Background gauge fields — finite particle density

hypermultiplets. These hypermultiplets arise from the lowest excitations of the strings stretching between the D7-branes and the background-generating D3-branes. The particles represented by the fundamental fields of theN = 2 hypermultiplets model the quarks in our system. Their massm_qis given by the asymptotic value of the separation of the D3- and D7-branes. Since the physics of the thermal D3/D7setup is determined by the ratio of quark mass to temperature, we use the parameterm, which is proportional to this ratio.

Here µ_q is the quark chemical potential, n_q is the quark density, m is the dimensionless quark mass parameter given in(3.15)andccis the quark con-densate mentioned earlier (but irrelevant in this work). We made use of the dimensionlessρ-coordinate that runs from the horizon value ρ = 1to the boundary atρ→ ∞. The chemical potential and density of baryons are simply

µB = µq

N_c, nB = nq

N_c . (3.34)

Once we have found the solutionsA¯₀ to the equations of motion for the gauge field, the valueµqof the chemical potential in the dual field theory can be extracted as

µ_q= lim

ρ→∞

A¯₀(ρ) = r◦

2πα⁰µ˜_q, (3.35)

where we introduced the dimensionless quantityµ˜for convenience. We apply the same normalization to the gauge field and distinguish the dimensionful quantityA¯from the dimensionless

A˜0 = 2πα⁰ r◦

A¯0 (3.36)

(we save the symbols without diacritics for later use). Analogously, the so-lutions of the embedding functions carry information about the quark mass parameterm,

m= lim

ρ→∞ ρ χ(ρ). (3.37)

We mentioned that for non-vanishing baryon density, there are no embed-dings of Minkowski type, and all embedembed-dings reach the black hole horizon.

This is due to the fact that a finite baryon density in an infinite volume of Minkowski spacetime requires an infinite number of strings in the dual super-gravity picture. These strings have one end on the stack of D3-branes and the other on the stack ofN_f probe D⁷-branes. These strings pull the brane towards the black hole[58]. Such spike configurations are common for configurations in which branes of different dimensionality connect[33].

Very recently, however, it was found that for a vanishing baryon number density, there may indeed be Minkowski embeddings if a constant vacuum expectation value ofA˜0 is present, which does not depend on the holographic coordinate[59, 60, 72–74]. The phase diagram found there is reproduced in fig-ure 3.1. In the shaded region, the baryon density vanishes (nB = 0) but temperature, quark mass and chemical potential can be nonzero. This low tem-perature region only supports Minkowski embeddings with the brane ending before reaching the horizon. In contrast, the unshaded region supports black hole embeddings with the branes ending on the black hole horizon. In this regime the baryon density does not vanish (nB 6= 0). At the low-µend of the

FIGURE3.1: The phase dia-gram of fundamental matter in the D3/D7setup. Horizontal axis: Chemical potential nor-malized to the quark mass. Ver-tical axism⁻¹ ∝ T /mq. In this work we analyze the white region of finite particle density d, for which we show some˜ lines of constant values ford.˜ The relation betweend˜andnB

is given by (3.43) ^{0.0 0.2 0.4 0.6 0.8 1.0 1.2}

0.0 0.2 0.4 0.6 0.8 1.0

µq/mq

1/m

d˜= 0

˜d=0.25

˜d

= 0.5

˜d

= 1

˜d= 2

d˜=4

line separatingnB = 0fromnB 6= 0in figure 3.1, there exists also a small region of multivalued embeddings, which are thermodynamically unstable[59]. In the black hole phase there is a phase transition between different black hole embeddings[58], resembling the meson melting phase transition for fundamen-tal matter at vanishing density. This first order transition occurs in a region of the phase diagram close to the separation line between the two regions with vanishing (shaded) and non-vanishing (unshaded) baryon density. This transition disappears above a critical value for the baryon densitynB given by

nB = N_f√ λT³ 2^5/2

d,˜ with criticald˜^∗ = 0.00315. (3.38) In this work we exclusively explore the region in whichnB >0,i. e.we examine thermal systems in the canonical ensemble. For a detailed discussion of this aspect see refs. 59, 60.

