulation, the product branching fraction is

$\displaystyle\mathcal{B}($ $\displaystyle J/\psi\to\eta
Y(2175),Y(2175)\to\phi f_{0}(980),$ $\displaystyle
f_{0}(980)\to\pi^{+}\pi^{-})=(1.20\pm 0.14)\times 10^{-4}.$

Figure 3: Result of the fit to the $\phi f_{0}(980)$ invariant mass
distribution described in the text. The circular dots with error bars show the
distribution in the signal region; the triangular dots with error bars show
the backgrounds estimated using sideband regions; the solid curve shows the
overall fit projection; the dotted curve shows the fit for the backgrounds;
and the dashed curve is for the sum of the direct decay of $J/\psi\to\eta\phi
f_{0}$ and backgrounds.

We also perform a fit to the ${\phi f_{0}(980)}$ invariant mass, allowing
interference between the $Y(2175)$ and the direct decay $J/\psi\to\eta\phi
f_{0}(980)$. An ambiguity in the phase angle occurs when a resonance
interferes with a varying continuum bukin . Thus, two solutions with different
relative phase angles, corresponding to constructive and destructive
interferences, are found. The final fit and the individual contributions of
each of the components are shown in Fig. 4(a), (b) for constructive and
destructive interference, respectively. The mass, width, and yields of the
$Y(2175)$ signal, as well as the relative phase angle, are shown in Table 1.
The statistical significance of the interference is $2.5\sigma$, which is
determined from the differences of the likelihood values and the degrees of
freedom between the fits with and without interference. In this analysis, the
fit results without considering interference are taken as the nominal values.

(a)(b)

Figure 4: The fit projections to the $\phi f_{0}(980)$ invariant mass distribution showing the (a) constructive and (b) destructive solutions. The short-dashed line denotes the signal distribution; the dot-dashed curve shows the fit to the backgrounds estimated by the sidebands; the long-dashed line denotes the direct decay of $J/\psi\to\eta\phi f_{0}$; and the dotted line denotes the interference component.  Table 1: Two solutions of the fit to $M(\phi f_{0}(980))$, taking interference with the direct decay $\eta\phi f_{0}$ into account. Errors are statistical only.  Parameters | Constructive | Destructive
---|---|---
M (MeV$/c^{2}$) | $2171\pm 10$ | $2170\pm 9$
$\Gamma$ (MeV) | $128\pm 26$ | $126\pm 25$
Signal yields | $400\pm 167$ | $744\pm 40$
relative angle $\Phi$(rad) | $-0.51\pm 0.78$ | $0.60\pm 0.64$

##  V Measurement of $J/\psi\to\phi f_{1}(1285)$ and $\phi\eta(1405)$

The $\eta\pi^{+}\pi^{-}$ mass spectrum recoiling against the $\phi$ is shown
in Fig. 5. Besides the significant and well-known $f_{1}(1285)$ signal, a
small structure around 1.4 GeV$/c^{2}$, which is assumed to be the
$\eta(1405)$, is evident over a large non-resonant background. A fit to the
${\eta\pi^{+}\pi^{-}}$ invariant mass is performed with a PDF that includes
contributions from the $f_{1}(1285)$ and $\eta(1405)$ signals, the decay
$J/\psi\to\eta\phi\pi^{+}\pi^{-}$ (including the process $J/\psi\to\eta\phi
f_{0}(980)$), and backgrounds from non-$\eta$ and non-$\phi$ processes. In the
fit, the $f_{1}(1285)$ and $\eta(1405)$ signal shapes are described by Breit-
Wigner functions convoluted with Gaussian functions for their mass
resolutions. The mass and width of the $f_{1}(1285)$ signal are left free in
the fit, while those of the $\eta(1405)$ signal are fixed to the values in the
PDG PDG . The parameters of the Gaussian functions for the mass resolutions
are fixed to their MC values. The shape of the
$J/\psi\to\eta\phi\pi^{+}\pi^{-}$ decay is represented by a third-order
Chebychev polynomial function, and the corresponding parameters are allowed to
vary. The non-$\eta$ and non-$\phi$ background is estimated with the events in
the $\eta$-$\phi$ sideband regions, as shown by the dashed lines in Fig. 5,
and is fixed in the fit.

