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The white lines represent the cuts in the corresponding phases. The endpoints of the white lines correspond to the superconductor and superfluid vortices, respectively. The plot on the right shows the positions of the superconductor and the superfluid vortices. Notice the difference in the coordinate scales of the plotted regions in Fig.~\\ref{fig:absolute} and Fig.~\\ref{fig:phases}  (Ref.~[29]).","source":"arxiv","key":"eea94578d6e84c30a6d1e673ada3c754","url":"https://inspirehep.net/files/eea94578d6e84c30a6d1e673ada3c754"},{"filename":"rho0_arg_cuts_vortices.png","caption":"The same as in Fig.~\\ref{fig:absolute} but for the phases for the charged ($\\arg \\rho$) and neutral ($\\arg \\rho^{(0)}$) condensates, respectively. The white lines represent the cuts in the corresponding phases. The endpoints of the white lines correspond to the superconductor and superfluid vortices, respectively. The plot on the right shows the positions of the superconductor and the superfluid vortices. Notice the difference in the coordinate scales of the plotted regions in Fig.~\\ref{fig:absolute} and Fig.~\\ref{fig:phases}  (Ref.~[29]).","source":"arxiv","key":"867d7ac9b1296e4ec8be068c53e9f8eb","url":"https://inspirehep.net/files/867d7ac9b1296e4ec8be068c53e9f8eb"},{"filename":"3d_vortices_a.png","caption":"A $3d$ visualization of the ground state. (left) The honeycomblike structure of charged and neutral condensates. (right) The vortex lattice superimposed on the condensates. The magnetic field is directed out of the plane of the drawing.  The charged and neutral condensates approach their maxima in the violet and green regions, respectively.  The superconductor  and superfluid vortices are shown by, respectively, the large red and small blue arrows directed out of the plane. 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Generalizations to the non-extremal case are not analyzed explicitly, but can be pursued as indicated.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["hep-th"],"titles":[{"title":"p-branes on the waves"},{"source":"arXiv","title":"p-branes on the waves"}],"curated":true}},{"created":"2010-11-28T00:00:00+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/879303?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/879303?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/879303?format=latex-us","json":"https://inspirehep.net/api/literature/879303?format=json","json-expanded":"https://inspirehep.net/api/literature/879303?format=json-expanded","cv":"https://inspirehep.net/api/literature/879303?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A879303"},"id":"879303","updated":"2023-03-06T14:12:02.499232+00:00","metadata":{"citation_count":4,"publication_info":[{"journal_volume":"ICHEP2010","artid":"340","year":2010,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1213080"},"page_start":"340","journal_title":"PoS"},{"conference_record":{"$ref":"https://inspirehep.net/api/conferences/980692"},"parent_record":{"$ref":"https://inspirehep.net/api/literature/901471"},"cnum":"C10-07-21"}],"documents":[{"filename":"ICHEP 2010_340.pdf","attachment":{"content":"P\no\nS\n(\nI\nC\nH\nE\nP\n \n2\n0\n1\n0\n)\n3\n4\n0\n\nPossible splitting of deconfinement and chiral\ntransitions in strong magnetic fields in QCD\n\nEduardo S. Fraga∗\n\nInstituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro,\nRJ 21941-972, Brazil\nE-mail: fraga@if.ufrj.br\n\nAna Júlia Mizher\nCentro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ,\nBrazil\nE-mail: anajulia@cbpf.br\n\nM. N. Chernodub†\n\nCNRS, Laboratoire de Mathématiques et Physique Théorique, Université François-Rabelais,\nFédération Denis Poisson - CNRS, Parc de Grandmont, Université de Tours, 37200 France\nDepartment of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent,\nBelgium\nE-mail: maxim.chernodub@lmpt.univ-tours.fr\n\nWe show that finite-temperature deconfinement and chiral transitions can split in a strong enough\nmagnetic field. The splitting in critical temperatures of these transitions in a constant magnetic\nfield of a typical LHC magnitude is of the order of 10 MeV. A new deconfined phase with broken\nchiral symmetry appears.\n\n35th International Conference of High Energy Physics - ICHEP2010,\nJuly 22-28, 2010\nParis France\n\n∗Speaker.\n†On leave from ITEP, Moscow, Russia.