QCD at high energy (experiments)

a r X i v :h e p -p h /9602413v 1 27 F eb 1996QCD Phase Transition and Hadron Bubble Formation in theDual Ginzburg-Landau TheoryH.Ichie,H.Monden ∗,H.Suganuma and H.TokiResearch Center for Nuclear Physics (RCNP),Osaka University,Ibaraki 567,Osaka,Japan ∗Department of Physics,Tokyo Metropolitan University,Hachioji 192,Tokyo,JapanAbstract We study the QCD phase transition at finite temperature and discuss the hadron bubble formation in the dual Ginzburg-Landau theory,which is an ef-fective theory of QCD and describes the color confinement via QCD-monopole condensation.We formulate the effective potential at various temperatures and find that thermal effects reduce the QCD-monopole condensate and bring a first-order deconfinement phase transition.Based on this effective potential at finite temperature,we investigate properties of hadron bubbles created in the early Universe and discuss the hadron bubble formation process.I.INTRODUCTION At low temperature and low density,colored particles,quarks and gluons are confined in hadrons due to the nonperturbative nature of the QCD vacuum.However,as the tem-perature increases above a certain critical temperature,these colored particles would be liberated like free particles and the vacuum becomes quark gluon plasma (QGP)phase [1].This phenomenon is called QCD phase transition and many physists try to create QGPin high-energy heavy-ion collisions (e.g.RHIC at Brookhaven).After two heavy ions col-lide,and pass through each other,the huge energy deposition at central region leads to QGP [1].The QCD phase transition also happened as a real event in the early Universe,which strongly influenced the afterward nucleosynthesis.Thus,the QCD phase transition is interesting in the various fields.The QCD phase transition is caused by the change of QCD vacuum.In low energy region,the QCD vacuum is regarded as the dual version of superconductor [2].Condensation of the magnetic charge makes the color-electric field excluded from the QCD vacuum,which leads to the color confinement.This picture is constructed after abelian gauge fixing as ’t Hooft proposed [3].The most relevant gauge for the discussion of confinement is the abelian gauge,where QCD-monopoles appear as topological objects having the magnetic charges.This is modeled in the dual Ginzburg Landau (DGL)theory [4,5],which has a strong connection with QCD and are supported from the recent results of lattice QCD [6].II.DUAL GINZBURG-LANDAU THEORYThe DGL lagrangian[5,7]in the pure gauge system is written by using the dual gauge field,Bµ= Bµ· H=B3µT3+B8µT8,and the QCD-monopolefield,χ= a√√√√2(∂µBν−∂νBµ)2+[ˆDµ,χ]†[ˆDµ,χ]−λ(χ†χ−v2)2,(1)whereˆDµ=ˆ∂µ+igBµis the dual covariant derivative.The dual gaugefield Bµis defined on the dual gauge manifold U(1)m3×U(1)m8,which is the dual space of the maximal torus subgroup U(1)e3×U(1)e8embedded in the original gauge group SU(3)[5].The abelianfield strength tensor is written as Fµν=∗(∂∧B)µνso that the role of the electric and magnetic field are interchanged in comparison with the ordinary Aµdescription.The QCD-monopole has magnetic charge g α,where αis the root vector,and g is the dual gauge coupling constant. If the QCD-monopole condenses,|χα|=v,the dual gaugefield acquires mass m B=√π2 ∞0dkk2ln 1−e−√2T k2+m2χ/T ,(2) where m B=√λ¯χ.The effective potential is plotted in Fig.1and the behavior of the QCD-monopole condensate at various temperatures is shown in Fig.2.As the temperature increases,the QCD-monopole condensate in the physical vacuum,¯χphys(T), decreases and the broken dual gauge symmetry tends to be restored.Afirst order phase transition is found at the critical temperature,T c≃0.49GeV.This phase transition is regarded as the deconfinement phase transition,because there is no confining force among colored particles in the QCD vacuum with¯χphys(T)=0.The critical temperature T c= 0.49GeV,however,seems much larger than the recent lattice QCD prediction[8,9].We introduce therefore the temperature dependence on the parameterλas,λ(T)=λ T c−aTaccording to the asymptotic freedom property of QCD.The behavior of the QCD-monopole condensate is shown in Fig.3for the case where the parameter a are chosed so as to provide the critical temperature at T c=0.2GeV.Wefind a weakfirst order phase transition.Using this QCD-monopole condensate at various temperatures,we next consider the string tension of hadron(Fig.4)and the glueball mass(Fig.5).As the temperature increases, the string tension becomes smaller and drops rapidly near the critical temperature,which agrees with the lattice simulation data[10].This means that the hadron size atfinite temperature becomes larger as the temperature increases.Here,wefind also the large reduction of the glueball mass[7]near the critical temperature,where the glueball excites violently as its mass becomes small and it leads to the QCD phase transition.IV.HADRON BUBBLE FORMATION IN EARLY UNIVERSE Finally,we consider