Figure 5:  Fit to the $\eta\pi^{+}\pi^{-}$ invariant mass spectrum. The solid
lines show the total fit and the $f_{1}(1285)$ and $\eta(1405)$ components;
the dashed line denotes the non-$\eta$ and non-$\phi$ background estimated
using the $\eta$-$\phi$ sidebands; the dotted curve represents the
$J/\psi\to\eta\phi\pi^{+}\pi^{-}$ component; the solid histogram indicates the
shape of the $X(1835)$ (with arbitrary normalization); and the dash-dotted
histogram shows the predicted shape of the $X(1870)$ signal (with arbitrary
normalization).

The fit, shown in Fig. 5, yields $1154\pm 56$ $f_{1}(1285)$ signal events,
with a mass of $1281.7\pm 0.6$ MeV$/c^{2}$ and a width of $21.0\pm 1.7$ MeV.
The mass and width are in good agreement with world average values PDG . Using
a detection efficiency of $(22.14\pm 0.09)\%$, obtained from MC simulation,
the product branching fraction is measured to be:

$\displaystyle\mathcal{B}($ $\displaystyle J/\psi\to\phi f_{1}(1285),$
$\displaystyle f_{1}(1285)\to\eta\pi^{+}\pi^{-})=(1.20\pm 0.06)\times
10^{-4},$

where the error is statistical only.

For the $\eta(1405)$ signal, the fit yields $172\pm 50$ events with a
statistical significance of $3.6\sigma$, evaluated from the difference of the
likelihood values between the fits with and without the $\eta(1405)$ included.
The product branching fraction is
$\mathcal{B}(J/\psi\to\phi\eta(1405),\eta(1405)\to\eta\pi^{+}\pi^{-})$=$(2.01\pm
0.58)\times 10^{-5}$, where the error is statistical only. To determine the
upper limit on the $\eta(1405)$ production rate, a series of similar fits with
given numbers of $\eta(1405)$ events are performed, and the likelihood values
of the fits as a function of the number of $\eta(1405)$ events are taken as a
normalized probability function. The upper limit on the number of signal
events at the $90\%$ C.L., $N^{U.L.}$, is defined as the value that contains
$90\%$ of the integral of the normalized probability function. The fit-related
uncertainties on $N^{U.L.}$ are estimated by using different sideband regions
for the effect of the non-$\eta$ and non-$\phi$ background, different orders
of Chebychev polynomials for the shape of the
$J/\psi\to\eta\phi\pi^{+}\pi^{-}$ and changing the mass and width values of
the $\eta(1405)$ within one standard deviation from the central values for the
signal shape. Finally, after taking into account fit-related uncertainties, we
obtain $N^{U.L.}$ = $345$. This upper limit and the detection efficiency of
$(19.75\pm 0.12)\%$, estimated from MC simulation, are used to evaluate the
upper limit on the branching fraction:

$\begin{split}&\mathcal{B}(J/\psi\to\phi\eta(1405),\eta(1405)\to\eta\pi^{+}\pi^{-}))\\\
&<\frac{N^{U.L.}}{\epsilon\times
N_{J/\psi}\times\mathcal{B}(\eta\to\gamma\gamma)\times\mathcal{B}(\phi\to
K^{+}K^{-})\times(1-\sigma_{\mathrm{sys}})}\\\ &=4.45\times 10^{-5},\\\
\end{split}$ (2)

where $\sigma_{\mathrm{sys}}$ is the systematic error to be discussed in
detail below. Since the background uncertainty is taken into account in the
calculation of $N^{U.L.}$ by choosing the maximum event yield from the
variations of the background functions, the systematic uncertainty from this
source is excluded here. The final results on the upper limit of the branching
fraction are shown in Table 3.

In the $\eta\pi^{+}\pi^{-}$ mass spectrum shown in Fig. 5, we do not observe
obvious structures around 1.84 GeV/c2 or at 1.87 GeV/c2. Using the same
approach as was used for the $\eta(1405)$, we set 90% C.L. upper limits for
the $X(1835)$ and $X(1870)$ production rates, where the signal shape of the
$X(1835)$ or $X(1870)$ is described by a Breit-Wigner function convoluted with
a Gaussian function for the mass resolution, and the background is modeled by
a third-order Chebychev polynomial. The resonant parameters of the $X(1835)$
and $X(1870)$ are fixed to the values of previous BESIII measurements
X1835_pipietap2 ; liuk . The results are summarized in Table 2 and Table 3.

