Magnetofluid Unification in the Yang-Mills Lagrangian

PAQFT 2008 - Singapore, 27 29 November 2008 p. 1 Magnetofluid Unification in the Yang-Mills Lagrangian L.T. Handoko in collaboration with A. Fajarudin, A. Sulaiman, T.P. Djun handoko@teori.fisika.lipi.go.id Group for Theoretical & Computational Physics Indonesian Institute of Sciences - LIPI http://teori.fisika.lipi.go.id

PAQFT 2008 - Singapore, 27 29 November 2008 p. 2 Background Experimental discoveries : deconfined quark gluon matter behaves more like a quark gluon plasma liquid. (STAR & PHENIX @ RHIC, 2005)

PAQFT 2008 - Singapore, 27 29 November 2008 p. 3 Background (2) Motivates works in constructing non-abelian fluid models = magnetohydrodynamics (MHD). Unification of the electromagnetic and fluid fields. But, most works starts from the EOM of fluid-like system inspired by the classical fluid mechanics. Eg. The hybrid MHD : M µν F µν + m S q µν where S µν is the fluid strength tensor

PAQFT 2008 - Singapore, 27 29 November 2008 p. 4 Our proposal The relativistic plasma fluid is modelled as a fluid system of gluon cloud with matters (quarks and anti-quarks) inside.

PAQFT 2008 - Singapore, 27 29 November 2008 p. 4 Our proposal The relativistic plasma fluid is modelled as a fluid system of gluon cloud with matters (quarks and anti-quarks) inside. construct the model from first principle... Introduce a lagrangian involves non-abelian fluid fields U µ interacting with another Abelian / non-abelian gauge fields A µ with some matters inside. Certain gauge symmetry G(n) F G(n) G (eg. MHD is U(1) F U(n) G ).

PAQFT 2008 - Singapore, 27 29 November 2008 p. 5 The Model : G(n) F G(n) G with, L = L matter + L gauge + L int. L matter L gauge L int. = iψ / Ψ m Ψ ΨΨ = 1 4 Sa µν Saµν 1 4 F a µν F aµν = g F J a Fµ Uaµ g G J a Gµ Aaµ while, S a µν µ U a ν ν U a µ + g F f abc U b µu c ν J a Xµ ΨT a X γ µψ

PAQFT 2008 - Singapore, 27 29 November 2008 p. 6 EOM for relativistic fluid The EOM in term of fluid field U µ is, Introduce : ν S µν = g F J F µ for Abelian D ν S a µν = g F J a Fµ for non Abelian U a µ = (U a 0,U a ) u a µφ and u µ γ a (1, v a ) A µ ǫ µ e ip x Constraint : U µ must be massive gauge bosons, so there are still 3 degree of freedoms to represent the spatial velocity v!

PAQFT 2008 - Singapore, 27 29 November 2008 p. 7 EOM for relativistic fluid (2) Summing up all components, 0 ( µ U a 0 0U a µ) i ( µ U a i i U a µ where, F a µ f abc F i +i [ ( 0 UµU b 0 c ( TFU d d0 + g G ) i ( U b µu c i )] ( T d FU di + g G g F T d GA di ) ( µ T d g GA d0 U a F ) ( ) = gf JFµ a + F µ a 0 0 U a µ + g F f abc F U b µu c 0 ) ( µ U a i i U a µ + g F f abc F U b µu c i F a µ : additional force from the fluid self-interaction ) )

PAQFT 2008 - Singapore, 27 29 November 2008 p. 8 EOM for relativistic fluid (3) t Ua U a 0 = g F dt (J a F + F a ) = g F dx (J a F0 + F a 0 ) which always satisfies ω a U a = 0 irrortational fluid. Finally, t (γa v a φ) + (γ a φ) = g F dx (J a F0 + F a 0 ) a general relativistic fluid equation?

PAQFT 2008 - Singapore, 27 29 November 2008 p. 9 EOM for relativistic fluid (4) Reason At non-relativistic limit, γ 1 + 1/2 v 2 up to O( v 2 ) accuracy and φ 1 : v a t + 1 2 va 2 = g F dx (JF0 a + F 0 a ) non rel. and v a t + (va )v a = g F dx (JF0 a + F 0 a ) non rel. just the classical irrotational fluid equation!

PAQFT 2008 - Singapore, 27 29 November 2008 p. 10 Modelling the QGP Let s consider SU(3) F U(1) G Describe macroscopically non-abelian fluid formed by dense gluon surrounding the (quarks and anti-quarks) matters in an electromagnetic field. L = iq / Q m Q QQ 1 4 Sa µνs aµν 1 4 F µνf µν +g F J a Fµ Uaµ + qj G µa µ and, its relativistic fluid dynamics obeys, t (γa v a φ) + (γ a φ) = g F dx (ρ a F + F a 0 )

PAQFT 2008 - Singapore, 27 29 November 2008 p. 11 Modelling the QGP (2) According to the experiments at RHIC : the hot QGP is dense and flows with tiny viscosity approximating to the ideal fluid. This draws : U a 0 affecting some terms in Fµ a. No turbulence, i.e. ω a 0, that is always satisfied. Also, since q/g F α/α s O(10 1 ) the EM force is negligible (no long range EM forces in non-abelian plasmas) the pure SU(3) F model : L = 1 4 Sa µνs aµν g F J a Fµ Uaµ

PAQFT 2008 - Singapore, 27 29 November 2008 p. 12 Consequences For instance, one can deduce the Hamiltonian density : H = g2 F 2 ( dx (ρ a F + F0 a ) +g F γ a φ (ρ a F + v a J a F) 2 + 1 2 ) f abc U b U c 2 Another type of energy momentum tensor in general relativity etc. Perform numeric calculation using lattice gauge theory.

PAQFT 2008 - Singapore, 27 29 November 2008 p. 13 Summary The magnetofluid is described using gauge invariant lagrangian. It provides alternative insight into macroscopic dynamics of relativistic fluids relevent for plasma etc.