Chung-Wen Kao Chung Yuan Christian University Taiwan QCD Chiral restoration at finite T and B A study based on the instanton model

Collaboration and Reference In Collaboration with Dr. Nam Seung-il This talk is based on the following works: (1) S.i.Nam and CWK, Phys.Rev. D.83:096009,2011. (2) S.i.Nam and CWK, Phys.Rev. D.82:096001,2010.

QCD and its symmetries QCD with N f massless quarks is a very beautiful theory with many symmetries. SU(3)c : Colour Gauge Symmetry. SU(N f ) L x SU(N f ) R : Chiral symmetry. U(1) B : Baryon number Conservation. U(1) A : Axial symmetry, broken by axial anomaly.

QCD vacuum structure QCD vacuum owns very complicated structure, due to the underlying strong interaction between quarks and gluons. QCD vacuum spontaneously break chiral symmetry into vectorial SU(N f ) V symmetry. However, when T or μ increases, chiral symmetry will restore. The order parameter of chiral phase transition is chiral condensates Chiral condensate is nonzero when chiral symmetry is spontaneously broken. When chiral symmetry is restored the condensate becomes zero.

Phase structure of QCD χSB Chiral symmetric phase ≠0≠0 =0

Universal class of Chiral phase transition At high T, QCD with two-flavor massless quarks is belonged to the universal class of O(4) spin model in three dimension, therefore it is second-order transition. However, with the small quark mass, the second-order transition is replaced by a smooth crossover. At low T, the chiral restoration in the chiral limit (m q = 0) is 1st- order transition. If the quark masses are nonzero, there should be a critical end point (CEP) in the phase diagram of QCD at which the rst-order transition line terminates and followed by the crossover

Many Approaches … maybe too many? NJL model. Dyson-Schwinger method. PNJL(NJL with Polyakov Loop). Gauge-gravity duality mode. Hidden-local symmetry (HLS) model QCD sum rules Functional renormalization-group method. Chiral Perturbation Theory Instanton Vacuum Model

Instanton in QCD Yang-Mills instanton is a solution of the classical Euclidean equations of motion In a given topological sector, instanton minimizes the action has either a self- dual or anti-selfdual field strength. The potential energy of the YM fields is periodic in the particular coordinate called the Chern–Simons number.

Instanton induced interaction Since quarks scatter off the same (anti)instanton, one can average over their positions and orientations in colour space induce certain correlations between quarks. The result is an effective action for quarks which contains instanton effects in induced multi-quark interactions. The consequent interactions are vertices involving 2Nf quarks, commonly cited as ’t Hooft interactions. When Nf=2, one can apply bosonization to construct The effective actions as functional of quarks and mesons

Instanton Vacuum model Effective Chiral Action derived from the instanton configuration in the leading order (LO) of the 1/Nc expansion : The quarks are moving inside the (anti)instanton ensemble and flipping their helicities. It results in that (anti)quarks acquire the momentum-dependent effective masses dynamically, i.e. constituent-quark masses M(k).

Effective quark mass M(k) Assuming that the zero modes dominate the low-energy phenomena, we can write the Dirac equation for a quark for the (anti)instanton background By making Fourier transform o f instanton size ≈1/3 ≈, One obtains

Meson Loop Corrections Meson Loop Correction NLO in large N C expansion

Instanton at finite T Harrington-Shepard caloron for the parameter modifications at finite T. The instanton density N/V becomes more dilute as T increases. The instanton size also shrinks. We also make use of the fermionic Matsubara formula

Chiral condensates v.s. T Without Meson Loop Corrections With Meson Loop Corrections

Chiral condensates v.s. m q Without Meson Loop CorrectionsWith Meson Loop Corrections

Effective quark mass Without Meson Loop CorrectionsWith Meson Loop Corrections Effective quark mass is very sensitive to MLC !

QCD under strong magnetic field QCD matter under intensive magnetic field can be found in several systems. For example, A magnetar is a type of neutron star with an extremely powerful magnetic field, the decay of which powers the emission of copious amounts of high- energy electromagnetic radiation, particularly X-rays and gamma rays. B ~ G at the surface, much higher in the core.

QCD under strong magnetic field Non-central RHIC collisions: eB~ MeV 2 B~ G. Therefore it is interesting to see what happens when QCD is under intensive B! [Au-Au, 200 GeV] [Kharzeev, McLerran & Warringa (2007)]

Studies on Chiral Phase Transition NJL: Klevansky & Lemmer (1989) Gusynin, Miransky & Shovkovy (1994/1995) Klimenko et al. ( ) Hiller, Osipov, … ( ) χPT: Shushpanov & Smilga (1997) Agasian & Shushpanov (2000) Cohen, McGady & Werbos (2007) Large-N QCD: Miransky & Shovkovy (2002) Quark model: Kabat, Lee & Weinberg (2002)

Effective quark mass v.s. B at T=0 m q =0m q ≠0 u quark d quark

Effective quark mass v.s. B at finite T u quark d quark

Condensate v.s. T under B in chiral limit d quark u quark

Condensate v.s. T under B d quark u quark

Iso-breaking of condensates

Pion mass & weak decay constant v.s. T FπFπ mπmπ

Conclusion and Outlook Inclusion of Meson Loop Correction is crucial to have the correct universal classes of chiral phase transitions. The constituent quark masses of u and d become equal under certain B field. Magnetic catalysis effect is observed. Under strong magnetic field it is more difficult to have chiral restoration. We plan to explore μ-T phase diagram of QCD, with/without Magnetic field.