\n\nc© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/\n\nmailto:fraga@if.ufrj.br\nmailto:anajulia@cbpf.br\nmailto:maxim.chernodub@lmpt.univ-tours.fr\n\n\nP\no\nS\n(\nI\nC\nH\nE\nP\n \n2\n0\n1\n0\n)\n3\n4\n0\n\nPossible splitting of deconfinement and chiral transitions Eduardo S. Fraga\n\nThe thermal quark-hadron transition can be dramatically modified in the presence of a strong\nmagnetic field [1, 2, 3, 4, 5, 6]. We work in the two-flavor linear sigma model coupled to quarks\nand to the Polyakov loop (PLSMq) in the presence of an external magnetic field [5]. The confining\nproperties of QCD are described by the complex-valued Polyakov loop variable L. The expectation\nvalue of the Polyakov loop L is an exact order parameter of the color confinement in the limit of\ninfinitely massive quarks:\n\nConfinement :\n\n{\n〈L〉 = 0 , low T\n〈L〉 6= 0 , high T\n\n, L(x) =\n1\n3\n\nTrP exp\n[\ni\n\n1/T∫\n0\n\ndτ A4(~x,τ)\n]\n, (1)\n\nwhere A4 = iA0 is the matrix-valued temporal component of the Euclidean gauge field Aµ and the\nsymbol P denotes path ordering. The integration takes place over compactified imaginary time τ .\n\nThe chiral features of the model are encoded in the dynamics of the O(4) chiral field, the\nsinglet component of which is an exact order parameter in the chiral limit (i.e., when quarks and\npions are massless degrees of freedom):\n\nChiral symmetry :\n\n{\n〈σ〉 6= 0 , low T\n〈σ〉 = 0 , high T\n\n,\nφ = (σ ,~π) ,\n~π = (π+,π0,π−) .\n\n(2)\n\nHere ~π is the isotriplet of the pseudoscalar pion fields and σ is the chiral scalar field which plays\nthe role of an approximate order parameter of the chiral transition in QCD.\n\nThe quark field ψ provides interaction between the Polyakov loop L and the chiral field φ ,\nmaking a bridge between confining and chiral properties. Quarks are also coupled to the external\nmagnetic field since the u and d quarks are electrically charged. Thus, it is clear that the external\nmagnetic field will affect the chiral dynamics and the confining properties of the model.\n\nThe Lagrangian of PLSMq describes the constituent quarks ψ , which interact with the meson\nfields σ , π± = (π1± iπ2)/\n\n√\n2 and π0 = π3, the Abelian gauge field aµ = (a0,~a) = (0,−By,0,0),\n\nand the SU(3) gauge field Aµ (related to the Polyakov loop L, see Eq. (1)) via the covariant deriva-\ntive /D = γµ(∂µ − iQaµ − iAµ) with the charge matrix Q = diag(+2e/3,−e/3):\n\nL = ψ [i /D−g(σ + iγ5~τ ·~π)]ψ +\n1\n2\n[\n(∂µσ)2 +(∂µπ\n\n0)2]+ |D(π)\nµ |2−Vφ (σ ,~π)−VL(L,T ) , (3)\n\nwhere D(π)\nµ = ∂µ + ieaµ is the covariant derivative acting on colorless pions. The chiral potential is\n\nVφ (σ ,~π) =\nλ\n\n4\n(σ2 +~π2− v2)2−hσ , h = fπm2\n\nπ , v2 = f 2\nπ −m2\n\nπ/λ , λ = 20 , (4)\n\nwhere fπ ≈ 93MeV and mπ ≈ 138MeV. The constituent quark mass is given by mq ≡ mq(〈σ〉) =\ng〈σ〉, and, choosing g = 3.3 at T = 0, one obtains for the constituent quarks in the vacuum mq ≈\n310 MeV. At low temperatures quarks are not excited, and the model reproduces results from the\nusual linear σ -model without quarks.\n\nThe Polyakov potential is given by [7]\n\nVL(L,T )\nT 4 =−L∗L\n\n2\n\n2\n\n∑\nl=0\n\nal\n\n(\nT0\n\nT\n\n)l\n\n+b3\n\n(\nT0\n\nT\n\n)3\n\nln\n[\n1−6L∗L+4\n\n(\nL∗3 +L3\n\n)\n−3(L∗L)2\n\n]\n, (5)\n\n2\n\n\n\nP\no\nS\n(\nI\nC\nH\nE\nP\n \n2\n0\n1\n0\n)\n3\n4\n0\n\nPossible splitting of deconfinement and chiral transitions Eduardo S. Fraga\n\nT = 0.8Tc, eB = 0 T = 0.8Tc, eB = 9T 2\n\nT = 1.2Tc, eB = 0 T = 1.2Tc, eB = 9T 2\n\nFigure 1: Effects of temperature and magnetic field on quark confinement: The Polyakov loop potential at\nT = 0.8T0 (top) and T = 1.2T0 (bottom) and at zero magnetic field (left) and at eB = 9T 2 (right).\n\nwhere T0≡TSU(3) = 270MeV is the critical temperature in the pure gauge case and a0 = 16π2/45≈\n3.51, a1 =−2.47, a2 = 15.2, and b3 =−1.75. Below we follow a mean-field analysis in which the\nmesonic sector is treated classically whereas quarks represent fast degrees of freedom.\n\nIn the confining sector, the strong magnetic field affects the potential for the expectation value\nof the Polyakov loop via the intermediation of the quarks in three ways [5]:\n\n(i) the presence of the magnetic field intensifies the breaking of the global Z3 symmetry and\nmakes the Polyakov loop real-valued (see Fig. 1);\n\n(ii) the thermal contribution from quarks tends to destroy the confinement phase by increasing\nthe expectation value of the Polyakov loop;\n\n(iii) on the contrary, the vacuum quark contribution tends to restore the confining phase by low-\nering the expectation value of the Polyakov loop.