the application of the DGL theory to big bang.As Witten proposed [11],if the QCD phase transition is offirst order,the hadron and the QGP phase should coexist in early Universe.Such a mixed phase may cause the inhomogeneity of the Universe in the baryon number distribution.This inhomoginity affects the primordial nucleo-synthesis [12].As a result of the1st order phase transition,hadron bubbles appear in the QGP phase near the critical temperature.We now consider how hadron bubbles are formed in the DGL theory.In the supercooling system,the free energy of the hadron bubble with radius R profile¯χ(r;R)is written using the effective potential atfinite temperature,E[¯χ(r;R)]=4π ∞0drr2{3(d¯χ(r;R)3πR3c h/T).Thus,the only small bubbles are created practically,although its radius should be larger than R c energitically.The temperature dependence of the critical radius and the bubble formation rate is shown in Fig.9and Fig.10respectively.In the temperature region of the supercoolingstate,i.e,T low<T<T c,the hadron bubbles are created.As the temperature decreases,the size of hadron bubble becomes smaller,but the bubble formation rate becomes larger.From these results,we can imagine how the QCD phase transition happens in the big bang scenario[13].At thefirst stage slightly below T c,only large bubbles are created but its rate is quite small.As temperature is lowered,smaller bubbles are created with much formation rate.During this process,the created hadron bubbles expand with radiating shock wave which reheats QGP phase[13].Near T low many small bubbles are violently created. Finally QGP phase is isolated like the bubble[13].Such an evolution of the hadron bubble can be obtained from the numerical simulation using the DGL theory.We study the QCD phase transition atfinite temperature in the DGL theory.Thermal effect reduces the QCD-monopole condensate,and the QCD vacuum is changed into the QGP phase.According to the reduction of the QCD-monopole condensate at high tem-perature,the string tension becomes smaller,the hadron size becomes larger and glueball mass becomes smaller.We apply the DGL theory to the hadron bubble formation in early ing the effective potential,we estimate the size of hadron bubble at various temperatures.We would like to thank Prof.Kajino for stimulating discussions on the hadron bubble formation.Figure CaptionFig.1The effective potential at various temperatures as a function of the QCD-monopole condensate.The minima of V eff(¯χ;T)are plotted by×.Fig.2The QCD-monopole condensate¯χphys(T)at the minimum of the effective potential as a function of temperature T.Fig.3The QCD-monopole condensate¯χphys(T)at the minimum of the effective potential as a function of the temperature in the case of variableλ(T),which reproduces T c=0.2GeV. Fig.4The string tensions k(T)for a constantλand a variableλ(T)as functions of the temperature T.The lattice QCD results[10]in the pure gauge are shown by black dots near and below the critical temperature.Fig.5The masses m B and mχof the dual gaugefield Bµand the QCD-monopolefieldχas the function of temperature.Fig.6The profile of the QCD-monopole condensate in the hadron bubble.Outside of bubble, QCD-monopole does not condense(QGP phase),while inside the bubble,QCD-monopole condenses(hadron phase).Fig.7The energy density of the hadron bubble.It is negative inside and positive near the boundary surface.Fig.8The total energy of the bubble is plotted as a function of the hadron bubble radius R. The total energy is roughly estimated as the sum of the volume term and the surface term. Fig.9The critical radius R c,corresponding to maximum of the energy in Fig.8,is plotted as a function of temperature.Fig.10The dominant factor of the bubble formation rate P(R c;T)≡exp(−4πR3c h(T)/3T) is plotted as a function of temperature.As the temperature decreases,the radius of the created hadron bubble becomes smaller,but the bubble formation rate larger.REFERENCES[1]W.Greiner and A.Sch¨a fer,“Quantum Chromodynamics”,(Springer-Verlag,1994)1.[2]Y.Nambu,Phys.Rev.D10(1974)4262.[3]G.’t Hooft,Nucl.Phys.B190(1981)455.[4]S.Maedan and T.Suzuki,Prog.Theor.Phys.81(1989)229.T.Suzuki,Prog.Theor.Phys.80(1988)929;81(1989)752.[5]H.Suganuma,S.Sasaki and H.Toki,Nucl.Phys.B435(1995)207.[6]A.S.Kronfeld,G.Schierholz and U.-J.Wiese,Nucl.Phys.B293(1987)461.[7]H.Ichie,H.Suganuma and H.Toki,H.Suganuma,S.Sasaki,H.Toki and H.Ichie,Prog.Theor.Phys.(Suppl.)120(1995)57.[8]J.Engels,F.Karsh,H.Satz and I.Montvay,Phys.Lett.B102(1981)332.[9]For instance,H.J.Rothe,“Lattice Gauge Theories”,(World Scientific Singapore,1992)1.[10]M.Gao,Nucl.Phys.B9(Proc.Suppl.)(1989)368.[11]E.Witten,Phys.Rev.D30(1984)272.[12]T.Kajino,this proceedings.[13]G.M.Fuller,G.J.Mathews,and C.R.Alcock,Phys.Rev.D37(1988)1380.This figure "fig1-1.png" is available in "png" format from: /ps/hep-ph/9602413v1This figure "fig2-1.png" is available in "png" format from: /ps/hep-ph/9602413v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-ph/9602413v1This figure "fig2-2.png" is available in "png" format from: /ps/hep-ph/9602413v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-ph/9602413v1This figure "fig2-3.png" is available in "png" format from: /ps/hep-ph/9602413v1。