Table 2:  Measurements of the number of events, statistical significances, and efficiencies.  Resonance | $N_{obs}$ | Significance | Efficiency($\%$)
---|---|---|---
$Y(2175)$ | $471\pm 54$ | $>10\sigma$ | $9.10\pm 0.01$
$f_{1}(1285)$ | $1154\pm 56$ | $-$ | $22.14\pm 0.09$
$\eta(1405)$ | $172\pm 50(<345)$ | $3.6\sigma$ | $19.75\pm 0.12$
$X(1835)$ | $394\pm 360(<1522)$ | $1.1\sigma$ | $13.85\pm 0.14$
$X(1870)$ | $25\pm 73(<330)$ | $0.8\sigma$ | $13.73\pm 0.14$
Table 3:  Measurements of the branching fractions for the decay modes. Upper limits are given at the $90\%$ C.L. Decay mode | Branching fraction $\mathcal{B}$
---|---
$J/\psi\to\eta Y(2175)$, $Y(2175)\to\phi f_{0}(980)$, $f_{0}(980)\to\pi^{+}\pi^{-}$ | $(1.20\pm 0.14\pm 0.37)\times 10^{-4}$
$J/\psi\to\phi f_{1}(1285)$, $f_{1}(1285)\to\eta\pi^{+}\pi^{-}$ | $(1.20\pm 0.06\pm 0.14)\times 10^{-4}$
$J/\psi\to\phi\eta(1405)$, $\eta(1405)\to\eta\pi^{+}\pi^{-}$ | $(2.01\pm 0.58\pm 0.82)(<4.45)\times 10^{-5}$
$J/\psi\to\phi X(1835)$, $X(1835)\to\eta\pi^{+}\pi^{-}$ | $<2.80\times 10^{-4}$
$J/\psi\to\phi X(1870)$, $X(1870)\to\eta\pi^{+}\pi^{-}$ | $<6.13\times 10^{-5}$

##  VI Systematic errors

The sources of systematic error include: the efficiency difference between
data and MC simulation for the track reconstruction, the PID, the photon
detection, and the kinematic fit; the fitting procedure; the ambiguity in the
interference; and the number of $J/\psi$ events. Their effects on the
measurement of the resonance parameters and the branching fractions are
discussed in detail below.

_a. MDC Tracking efficiency_ The tracking efficiency has been investigated
using the almost background-free control samples of
$J/\psi\to\pi^{+}\pi^{-}p\overline{p}$ and $J/\psi\to K_{S}^{0}K\pi$
trackerror . The difference in tracking efficiency between data and MC is
found to be 2% per charged kaon and pion. Therefore, 8% is taken as the total
systematic error for the detection efficiency of four charged tracks.

_b. PID efficiency_ To evaluate the PID efficiency uncertainty, we have
studied the kaon and pion PID efficiencies using the control samples of
$J/\psi\to K^{*\pm}K^{\mp}$ and $J/\psi\to\rho\pi$ trackerror , respectively.
The difference in PID efficiency between data and MC is 1% per kaon and pion.
Hence, 4% is taken as the total systematic error from the PID efficiency.

_c. Photon detection efficiency_ The photon detection efficiency has been
studied using a control sample of $J/\psi\to\rho\pi$ trackerror . The
difference between data and MC is found to be 1% per photon. Therefore, 2% is
taken as the total systematic error for the efficiency of the detection of the
two photons.

_d. Kinematic fit_ To estimate the uncertainty associated with the kinematic
fit, a control sample of
$J/\psi\to\phi\eta^{\prime}(\eta^{\prime}\rightarrow\eta\pi^{+}\pi^{-})$,
which has exactly the same final state as the signal, is first selected
without a kinematic fit. The kinematic fit efficiency is then evaluated from
the ratio of the $\eta^{\prime}$ yields with and without the kinematic fit
requirement, where the $\eta^{\prime}$ yield is extracted from the fit to the
$\eta^{\prime}$ signal in the $\eta\pi^{+}\pi^{-}$ invariant mass. The
difference of the kinematic fit efficiency between data and MC, $0.4\%$, is
taken as the systematic error for the kinematic fit.

_e. Uncertainties of $\mathcal{B}(\eta\rightarrow\gamma\gamma)$ and
$\mathcal{B}(\phi\rightarrow K^{+}K^{-})$ _ The branching fractions of
$\eta\rightarrow\gamma\gamma$ and $\phi\rightarrow K^{+}K^{-}$ are taken from
the PDG PDG . The uncertainties of these branching fractions, $0.5\%$ and
$1.0\%$, are taken as the systematic errors.