\n\nThe vacuum correction from quarks has a crucial impact on the phase structure. If one ignores\nthe vacuum contribution from the quarks, then one finds that the confinement and chiral phase\ntransition lines coincide, and in this case the increasing magnetic field lowers the common chiral-\nconfinement transition temperature [5]. However, if one includes the vacuum contribution, then the\nconfinement and chiral transition lines split, and both chiral and deconfining critical temperatures\nbecome increasing functions of the magnetic field, Fig. 2. The vacuum contribution from the quarks\naffect drastically the chiral sector as well. Our calculations also show that the vacuum contribution\nseems to soften the order of the phase transition: the first-order phase transition – which would be\nrealized in the absence of the vacuum contribution – becomes a smooth crossover in the system\nwith vacuum quark loops included.\n\n3\n\n\n\nP\no\nS\n(\nI\nC\nH\nE\nP\n \n2\n0\n1\n0\n)\n3\n4\n0\n\nPossible splitting of deconfinement and chiral transitions Eduardo S. Fraga\n\nno vacuum corrections with vacuum corrections\n\n0 10 20 30 40 50\n\neB(m\nπ\n\n2\n)\n\n100\n\n125\n\n150\n\n175\n\n200\n\nT\nc(M\n\neV\n) \n\nChiral transition\nDeconfinement transition\n\nWithout vacuum corrections\n\n0 10 20 30 40 50\n\neB(m\nπ\n\n2\n)\n\n200\n\n250\n\n300\n\nT\n (\n\nM\neV\n\n)\n\nChiral transition\nDeconfinement transition\nExpected magnetic field generated in the LHC\n\nWith vacuum corrections\n\nDeconfinement transition\n\nLHC\nfield\n\nChiral\ntransition   \n\nFigure 2: Phase diagram in the B-T plane. (left) Without vacuum corrections: the critical temperatures of\nthe deconfinement (the dash-dotted line) and chiral (the dashed line) transitions coincide all the way, and\ndecrease with B. (right) With vacuum corrections: the critical temperatures of the deconfinement (the dash-\ndotted line) and chiral (the dashed line) coincide at B = 0 and split at higher values of the magnetic field. A\ndeconfined phase with broken chiral symmetry appears. The vertical line represents a typical magnitude of\nthe magnetic field that expected to be realized at LHC heavy-ion collisions [8].\n\nEither scenario is very exciting, and brings new possibilities for the phase diagram of strong\ninteractions: a first-order order transition, the splitting of coexistence lines, new phases, the explicit\nbreaking of the center Z3 symmetry by the magnetic field, and so on.\n\nAn independent investigation in the Nambu-Jona-Lasinio model agrees with the second sce-\nnario [6], which was expected since by construction this model includes quark degrees of freedom\nin the vacuum. Preliminary numerical simulations of lattice QCD [9] seems to favor the second\nscenario as well indicating that the critical temperatures of both transitions are rising with strength\nof the background magnetic field.\n\nAcknowledgments\n\nThis work was partially supported by CAPES-COFECUB, CNPq, FAPERJ, and FUJB/UFRJ.\n\nReferences\n\n[1] N. O. Agasian and S. M. Fedorov, Phys. Lett. B 663, 445 (2008).\n\n[2] E. S. Fraga and A. J. Mizher, Phys. Rev. D 78, 025016 (2008).\n\n[3] J. K. Boomsma and D. Boer, Phys. Rev. D 81, 074005 (2010).\n\n[4] K. Fukushima, M. Ruggieri and R. Gatto, Phys. Rev. D 81, 114031 (2010).\n\n[5] A. J. Mizher, M. N. Chernodub and E. S. Fraga, Phys. Rev. D 82, 105016 (2010).\n\n[6] R. Gatto and M. Ruggieri, Phys. Rev. D 82, 054027 (2010). [arXiv:1007.0790 [hep-ph]].\n\n[7] S. Roessner, T. Hell, C. Ratti and W. Weise, Nucl. Phys. A 814, 118 (2008).\n\n[8] V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925 (2009).\n\n[9] M. D’Elia, S. Mukherjee and F. Sanfilippo, Phys. Rev. D 82, 051501 (2010).\n\n4"},"fulltext":true,"source":"PoS","key":"6b46fd67f91c3621405cca3ea6ca89e0","url":"https://inspirehep.net/files/6b46fd67f91c3621405cca3ea6ca89e0"}],"authors":[{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/03490as77"}],"full_name_unicode_normalized":"fraga, eduardo s.","full_name":"Fraga, Eduardo S.","curated_relation":true,"record":{"$ref":"https://inspirehep.net/api/authors/1009599"},"last_name":"Fraga","ids":[{"schema":"ORCID","value":"0000-0001-5340-156X"},{"schema":"INSPIRE BAI","value":"E.S.Fraga.1"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903160"},"value":"Rio de Janeiro Federal U."