Research ObjectivesThe MILC Collaboration is engaged in a broad research program in Quantum Chromodynamics (QCD).This research addresses fundamental questions in high energy and nuclear physics,and is directly related to major experimental programs in thesefields.It includes studies of the mass spectrum of strongly interacting particles,the weak interactions of these particles,and the behavior of strongly interacting matter under extreme conditions.The Standard Model of High Energy Physics encompasses our current knowledge of the funda-mental interactions of subatomic physics.It consists of two quantumfield theories:the Weinberg-Salaam theory of electromagnetic and weak interactions,and QCD,the theory of the strong interac-tions.The Standard Model has been enormously successful in explaining a wealth of data produced in accelerator and cosmic ray experiments over the past thirty years;however,our knowledge of it is incomplete because it has been difficult to extract many of the most interesting predictions of QCD,those that depend on the strong coupling regime of the theory,and therefore require non-perturbative calculations.At present,the only means of carrying out non-perturbative QCD calculations fromfirst principles and with controlled errors,is through large scale numerical sim-ulations within the framework of lattice gauge theory.These simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions,to de-termine a number of the fundamental parameters of the Standard Model,and to make precise tests of the Standard Model’s range of validity.Despite the many successes of the Standard Model,it is believed by high energy physicists that to understand physics at the shortest distances,a more general theory,which unifies all four of the fundamental forces of nature,will be required.The Standard Model is expected to be a limiting case of this more general theory,just as classical mechanics is a limiting case of the more general quantum mechanics.A central objective of the experimental program in high energy physics,and of lattice QCD simulations,is to determine the range of validity of the Standard Model,and to search for new physics beyond it.Thus,QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.QCD is formulated in the four-dimensional space-time continuum;however,in order to carry out numerical calculations one must reformulate it on a lattice or grid.It should be emphasized that the lattice formulation of QCD is not merely a numerical approximation to the continuum formu-lation.The lattice regularization of QCD is every bit as valid as continuum regularizations.The lattice spacing a establishes a momentum cutoffπ/a that removes ultraviolet divergences.Stan-dard renormalization methods apply,and in the perturbative regime they allow a straightforward conversion of lattice results to any of the standard continuum regularization schemes.Lattice QCD calculations proceed in two steps.In thefirst,one uses importance sampling tech-niques to generate gauge configurations,which are representative samples from the Feynman path integrals that define QCD.These configurations are saved,and in the second step they are used to calculate a wide variety of physical quantities.It is necessary to generate configurations with a range of lattice spacings,and then perform extrapolations to the zero lattice spacing limit.Fur-thermore,the computational cost of calculations rises as the masses of the quarks,the fundamental constituents of strongly interacting matter,decrease.Until recently,it has been too expensive to carry out calculations with the masses of the two lightest quarks,the up and the down,set to their physical values.Instead,one has performed calculations for a range of up and down quark masses, and extrapolated to their physical values guided by chiral perturbation theory,an effectivefield theory that determines how physical quantities depend on the masses of the lightest quarks.The extrapolations in lattice spacing(continuum extrapolation)and quark mass(chiral extrapolation) are the major sources of systematic errors in QCD calculations,and both must be under control in order to obtain trustworthy results.In our current simulations,we are,for thefirst time,working at or near the physical masses of the up and down quarks.The gauge configurations produced in these simulations greatly reduce,and will eventually eliminate,the systematic errors associatedwith the chiral extrapolation.A number of different formulations of QCD on the lattice are currently in use by lattice gauge theorists,all of which are expected to give the same results in the continuum limit.In recent years, major progress has been made in thefield through the development of improved