Drag Force in a Chiral Plasma

Transcription

1 Drag Force in a Chiral Plasma A.V. Sadofyev MI January 23, 2015 A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

2 Probe string As it was argued a long time ago 1 one could calculate drag force for a heavy quark moving through strongly coupled holographic plasma by considering a probe string, described by NG action S = 1 2πα d 2 σ det g, g αβ = G µν α X µ β X ν where the bulk AdS-BH metric describing the plasma is ds 2 = H 1/2 ( f (r)dt 2 + dx 2 ) + H 1/2 ( dr 2 and f (r) = 1 M r 4, H(r) = r 4. EOMs are α P α µ = 0, P α µ = 1 2πα G µν α X ν f (r) + dω2 5 1 C. Herzog et al, JHEP 0607, 013 (2006); S. Gubser, Phys.Rev.D 74, (2006) A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13 )

3 Probe string A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

4 Probe string For a trailing string the action is reduced in the static gauge to S = 1 2πα dtdr 1 + f (r) H(r) x 2 ẋ 2 f (r) and the corresponding ansatz for the string profile is x(t, r) = vt + ξ(r) + o(t) where v is a late-time velocity of a quark and for the solution we have ξ H(r) f (r) v = ±π 2 ξ f (r) f (r) πξ 2H(r), π ξ = v r h 2 1 v 2 and finally ξ = v ( tan 1 r ) r + rh + log 2r h r h r r h, dp x dt = gp r x = r 2 h 2πα p x m A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

5 Probe string o conserve energy momentum tensor one has to exert an external force upon the heavy quark to keep constant its velocity: ν νµ = f µ (t)δ (3) ( x vt), f µ (t) = lim n M d 3 x g Mµ r where f µ (t) is the drag force on the quark (the endpoint of the string) and n M is the unit normal to the boundary. and f µ (t) = dpµ (t) = lim dt r ηµν πν(t, r r) λ πµ(t, r r) = 2π G 1 ( µn g tr t X N g tt r X N) g Finally, from the string solution the drag force is λ (0) = π 2 2 2π γ (sw µ + u µ ) f µ where w µ = γ(1, v) and s = u w = γ A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

6 Hydrodynamics and holography Now let s turn to the simplest generaliztion of the setup. he plasma above was static but it is well known how to include hydrodynamic perturbations to the theory 2. Here the starting point is the five dimensional Einstein-Maxwell action: S = 1 ( g ( R + 12 F 2 ) + κɛ MNOPQ A M F NOF PQ) d 5 x 16πG 5 It was shown that the theory at the boundary could be described by relativistic hydrodynamics: µ µν = 0, µν = wu µ u ν + Pg µν + τ (1)µν µ J µ = 0, J µ = nu µ + ν (1)µ, where ν (1) and τ (1)µν are corrections of the first order in spatial gradients. 2 J. Erdmenger et al, JHEP 0901, 055 (2009) A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

7 Hydrodynamics and holography Solving Einstein-Maxwell equations by the gradient expansion we find the bulk metric as a perturbative series: ds 2 = r 2 k(r)u µ u ν dx m udx ν + r 2 h(r)p µν dx µ dx ν + r 2 π µν (r)dx µ dx ν +r 2 j σ (r) ( P σ µ u ν + P σ ν u µ ) dx µ dx ν 2S(r)u µ dx µ dr and up to the first order in gradients it is (in h(r) = 1 gauge) S(r) = 1, k(r) = f (r) + 2 3r u, π µν(r) = F (r)σ µν j σ = 1 r (u )u σ + 3 3Q 3 κ 2 2Mr l µ 6 σ + J(r) σ where we should add also solution for the gauge field which however is not required for our consideration now. A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

8 Drag force gradient corrections In the same manner one could use the gradient expansion procedure to find corrections to the drag force on the heavy quark running through a perturbed plasma 3 at µ = 0, then ( x(t, r) = x 0 (t, r) + x 1 (t, r), x 0 (t, r) = v t 1 ( tan 1 r π π π ) ) 2 and one can check that x (1) = t D t x 0 (t, r) t=0 + g(r) solves EOMs. After some algebra the correction to the instantaneous drag force reads λ f µ (1) π ( = c1 (s)(u µ (w )s s µ s s(su α + w α ) α U µ ) 2π γ +c 2 (s)u µ ( u) ) s(u )U µ where c 1 (s) = π/2 tan 1 ( s π ) πf (s), c2 (s) = 1 3 ( s + (1 + s 2 )c 1 (s)) 3 M. Lekaveckas et al, JHEP 1402, 068 (2014) A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

9 Drag force gradient corrections We can now turn on chemical potential and expand in powers of µ. he first non-zero contribution to the drag force appears at the second order in this expansion and it is f (0) µ = λ π 2 2 2π γ (sw µ + u µ ) ( (1 + 3s) ( µ ) ) π 2 s f µ (1,2) = µ2 ( λ 2c5(s) ( (w ) log µ ) ) u 48γπ 2 µ + c 3(s) (u µ(w )s + s µs) s µ2 λ ( 48γπ 2 Uµ c 6(s)(w ) log µ ) c4(s)( u) + c7(s)(suα + w α ) αs + c 10(s)(w )s µ2 λ ( c 8(s)(u )U 48γπ 2 µ + c 9(s)(w )U µ + 2c 5(s) µ log µ ) where c i (s) describe kinematic properties and we won t bring them here. A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

10 Anomalous drag More interesting physics appears if we concentrate on the anomalous contributions to the bulk metric. In the presence of the external magnetic field and vorticity the anomalous contribution takes form j σ = 3κ ( µ ) 2 2 CB (r)b 2π 4 2 σ + π2 5 κ ( µ ) 3 2 lσ 2r 6 and for the drag force it means f µ = λ 2π ( µ ) 2 3s 2 κ C B (π s) 4 (B 2π 2 µ + (B w)w µ ) γ κ λµ 3 4 l µ + (l w)w µ 2γπ 3 2 s A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

11 Anomalous drag New chiral effect that follows directly from the anomaly Anomalous effect for heavy particles New contribution to drag force, in the direction of (B, Ω), same direction for all quarks independent of their charge, even for heavy quarks at rest Phenomenological consequence in correlation between CME current and heavy quark momenta. In an event in which CME pushes positive light quarks up, all heavy quarks dragged upward. Corrections to CME and CVE in presence of non-zero heavy particle density A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

12 Anomalous drag Adding gravitational CS term to our consideration S gcs = 1 gd 5 x κ g ɛ MNPQR A M RBNP A 16πG Rb AQR we expect to gain 2 term in the axial current along vorticity and similarly one finds contribution to the vortical part of the chiral drag force f = κ g λµ γπ 3 C V (s) l and it is interesting to compare κ ( µ ) 2 B and κg µl µ to have an idea whether it is possible to find a non-negligible vortical contribution, here κ κ g = 24 A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13

13 Anomalous drag Starting at rest single quark would be trailed along magnetic field by the chiral drag force. Its motion through the plasma causes a decelerating force (ordinary drag) on the particle. hus there is a terminal velocity ( µ w x = and correction to the net CME ) 2 3κ 4 2π 4 B x π 2 2 δj ( µ = ) 2 3κ 4 2π 4 B π 2 2 (e h n h ) where n h is a density of heavy particles and e h is the charge of the given species. A.V. Sadofyev (MI)Drag Force in a Chiral Plasma January 23, / 13