}],"signature_block":"FRAGe","first_name":"Eduardo S.","uuid":"bd41f6bd-eece-4f30-b14a-76636c4e8cfb","recid":1009599},{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/02wnmk332"}],"full_name_unicode_normalized":"mizher, ana julia","full_name":"Mizher, Ana Julia","record":{"$ref":"https://inspirehep.net/api/authors/1032383"},"last_name":"Mizher","ids":[{"schema":"INSPIRE BAI","value":"A.J.Mizher.1"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903158"},"value":"Rio de Janeiro, CBPF"}],"signature_block":"MASARa","first_name":"Ana Julia","uuid":"34196b56-bd4c-4a22-a6db-1b373dbf4038","recid":1032383},{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/02wwzvj46"},{"schema":"ROR","value":"https://ror.org/00cv9y106"}],"full_name_unicode_normalized":"chernodub, m.n.","full_name":"Chernodub, M.N.","curated_relation":true,"record":{"$ref":"https://inspirehep.net/api/authors/1013801"},"last_name":"Chernodub","ids":[{"schema":"ORCID","value":"0000-0003-2101-4914"},{"schema":"INSPIRE BAI","value":"Maxim.N.Chernodub.1"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/909619"},"value":"Tours U., CNRS"},{"record":{"$ref":"https://inspirehep.net/api/institutions/902815"},"value":"Gent U."}],"signature_block":"CARNADABm","first_name":"M.N.","uuid":"6294a3be-a40a-44bb-ac6e-82fd96b06e59","recid":1013801}],"citation_count_without_self_citations":4,"citeable":true,"$schema":"https://inspirehep.net/schemas/records/hep.json","keywords":[{"schema":"INSPIRE","value":"transition: chiral"},{"schema":"INSPIRE","value":"magnetic field: constant"},{"schema":"INSPIRE","value":"symmetry: chiral"},{"schema":"INSPIRE","value":"deconfinement"},{"schema":"INSPIRE","value":"splitting"},{"schema":"INSPIRE","value":"quantum chromodynamics"},{"schema":"INSPIRE","value":"finite temperature"},{"schema":"INSPIRE","value":"critical phenomena"},{"schema":"INSPIRE","value":"CERN LHC Coll"}],"references":[{"reference":{"arxiv_eprint":"0803.3156","publication_info":{"journal_volume":"663","artid":"445","page_start":"445","journal_title":"Phys.Lett.B"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/781834"}},{"reference":{"arxiv_eprint":"0804.1452","publication_info":{"journal_volume":"78","artid":"025016","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/783096"}},{"reference":{"arxiv_eprint":"0911.2164","publication_info":{"journal_volume":"81","artid":"074005","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/836661"}},{"reference":{"arxiv_eprint":"1003.0047","publication_info":{"journal_volume":"81","artid":"114031","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/847179"}},{"reference":{"arxiv_eprint":"1004.2712","publication_info":{"journal_volume":"82","artid":"105016","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/852226"}},{"reference":{"arxiv_eprint":"1007.0790","publication_info":{"journal_volume":"82","artid":"054027","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/860719"}},{"reference":{"arxiv_eprint":"0712.3152","publication_info":{"journal_volume":"814","artid":"118","page_start":"118","journal_title":"Nucl.Phys.A"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/771085"}},{"reference":{"arxiv_eprint":"0907.1396","publication_info":{"journal_volume":"24","artid":"5925","page_start":"5925","journal_title":"Int.J.Mod.Phys.A"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/825229"}},{"reference":{"arxiv_eprint":"1005.5365","publication_info":{"journal_volume":"82","artid":"051501","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/856516"}}],"number_of_pages":4,"legacy_creation_date":"2010-11-28","preprint_date":"2010-11","author_count":3,"public_notes":[{"source":"arXiv","value":"4 pages, 6 figures/ talk given by E. 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(left) Without vacuum corrections: the critical temperatures of the deconfinement (the dash-dotted line) and chiral (the dashed line) transitions coincide all the way, and decrease with $B$. (right) With vacuum corrections: the critical temperatures of the deconfinement (the dash-dotted line) and chiral (the dashed line) coincide at $B=0$ and split at higher values of the magnetic field. A deconfined phase with broken chiral symmetry appears. The vertical line represents a typical magnitude of the magnetic field that expected to be realized at LHC heavy-ion collisions~\\cite{Skokov:2009qp}.","source":"arxiv","key":"a0b350ffa6ce2f5ecd562faa8c3930d2","url":"https://inspirehep.net/files/a0b350ffa6ce2f5ecd562faa8c3930d2"},{"filename":"vac_phase_diagram.png","caption":"Phase diagram in the $B$-$T$ plane. (left) Without vacuum corrections: the critical temperatures of the deconfinement (the dash-dotted line) and chiral (the dashed line) transitions coincide all the way, and decrease with $B$. (right) With vacuum corrections: the critical temperatures of the deconfinement (the dash-dotted line) and