_f. Uncertainty of the number of $J/\psi$ events _ The total number of
$J/\psi$ events is determined from an analysis of inclusive $J/\psi$ hadronic
decays, and the uncertainty of the number of $J/\psi$ events, $1.2\%$ jpsiN ,
is taken as the systematic error from the number of $J/\psi$ events.

_g. Background uncertainty_ In the measurement of the resonance parameters and
branching fractions of the $Y(2175)$, a fit is performed to the $\phi
f_{0}(980)$ invariant mass spectrum. In the fit, the shape and amplitude of
the background from the non-$\phi$ and non-$f_{0}(980)$ are fixed to the
estimation from the $\phi-f_{0}(980)$ sideband regions. To estimate its impact
on the final results, we use different $\phi-f_{0}(980)$ sideband regions to
estimate the background, and follow the same fit procedure. The maximum
changes on both the $Y(2175)$ resonance parameters and its signal yield are
taken as the systematic errors. The uncertainty due to the background on the
mass and width of the $Y(2175)$ are $\pm 4.0$ MeV$/c^{2}$ and $\pm 14.0$ MeV,
respectively.

For the branching fraction of $J/\psi\to\phi f_{1}(1285)/\eta(1405)$ with
$f_{1}(1285)/\eta(1405)\to\eta\pi^{+}\pi^{-}$, the non-$\eta$ and non-$\phi$
backgrounds are estimated with the events in the $\eta-\phi$ sideband regions.
Analogous to the evaluation of the $Y(2175)$ errors, we define different
sideband regions to estimate the backgrounds and follow the same fit
procedure. The largest changes are taken as the uncertainty from the
background for these measurements. Compared to the number of $f_{1}(1285)$
events, the fluctuation of background shape under the $\eta(1405)$ peak has a
large impact on the signal yields in the fit due to the limited statistics.

_h. Impact from possible extra resonances_ In the invariant mass spectrum of
$\phi f_{0}(980)$, a small structure around 2.35 GeV/$c^{2}$ is found (Fig.
3). To evaluate its impact on the $Y(2175)$ measurement, we perform a fit with
an additional signal around $2.35$ GeV/$c^{2}$, which is described with a
Breit-Wigner function convoluted with a Gaussian function for the mass
resolution. The fit results show that the significance of the structure around
2.35 GeV/c2 is only 3.8$\sigma$. It is therefore not considered in the nominal
final results. However, the impact on the $Y(2175)$ measurement is taken as
the systematic error. The uncertainty due to the possible extra resonance on
the mass and width of the $Y(2175)$ are $\pm 3.0$ MeV$/c^{2}$ and $\pm 5.0$
MeV, respectively.

In the measurement of the branching fraction of $J/\psi\to\phi f_{1}(1285)$
with $f_{1}(1285)\to\eta\pi^{+}\pi^{-}$, we perform a fit without the
$\eta(1405)$ signal. The difference of results with or without the
$\eta(1405)$ signal included in the fit is taken as the systematic error on
the $f_{1}(1285)$ measurement from the impact of the $\eta(1405)$.

_i. Parameterization of the $f_{0}(980)$ _ The systematic error from the
$f_{0}(980)$ shape is estimated by comparing the detection efficiencies from
the signal MC samples simulated with different parameterizations of the
$f_{0}(980)$. We use the resonant parameters of the $f_{0}(980)$ from Ref.
otherpara , instead of the nominal values from the measurements of BESII
phipipi  mentioned in Section II, to describe the $f_{0}(980)$ shape. This
leads to a difference in the detection efficiency of $7.6\%$, and is taken as
the systematic uncertainty on the $Y(2175)$ branching fraction measurement
from the $f_{0}(980)$ parameterization.

_j. Uncertainty from fixed mass and width values on the branching ratio of
$J/\psi\to\phi\eta(1405)$ with $\eta(1405)\to\eta\pi^{+}\pi^{-}$ _ The mass
and width of the $\eta(1405)$ are fixed to their PDG values in the fit to the
$\eta(1405)$ signal. We change the mass and width values by one standard
deviation from their central values in the fitting procedure. The maximum
change on the branching fraction is determined to be $7.0\%$ when the mass and
width values are fixed at one negative standard deviation from the central
values.