formulations(im-proved actions)which reducefinite lattice spacing artifacts.Approximately twelve years ago,we developed one such improved action called asqtad[1],which significantly increased the accuracy of our simulations for a given amount of computing resources.We have used the asqtad action to generate an extensive library of gauge configurations with small enough lattice spacings and light enough quark masses to perform controlled calculations of a number of physical quantities. Computational resources provided by the DOE and NSF have enabled us to complete our program of generating asqtad gauge configurations.These configurations are publicly available,and have been used by us and by other groups to study a wide range of physical phenomena of importance in high energy and nuclear physics.Ours was thefirst set of full QCD ensembles that enabled control over both the continuum and chiral extrapolations.We have published a review paper describing the asqtad ensembles and the many calculations that were performed with them up to2009[2]. Over the last decade,a major component of our work has been to use our asqtad gauge config-urations to calculate quantities of importance to experimental programs in high energy physics. Particular emphasis was placed on the study of the weak decays and mixings of strongly interact-ing particles in order to determine some of the least well known parameters of the standard model and to provide precise tests of the standard model.The asqtad ensembles have enabled the calcu-lation of a number of physical quantities to a precision of1%–5%,and will enable many more quantities to be determined to this precision in the coming years.These results are already having an impact on experiments in high energy physics;however,in some important calculations,partic-ularly those related to tests of the standard model,higher precision is needed than can be provided by the existing asqtad ensembles.In order to obtain the required precision,we are now working with the Highly Improved Staggered Quark(HISQ)action developed by the HPQCD Collabora-tion[3].We have performed tests of scaling in the lattice spacing using HISQ valence quarks with gauge configurations generated with HISQ sea quarks[4].We found that lattice artifacts for the HISQ action are reduced by approximately a factor of2.5from those of the asqtad action for the same lattice spacing,and taste splittings in the pion masses are reduced by approximately a factor of three,which is sufficient to enable us to undertake simulations with the mass of the Goldstone pion at or near the physical pion mass.(“Taste”refers to the different ways one can construct the same physical particle in the staggered quark formalism.Although particles with different tastes become identical in the continuum limit,their masses can differ atfinite lattice spacing).More-over,the improvement in the quark dispersion relation enables us to include charm sea quarks in the simulations.The properties of the HISQ ensembles are described in detail in Ref.[5],and the first physics calculations using the physical quark mass ensembles in Refs.[6,7,8].The current status of the HISQ ensemble generation project is described at the link HISQ Lattice Generation and some initial calculations with them at Recent Results.The HISQ action also has major advan-tages for the study of QCD at high temperatures,so we have started to use it in our studies of this subject.Projects using the HISQ action will be a major component of our research for the next several years.Our research is currently focused on three major areas:1)the properties of light pseudoscalar mesons,2)the decays and mixings of heavy-light mesons,3)the properties of strongly interacting matter at high temperatures.We briefly discuss our research in each of these areas at the link Recent Results.References[1]The MILC Collaboration:C.Bernard et al.,Nucl.Phys.(Proc.Suppl.),60A,297(1998);Phys.Rev.D58,014503(1998);G.P.Lepage,Nucl.Phys.(Proc.Suppl.),60A,267(1998);Phys.Rev.D59,074501(1999);Kostas Orginos and Doug Toussaint(MILC),Nucl.Phys.(Proc.Suppl.),73,909(1999);Phys.Rev.D59,014501(1999);Kostas Orginos,Doug Tou-ssaint and R.L.Sugar(MILC),Phys.Rev.D60,054503(1999);The MILC Collaboration:C.Bernard et al.,Phys.Rev.D61,111502(2000).[2]The MILC Collaboration: A.Bazavov et al.,Rev.Mod.Phys.82,1349-1417(2010)[arXiv:0903.3598[hep-lat]].[3]The HPQCD/UKQCD Collaboration: E.Follana et al.,Phys.Rev.D73,054502(2007)[arXiv:hep-lat/0610092].[4]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D82,074501(2010)[arXiv:1004.0342].[5]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D87,054505(2013)[arXiv:1212.4768].[6]The MILC Collaboration: A.Bazavov et al.,Phys.Rev.Lett.110,172003(2013)[arXiv:1301.5855].[7]The Fermilab Lattice and MILC Collaborations:A.Bazavov,et al.,Phys.Rev.Lett.112,112001(2014)[arXiv:1312.1228].[8]The MILC Collaboration:A.Bazavov et al.,Proceedings of Science(Lattice2013)405(2013)[arXiv:1312.0149].。