chiral (the dashed line) coincide at $B=0$ and split at higher values of the magnetic field. A deconfined phase with broken chiral symmetry appears. The vertical line represents a typical magnitude of the magnetic field that expected to be realized at LHC heavy-ion collisions~\\cite{Skokov:2009qp}.","source":"arxiv","key":"10cf11fb06c50a2139780da7d0bccb84","url":"https://inspirehep.net/files/10cf11fb06c50a2139780da7d0bccb84"}],"legacy_version":"20180514104207.0","inspire_categories":[{"term":"Lattice"},{"term":"Phenomenology-HEP"},{"term":"Theory-HEP"}],"urls":[{"description":"Proceedings of Science Server","value":"http://pos.sissa.it//archive/conferences/120/340/ICHEP%202010_340.pdf"}],"first_author":{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/03490as77"}],"full_name":"Fraga, Eduardo S.","last_name":"Fraga","ids":[{"schema":"ORCID","value":"0000-0001-5340-156X"},{"schema":"INSPIRE BAI","value":"E.S.Fraga.1"}],"first_name":"Eduardo S.","recid":1009599},"control_number":879303,"dois":[{"value":"10.22323/1.120.0340"}],"document_type":["conference paper"],"texkeys":["Fraga:2010qb"],"abstracts":[{"source":"arXiv","value":"We show that finite-temperature deconfinement and chiral transitions can split in a strong enough magnetic field. The splitting in critical temperatures of these transitions in a constant magnetic field of a typical LHC magnitude is of the order of 10 MeV. A new deconfined phase with broken chiral symmetry appears.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["hep-ph"],"titles":[{"title":"Possible splitting of deconfinement and chiral transitions in strong magnetic fields in QCD"},{"source":"arXiv","title":"Possible splitting of deconfinement and chiral transitions in strong magnetic fields in QCD"}],"curated":true}},{"created":"2010-11-16T00:00:00+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/877492?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/877492?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/877492?format=latex-us","json":"https://inspirehep.net/api/literature/877492?format=json","json-expanded":"https://inspirehep.net/api/literature/877492?format=json-expanded","cv":"https://inspirehep.net/api/literature/877492?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A877492"},"id":"877492","updated":"2023-03-06T16:19:37.968229+00:00","metadata":{"citation_count":1,"publication_info":[{"journal_volume":"LC2010","artid":"071","year":2010,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1213080"},"page_start":"071","journal_title":"PoS"},{"conference_record":{"$ref":"https://inspirehep.net/api/conferences/980632"},"parent_record":{"$ref":"https://inspirehep.net/api/literature/893419"},"cnum":"C10-06-14.2"}],"documents":[{"filename":"LC2010_071.pdf","attachment":{"content":"P\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate\n\nDavid Vercauteren∗1\n1Department of Physics and Astronomy, Ghent University\nE-mail: David.Vercauteren@UGent.be\n\nDavid Dudal1\n\nE-mail: David.Dudal@UGent.be\n\nJohn Gracey2\n\n2Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool\nE-mail: gracey@liv.ac.uk\n\nNele Vandersickel1\n\nE-mail: Nele.Vandersickel@UGent.be\n\nHenri Verschelde1\n\nE-mail: Henri.Verschelde@UGent.be\n\nI present recent analytical work concerning the electric-magnetic asymmetry in the dimension\ntwo condensate 〈A2\n\nµ〉 in pure Yang–Mills theory. We reproduce qualitatively the lattice results\npreviously found by Chernodub and Ilgenfritz.\n\nLight Cone 2010 - LC2010\nJune 14-18, 2010\nValencia, Spain\n\n∗Speaker.\n\nc© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/\n\nmailto:David.Vercauteren@UGent.be\nmailto:David.Dudal@UGent.be\nmailto:gracey@liv.ac.uk\nmailto:Nele.Vandersickel@UGent.be\nmailto:Henri.Verschelde@UGent.be\n\n\nP\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate David Vercauteren\n\n1. Introduction\n\nRecent years have witnessed a great deal of interest in the possible existence of mass dimension\ntwo condensates in gauge theories, see for example [1, 2, 3, 4, 6, 7, 5, 8, 9, 10, 11, 13, 14, 12] and\nreferences therein for approaches based on phenomenology, operator product expansion, lattice\nsimulations, an effective potential and the string perspective. There is special interest in the operator\n\nA2\nmin = min\n\nU∈SU(N)\nV −1\n\n∫\nd4x(AU\n\nµ )2 , (1.1)\n\nwhich is gauge invariant due to the minimization along the gauge orbit. It should be mentioned that\nobtaining the global minimum is delicate due to the problem of gauge (Gribov) ambiguities [15].