_k. Uncertainty from parameter sets in the generation of $J/\psi\to\phi
f_{1}(1285)$ _ The parameters used in the generation of the signal MC sample
of $J/\psi\to\phi f_{1}(1285)$ are taken from the angular distribution of the
$\phi$ in the rest frame of the $J/\psi$ found in real data. The impact of the
uncertainty of these parameters on the efficiency, $3.2\%$, is taken as a
source of systematic error on the branching fraction.

Table 4: Summary of systematic errors (in %) for the branching fraction measurements. The fourth column shows the sources of systematic errors on the branching fraction of $J/\psi\to\phi\eta(1405)$ with $\eta(1405)\to\eta\pi^{+}\pi^{-}$, while the fifth column shows those on the upper limits of the branching fractions of $J/\psi\to\phi\eta(1405)$, $\phi X(1835)$, $\phi X(1870)$ with $\eta(1405)/X(1835)/X(1870)\to\eta\pi^{+}\pi^{-}$.  Sources |  $Y(2175)$ |  $f_{1}(1285)$ |  $\eta(1405)$ |  $\eta(1405)/X(1835)/X(1870)$
---|---|---|---|---
MDC tracking | $8.0$
Photon detection | $2.0$
PID | $4.0$
Kinematic fit | $0.4$
$\mathcal{B}(\eta\to\gamma\gamma)$ | $0.5$
$\mathcal{B}(\phi\to K^{+}K^{-})$ | $1.0$
Number of $J/\psi$ events | $1.2$
$f_{0}(980)$ selection | $7.6$ | $-$ | $-$ | $-$
Background uncertainty | $19.1$ | $4.1$ | $39.3$ | $-$
The fixed $M/\Gamma$ of $\eta(1405)$ | $-$ | $-$ | $7.0$ | $-$
Parameters of $\phi f_{1}(1285)$ generation | $-$ | $3.2$ | $-$ | $-$
Extra resonance | $21.4$ | $4.0$ | $-$ | $-$
Total | $31.1$ | $11.4$ | $41.0$ | $9.4$

In Table 4, a summary of all contributions to the systematic errors on the
branching fraction measurements is shown. In each case, the total systematic
uncertainty is obtained by adding the individual contributions in quadrature.
For the uncertainties on the $Y(2175)$ resonant parameters, we find that the
dominant systematic uncertainties are from the background shape and a possible
additional resonance around 2.35 GeV/$c^{2}$. Adding the various systematic
uncertainties in quadrature, the total systematic errors on the mass and width
of the $Y(2175)$ are $\pm 5.0$ MeV$/c^{2}$ and $\pm 14.8$ MeV, respectively.

##  VII Summary

In summary, we present an analysis of $J/\psi\to\eta\phi\pi^{+}\pi^{-}$ based
on $(225.3\pm 2.8)\times 10^{6}$ $J/\psi$ events collected with the BESIII
detector. The $Y(2175)$ resonance is observed in the invariant mass spectrum
of $\phi f_{0}(980)$ with a statistical significance of greater than
$10\sigma$. The mass and width of the $Y(2175)$ are measured and are in good
agreement with previous experimental results (Table 5). Neglecting the effects
of interference with the direct decay $J/\psi\to\eta\phi f_{0}(980)$, the
product branching fraction is measured to be $\mathcal{B}(J/\psi\to\eta
Y(2175)$, $Y(2175)\to\phi f_{0}(980)$, $f_{0}(980)\to\pi^{+}\pi^{-})=(1.20\pm
0.14\pm 0.37)\times 10^{-4}$. We also perform a fit taking the interference
between the $Y(2175)$ and the direct decay. The corresponding results are
shown in Table 1.