\nAs is well known, local gauge invariant dimension two operators do not exist in Yang-Mills gauge\ntheories. The nonlocality of (1.1) is best seen when it is expressed as [16]1\n\nA2\nmin =\n\n∫\nd4x\n\n[\nAa\n\nµ\n\n(\nδµν −\n\n∂µ∂ν\n\n∂ 2\n\n)\nAa\n\nν −g f abc\n(\n\n∂ν\n\n∂ 2 ∂Aa\n)(\n\n1\n∂ 2 ∂Ab\n\n)\nAc\n\nν\n\n]\n+O(A4) . (1.2)\n\nThe relevance of the condensate 〈A2\nµ〉min was discussed in [1, 2], where it was shown that it can\n\nserve as a measure for the monopole condensation in the case of compact QED.\nAll efforts so far have concentrated on the Landau gauge ∂µAµ = 0. The preference for this\n\nparticular gauge fixing is obvious since the nonlocal expression (1.2) reduces to the local opera-\ntor A2\n\nmin = A2\nµ . In the case of a local operator, the Operator Product Expansion (OPE) becomes\n\napplicable, and consequently a measurement of the soft (infrared) part 〈A2\nµ〉OPE becomes possible.\n\nSuch an approach was followed in e.g. [8] by analyzing the appearance of 1/q2 power corrections\nin (gauge variant) quantities like the gluon propagator or strong coupling constant, defined in a\nparticular way, from lattice simulations. Let us mention that already three decades ago attention\nwas paid to the condensate 〈A2\n\nµ〉 in the OPE context [17].\nRecently, Chernodub and Ilgenfritz [12] have considered the asymmetry in the dimension two\n\ncondensate. They performed lattice simulations, computing the expectation value of the electric-\nmagnetic asymmetry in Landau gauge, which they defined as\n\n∆A2 = 〈g2A2\n0〉−\n\n1\nd−1\n\nd−1\n\n∑\ni=1\n\n〈g2A2\ni 〉 . (1.3)\n\nAt zero temperature, this quantity must, of course, be zero due to Lorentz invariance2. Necessarily\nit cannot diverge as divergences at finite T are the same as for T = 0, hence this asymmetry is,\nin principle, finite, and it can be computed without renormalization, for all temperatures. Their\nresults are depicted in Figure 1. At high temperatures, general thermodynamic arguments predict a\npolynomial behavior ∝ T 2, and this is also what the authors of [12] found3. For the low-temperature\n\n1We will always work in Euclidean spacetime.\n2We shall deliberately use the term Lorentz invariance, though we shall be working in Euclidean space throughout\n\nthis paper.\n3A perturbative computation gives a positive proportionality constant, in contrary to what is erronously [18] found\n\nin [12]. The lattice computations for T < 6 Tc find a negative proportionality constant, so one would expect the real\nhigh-temperature behavior to start yet later.\n\n2\n\n\n\nP\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate David Vercauteren\n\nFigure 1: Lattice results for the electric-magnetic assymetry ∆A2 as function of the temperature for SU(2)\npure gauge theory, as found in [12].\n\nbehavior, however, one would expect an exponential fall-off with the lowest glueball mass in the\nexponent, ∆ ∼ e−mglT . Instead, they found an exponential with a mass m significantly smaller than\nmgl.\n\n2. 〈A2\nµ〉 and ∆A2 in the LCO formalism\n\nIn order to get more insight in the behavior of the asymmetry, we have investigated it using\nthe formalism presented in [3]. A meaningful effective potential for the condensation of the Local\nComposite Operator (LCO) A2\n\nµ was constructed by means of the LCO method. This is a nontrivial\ntask due to the compositeness of the considered operator. We consider pure Euclidean SU(N)\nYang–Mills theories with action\n\nSYM+gf =\n∫\n\nd4x\n(\n\n1\n4\n(Fa\n\nµν)2 +Sgf +ba\n∂µAa\n\nµ + c̄a\n∂µDab\n\nµ cb\n)\n\n. (2.1)\n\nWe couple the operator A2\nµ to the Yang–Mills action by means of a source J:\n\nSJ = SYM +\n∫\n\nd4x\n(\n\n1\n2\n\nJ(Aa\nµ)2− 1\n\n2\nζ J2\n\n)\n. (2.2)\n\nThe last term, quadratic in the source J, is necessary to kill the divergences in vacuum correlators\nlike 〈A2(x)A2(y)〉 for x → y, or equivalently in the generating functional W [J], defined as\n\ne−W [J] =\n∫\n\n[fields]e−SJ . (2.3)\n\nThe presence of the LCO parameter ζ ensures a homogenous renormalization group equation for\nW [J]. Its arbitrariness can be overcome by making it a function ζ (g2) of the strong coupling\nconstant g2, allowing one to fix ζ (g2) order by order in perturbation theory in accordance with the\nrenormalization group equation.\n\nIn order to access the electric-magnetic asymmetry, a second source Kµν is coupled to the\ntraceless part of Aa\n\nµAa\nν . This second operator will not mix with A2\n\nµ itself, which allows control\n\n3\n\n\n\nP\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate David Vercauteren\n\nover the renormalization group of these two operators. Again a term quadratic in the new source\nmust be added, introducing a second parameter ω(g2) which can, again, be fixed order by order\nin accordance with the renormalization group equation. We have proven the all-order perturbative\nrenormalizability of this extention of the formalism using the algebraic method based on the Ward\nidenties [19].\n\nIn order to recover an energy interpretation, the terms ∝ J2 and k2 can be removed by employ-\ning a Hubbard–Stratonovich transformation, amounting to inserting the following unities into the\npath integral:\n\n1 =\n∫\n\n[dσ ]e−\n1\n\n2ζ\n\n∫\nddx( σ\n\ng + 1\n2 A2\n\nµ−ζ J)2\n\n=\n∫\n\n[dϕµν ]e−\n1\n\n2ω\n\n∫\nddx( 1\n\ng ϕ+ 1\n2 Aµ Aν−ω kµν)2\n\n, (2.4)\n\nwith ϕµν a traceless field, leading to the action\n\nS = SYM +\n∫\n\nddx\n\n[\n1\n\n2ζ\n\nσ2\n\ng2 +\n1\n\n2ζ g\nσA2\n\nµ +\n1\n\n8ζ\n(A2\n\nµ)2 +\n1\n\n2ω\n\nϕ2\nµν\n\ng2 +\n1\n\n2ωg\nϕµνAµAν +\n\n1\n8ω\n\n(Aa\nµAa\n\nν)2\n\n]\n.\n\n(2.5)\nStarting from this, it is possible to compute the effective potential V (σ ,ϕµν), given the correspon-\ndences\n\n〈σ〉=−g\n2\n〈A2\n\nµ〉 , 〈ϕµν〉=−g\n2\n\n〈\nAµAν −\n\nδµν\n\nd\nA2\n\nλ\n\n〉\n. (2.6)\n\nNow we determine the values of ζ and ω from the renormalization group equations for the\nsources J and Kµν . For this, some anomalous dimensions and renormalization factors have to be\ncomputed up to one loop order higher than the intended loop order we are interested in. We have\ndone this using the MINCER algorithm. The final result is up to one-loop order:\n\nζ =\nN2−1\n16π2\n\n[\n9\n\n13\n16π2\n\ng2N\n+\n\n161\n52\n\n]\n, ω =\n\nN2−1\n16π2\n\n[\n1\n4\n\n16π2\n\ng2N\n+\n\n73\n1044\n\n]\n. (2.7)\n\n3. Computation and minimalization of the action\n\nThe effective potential V (σ ,ϕµν) can now be computed using standard techniques. We have\ntaken the background fields σ and ϕµν to have space-time independent vacuum expectation values\nand ϕµν to be the traceless diagonal matrix diag(A,− 1\n\nd−1 A, . . . ,− 1\nd−1 A) [20].\n\n3.1 Low temperatures\n\nComputing the effective action up to one-loop order at zero temperature, yields only the mini-\nmum found in [3], which is what we expect. For finite but still not too high temperatures, the poten-\ntial can be minimized numerically. The result is depicted in Figure 2. We see that the asymmetry\nrises at low temperatures, which agrees qualitatively with the findings of [12]. The low-temperature\nexpansion of ∆A2 reads\n\n∆A2 = (N2−1)\ng2π2\n\n30\n\n(\n1− 85\n\n1044\ng2N\n\n(4π)2\n\n)\nT 4\n\nm2 , (3.1)\n\nand there is no correction to 〈A2\nµ〉 at this order. Remark that we find a polynomial behavior ∝\n\nT 4/m2 instead of an exponential. This does not agree with the lattice results, but in [12] the lowest\n\n4\n\n\n\nP\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate David Vercauteren\n\n0.1 0.2 0.3 0.4 0.5 0.6\nT�L\n\nMS\n\n-8\n\n-6\n\n-4\n\n-2\n\nFigure 2: The 〈g2A2\nµ〉 condensate (full line) and the assymetry ∆A2 (dashed line) as functions of the temper-\n\nature, in units ΛMS.\n\nFigure 3: Left: The diagrams giving the Debye mass in HTL resummation. The ghost loop is not necessary\n[22]. Right: More diagrams that need to be resummed, coming from the σ and ϕµν parts of the LCO\nLagrangian. The dotted line is the σ or φµν propagator, and the labeled vertex depicts the additional vertices\nintroduced by the formalism.\n\ntemperatures reached were T = 0.4 Tc, where our expansion is not valid anymore. This also does\nnot agree with the thermodynamic argument that, in a theory with a massgap, one would expect\nexponential scaling. However, the asymmetry does not get a gauge invariant meaning by going to\nthe Landau gauge.\n\n3.2 High temperatures\n\nAt temperatures higher than 0.67 ΛMS, the minimum disappears. This signals a phase transi-\ntion to the perturbative vacuum. In order to access this regime, it is possible to expand the effective\npotential for high temperatures, which yields\n\n〈g2A2\nµ〉= g2(N2−1)\n\nT 2\n\n4\n, ∆A2 = g2(N2−1)\n\nT 2\n\n12\n. (3.2)\n\nThis coincides with the perturbative result.