Table 5:  Comparison of $Y(2175)$ parameters as measured by different experiments. Collaboration | Process |  $M$ (MeV$/c^{2}$) |  $\Gamma$ (MeV)
---|---|---|---
BABAR aaaa_babar  | $e^{+}e^{-}\to\phi f_{0}~{}~{}(ISR)$ | $2175\pm 10\pm 15$ | $58\pm 16\pm 20$
BESII aaab_wanx  | $J/\psi\to\eta\phi f_{0}(980)$ | $2186\pm 10\pm 6$ | $65\pm 23\pm 17$
BELLE aaba_belle  | $e^{+}e^{-}\to\phi f_{0}~{}~{}(ISR)$ | $2079\pm 13^{+79}_{-28}$ | $192\pm 23^{+25}_{-61}$
BABAR(updated) babar_y2175  | $e^{+}e^{-}\to\phi f_{0}~{}~{}(ISR)$ | $2172\pm 10\pm 8$ | $96\pm 19\pm 12$
BESIII | $J/\psi\to\eta\phi f_{0}(980)$ | $2200\pm 6\pm 5$ | $104\pm 15\pm 15$

In addition, we investigate the $\eta\pi^{+}\pi^{-}$ mass spectrum recoiling
against the $\phi$ in the $J/\psi$ decay. A structure around 1.28 GeV$/c^{2}$
is clearly seen, and the fit results are in good agreement with the world
average values of the $f_{1}(1285)$ parameters. The product branching fraction
of $J/\psi\to\phi f_{1}(1285)$ with $f_{1}(1285)\to\eta\pi^{+}\pi^{-}$ is
measured to be $\mathcal{B}(J/\psi\to\phi
f_{1}(1285)\to\phi\eta\pi^{+}\pi^{-})=(1.20\pm 0.06\pm 0.14)\times 10^{-4}$. A
structure around $1.4$ GeV$/c^{2}$ seems to be present in the
$\eta\pi^{+}\pi^{-}$ mass spectrum. Assuming it to be the $\eta(1405)$, the
product branching fraction is calculated to be
$\mathcal{B}(J/\psi\to\phi\eta(1405)\to\phi\eta\pi^{+}\pi^{-})=(2.01\pm
0.58\pm 0.82)\times 10^{-5}$. We also present a $90\%$ C.L. upper limit on the
branching fraction
$\mathcal{B}(J/\psi\to\phi\eta(1405)$,$\eta(1405)\to\eta\pi^{+}\pi^{-})<4.45\times
10^{-5}$. In a previous experiment, the $\eta(1405)/\eta(1440)$ is observed in
both $\eta\pi\pi$ and $K\overline{K}\pi$ invariant mass spectra recoiling
against the $\gamma$ and $\omega$ in $J/\psi$ decays. However, no significant
structure around 1.4 GeV$/c^{2}$ is observed in the $\pi^{+}\pi^{-}\eta$ mass
spectrum recoiling against the $\phi$ in this analysis, which may imply that
$u$ and $d$ quarks account for more of the quark content in the $\eta(1405)$
than the $s$ quark. We also perform searches for the $X(1835)$ and $X(1870)$
in the vicinity of 1.8 GeV$/c^{2}$ in the $\eta\pi^{+}\pi^{-}$ mass spectrum,
and observe no evident structures. The corresponding upper limits at $90\%$
C.L. of branching fraction are measured. All of these measurements provide
information in understanding the nature of the $X(1835)$ and $X(1870)$.

## Acknowledgements

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Basic Research Program of China under Contract No. 2015CB856700; Joint
Funds of the National Natural Science Foundation of China under Contracts Nos.
11079008, 11179007, U1232201, U1332201; National Natural Science Foundation of
China (NSFC) under Contracts Nos. 10935007, 11121092, 11125525, 11235011,
11322544, 11335008, 11175189; the Chinese Academy of Sciences (CAS) Large-
Scale Scientific Facility Program; CAS under Contracts Nos. KJCX2-YW-N29,
KJCX2-YW-N45; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory
for Particle Physics and Cosmology; German Research Foundation DFG under
Contract No. Collaborative Research Center CRC-1044; Istituto Nazionale di
Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; Russian Foundation for Basic Research under Contract No.
14-07-91152; U.S. Department of Energy under Contracts Nos. DE-FG02-04ER41291,
DE-FG02-05ER41374, DE-FG02-94ER40823, DESC0010118; U.S. National Science
Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research
Foundation of Korea under Contract No. R32-2008-000-10155-0.