\nIn order to compute higher-order corrections to this, it is necessary to perform a Hard Thermal\n\nLoop (HTL) resummation [21], as nonresummed perturbation theory leads to a tachyonic mass in\nour case4. In ordinary pure Yang–Mills theory at this order, HTL amounts to giving the timelike\ngluon a Debye mass m2\n\nD = N\n3 g2T 2, which effectively resums the hard (high momentum) contribu-\n\ntions of the diagrams left in Figure 3. In our formalism, however, there are four additional vertices\ngiving rise to four extra diagrams that need to be resummed, shown at the right in Figure 3. Com-\nputing these additional diagrams, it turns out that they exactly cancel the lowest-order contribution\nfrom the condensate. Solving in the large-T limit yields\n\n〈g2A2\nµ〉= g2(N2−1)\n\n(\nT 2\n\n4\n− mDT\n\n4π\n+ · · ·\n\n)\n, ∆A2 = g2(N2−1)\n\n(\nT 2\n\n12\n− mDT\n\n36π\n+ · · ·\n\n)\n, (3.3)\n\n4The condensate and the mass generated by it differ by a negative multiplicative constant.\n\n5\n\n\n\nP\no\nS\n(\nL\nC\n2\n0\n1\n0\n)\n0\n7\n1\n\nThe asymmetry of the dimension two condensate David Vercauteren\n\nwhich is exactly what one would expect from perturbation theory.\n\n4. Conclusions\n\nThe temperature dependence of the nonperturbative dimension two condensate and its electric-\nmagnetic asymmetry have been studied analytically. At low temperatures, we find qualitative agree-\nment with the lattice results. At high temperatures, the perturbative vacuum is recovered.\n\nWe wish to thank M. N. Chernodub for encouraging this research.\n\nReferences\n\n[1] F. V. Gubarev, L. Stodolsky and V. I. Zakharov, Phys. Rev. Lett. 86 (2001) 2220.\n\n[2] F. V. Gubarev and V. I. Zakharov, Phys. Lett. B 501 (2001) 28.\n\n[3] H. Verschelde, K. Knecht, K. Van Acoleyen and M. Vanderkelen, Phys. Lett. B 516 (2001) 307.\n\n[4] D. Dudal, H. Verschelde and S. P. Sorella, Phys. Lett. B 555 (2003) 126.\n\n[5] D. Dudal, R. F. Sobreiro, S. P. Sorella and H. Verschelde, Phys. Rev. D 72 (2005) 014016.\n\n[6] D. Dudal, H. Verschelde, R. E. Browne and J. A. Gracey, Phys. Lett. B 562 (2003) 87.\n\n[7] D. Vercauteren and H. Verschelde, Phys. Lett. B 660 (2008) 432.\n\n[8] Ph. Boucaud, A. Le Yaouanc, J. P. Leroy, J. Micheli, O. Pene and J. Rodriguez-Quintero, Phys. Rev. D\n63 (2001) 114003.\n\n[9] S. Furui and H. Nakajima, PoS LAT2005 (2006) 291.\n\n[10] F. V. Gubarev and S. M. Morozov, Phys. Rev. D 71 (2005) 114514.\n\n[11] R. E. Browne and J. A. Gracey, JHEP 0311 (2003) 029.\n\n[12] M. N. Chernodub and E. M. Ilgenfritz, Phys. Rev. D 78 (2008) 034036.\n\n[13] O. Andreev, Phys. Rev. D 73 (2006) 107901.\n\n[14] E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 73 (2006) 097502.\n\n[15] V. N. Gribov, Nucl. Phys. B 139 (1978) 1.\n\n[16] M. Lavelle and D. McMullan, Phys. Rept. 279 (1997) 1.\n\n[17] M. J. Lavelle and M. Schaden, Phys. Lett. B 208 (1988) 297.\n\n[18] M. N. Chernodub, personal communication.\n\n[19] D. Dudal, J. A. Gracey, N. Vandersickel, D. Vercauteren and H. Verschelde, Phys. Rev. D 80 (2009)\n065017.\n\n[20] D. Vercauteren and H. Verschelde, arXiv:1007.2789 [hep-th].\n\n[21] J. O. Andersen and M. Strickland, Annals Phys. 317 (2005) 281.\n\n[22] E. Braaten and R. D. Pisarski, Nucl. Phys. B 337 (1990) 569.\n\n6"},"fulltext":true,"source":"PoS","key":"b46a3c926469ac6489b6080c86ec74a6","url":"https://inspirehep.net/files/b46a3c926469ac6489b6080c86ec74a6"}],"authors":[{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/00cv9y106"}],"full_name_unicode_normalized":"vercauteren, david","full_name":"Vercauteren, David","curated_relation":true,"record":{"$ref":"https://inspirehep.net/api/authors/1050215"},"ids":[{"schema":"ORCID","value":"0000-0002-4640-2385"},{"schema":"INSPIRE BAI","value":"D.Vercauteren.1"}],"last_name":"Vercauteren","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/902815"},"value":"Gent U."}],"signature_block":"VARCATARANd","uuid":"78705bab-64d2-474c-ba01-563b9093f5da","first_name":"David","recid":1050215},{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/00cv9y106"}],"full_name_unicode_normalized":"dudal, david","full_name":"Dudal, David","record":{"$ref":"https://inspirehep.net/api/authors/1026581"},"ids":[{"schema":"INSPIRE BAI","value":"D.H.Dudal.1"}],"last_name":"Dudal","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/902815"},"value":"Gent U."}],"signature_block":"DADALd","uuid":"add29bb8-7d8a-4609-bef0-3097f56c9e46","first_name":"David","recid":1026581},{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/04xs57h96"}],"full_name_unicode_normalized":"gracey, john","full_name":"Gracey, John","curated_relation":true,"record":{"$ref":"https://inspirehep.net/api/authors/1007785"},"ids":[{"schema":"ORCID","value":"0000-0002-9101-2853"},{"schema":"INSPIRE BAI","value":"J.A.Gracey.1"}],"last_name":"Gracey","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907986"},"value":"Liverpool U., Dept. 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