## References

  * (1) K. A. Olive _et al._ [Particle Data Group], Chin. Phys. C 38, 090001 (2014).
  * (2) B. Aubert _et al._ [BABAR Collaboration], Phys. Rev. D 74, 091103(R) (2006).
  * (3) M. Ablikim _et al._ [BES Collaboration], Phys. Rev. Lett. 100, 102003 (2008).
  * (4) C. P. Shen _et al._ [BELLE Collaboration], Phys. Rev. D 80, 031101(R) (2009).
  * (5) B. Aubert _et al._ [BABAR Collaboration], Phys. Rev. D 86, 012008 (2012).
  * (6) B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 95, 142001 (2005).
  * (7) C. Z. Yuan et al. [BELLE Collaboration], Phys. Rev. Lett. 99, 182004 (2007).
  * (8) G. J. Ding and M. L. Yan, Phys. Lett. B 650, 390 (2007).
  * (9) G. J. Ding and M. L. Yan, Phys. Lett. B 657, 49 (2007).
  * (10) Z. G. Wang, Nucl. Phys. A 791, 106 (2007).
  * (11) E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007).
  * (12) C. F. Qiao, Phys. Lett. B 639, 263 (2006).
  * (13) A. Martinez Torres, K. P. Khemchandani, L. S. Geng, M. Napsuciale and E. Oset, Phys. Rev. D 78, 074031 (2008).
  * (14) S. L. Zhu, Int. J. Mod. Phys. E 17, 283 (2008).
  * (15) T. Barnes, N. Black and P. R. Page, Phys. Rev. D 68, 054014 (2003).
  * (16) W. M. Yao et al., J. Phys. G 33, 1 (2006).
  * (17) J. J. Manak _et al._ , Phys. Rev. D 62, 012003 (2000).
  * (18) D. L. Scharre _et al._ , Phys. Lett. B 97, 329 (1980).
  * (19) H. J. Behrend _et al._ [CELLO Collaboration], Z. Phys. C 42, 367 (1989).
  * (20) I. Vodopianov _et al._ [L3 Collaboration], Acta Phys. Pol. B 31, 2453 (2000).
  * (21) M. Ablikim _et al._ [BES Collaboration], Phys. Rev. Lett. 95, 262001 (2005).
  * (22) M. Ablikim _et al._ [BESIII Collaboration], Phys. Rev. Lett. 106, 072002 (2011).
  * (23) M. Ablikim _et al._ [BESIII Collaboration], Phys. Rev. Lett. 107, 182001 (2011).
  * (24) C. S. Gao and S. L. Zhu, Commun. Theor. Phys. 42, 844 (2004).
  * (25) G. J. Ding and M. L. Yan, Phys. Rev. C 72, 015208 (2005).
  * (26) G. Hao, C. F. Qiao and A. L. Zhang, Phys. Lett. B 642, 53 (2006).
  * (27) B. A. Li, Phys. Rev. D 74, 034019 (2006).
  * (28) N. Kochelev and D. P. Min, Phys. Lett. B 633, 283 (2006).
  * (29) B. Kerbikov, A. Stavinsky and V. Fedotov, Phys. Rev. C 69, 055205 (2004).
  * (30) D. V. Bugg, Phys. Lett. B 598, 8 (2004).
  * (31) B. S. Zou and H. C. Chiang, Phys. Rev. D 69, 034004 (2004).
  * (32) M. Ablikim _et al._ [BESIII Collaboration], Nucl. Instrum. Meth. A 614, 345 (2010).
  * (33) Z. Y. Deng _et al._ , HEP&NP. 30, 371 (2006).
  * (34) S. Jadach, B. F. L. Ward and Z. Was, Comp. Phys. Commu. 130, 260 (2000).
  * (35) S. Jadach, B. F. L. Ward and Z. Was, Phys. Rev. D 63, 113009 (2001).
  * (36) K. T. Chao _et al._ , Modern Physics A 24, No.1 supp. (2009).
  * (37) R. G. Ping, Chin. Phys. C 32, 599 (2008).
  * (38) D. J. Lange, Nucl. Instrum. Meth. A 462, 152 (2001).
  * (39) J. C. Chen _et al._ , Phys. Rev. D 62, 034003 (2000).
  * (40) S. M. Flatté, Phys. Lett. B 63, 224 (1976).
  * (41) M. Ablikim _et al._ [BES Collaboration], Phys. Lett. B 607, 243 (2005).
  * (42) A. D. Bukin, arXiv:0710.5627.
  * (43) M. Ablikim _et al._ [BESIII Collaboration], Phys. Rev. D 83, 112005 (2011).
  * (44) M. Ablikim _et al._ [BESIII Collaboration], Chin. Phys. C 36 (10), 915 (2012).
  * (45) B. S. Zou and D.V. Bugg, Phys. Rev. D 48, R3948 (1993).