To determine the solutions of the supergravity fields on the probe branes we have to extremize the DBI action (3.30), we write shortly as

S_DBI=−T₇

N_f

X

k=1

Z d⁸ξ

q

|det(G+ ˜F^(k))|. (3.39) The induced metricG(ξ) on the stack ofNf coincident branes is given by (3.18),F˜is the dimensionless field strength tensor of the gauge fields on the brane.

For now we consider the simpler case of a baryonic chemical potential modeled by theU(1)subgroup ofU(N_f). In this case, the sum amounts to an overall factor ofN_f. In ref. 58 the dynamics of such a system of branes and gauge fields was analyzed in view of describing phase transitions at finite baryon density. Here, we use these results as a starting point which gives the background configuration of the probe branes’ embedding function and the gauge field values at finite baryon density. To examine vector meson spectra, we will then investigate the dynamics of fluctuations in this gauge field background.

In the coordinates introduced above, the actionS_DBIfor the embedding χ(%) and the field strength F is obtained by inserting the induced metric and the field strength tensor into(3.39). From now on we make use of the dimensionless coordinates and reproduce the action found in ref. 58. To do so, we remember that the only non-vanishing component of the background field is theρ-dependent time component. Therefore, the only non-vanishing components of the field strength tensor areF˜₄₀^(k) =−F˜₀₄^(k). We evaluate the determinant and arrive at

SDBI=−T₇Nf

Z d⁸ξ √

−G q

1 +G⁰⁰G⁴⁴ F˜40

2

(3.40) with componentsG^ab of the inverse metricG⁻¹. After inserting these compo-nents we get

SDBI=−N_fT7r_◦³ Z

d⁸ξ ρ³

4ff˜(1−χ²)

× s

1−χ²+ρ²χ⁰²−2 f˜

f²(1−χ²) ˜F40

2

, (3.41)

whereF˜40=∂ρA˜0is the field strength on the brane. The background fields χandA˜₀depend solely onρ. This action only depends on derivatives of the gauge field. We therefore can identify the constant of motiond˜satisfying

∂ρd˜= 0,

d˜= ∂SDBI

∂ ∂_ρA˜₀. (3.42)

Evaluation of this formula and insertion of the asymptotic expansion of A˜ reveal that this dimensionless constant is related to the parameters of our setup by[58]

d˜= 2^5/2nB

N_f√

λ T³. (3.43)

We can therefore think of the constantd˜as parametrizing the baryon density nB.

The equations of motion for the background fields are conveniently ob-tained after Legendre transforming the action(3.41)toSˆ=S−δS/δF40in order to eliminate dependence on the gauge field in favor of dependence on the constant of motiond˜[58]. Varying this Legendre transformed action with

2 4 6 8 10 0.0

0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ρ w

χ L

θ d˜= 0.05 d˜= 0.05

m= 3

m= 2

m= 1

m= 0.5

FIGURE3.2: Examples for the embedding functionχ(ρ)withχ= cosθand the according profileL(w). Matching colors indicate corresponding curves.

respect to the fieldχgives the equation of motion for the embeddingsχ(ρ),

∂ρ

"

ρ⁵ff˜(1−χ²)χ⁰ p1−χ²+ρ²χ⁰²

s

1 + 8 ˜d² ρ⁶f˜³(1−χ²)³

#

=− ρ³ff χ˜ p1−χ²+ρ²χ⁰²

s

1 + 8 ˜d² ρ⁶f˜³(1−χ²)³

×

"

3(1−χ²) + 2ρ²χ⁰²−24 ˜d² 1−χ²+ρ²χ⁰² ρ⁶f˜³(1−χ²)³+ 8 ˜d²

# .

(3.44)

This equation for χ(ρ) can be solved numerically for given d˜and initial valueχ0. We impose boundary conditions such that the branes cross the horizon perpendicularly

χ(ρ= 1) =χ0, ∂ρχ(ρ)

ρ=1= 0. (3.45)

Figure 3.2 shows some examples. The embeddings at finite density resemble the largeρasymptotics of the embeddings found at zero density. For smallρ however, at finite particle density there always is the spike reaching down to the event horizon. The initial value ofχ0determines the position on which the brane reaches the horizon and in this way determines the quark mass parameter m,cf.equation(3.37). It is zero forχ₀ = 0and tends to infinity forχ₀ →1.

Figure 3.3 shows this dependence ofmonχ0for different values of the baryon densityd. In general, a small (large)˜ χ0is equivalent to a small (large)m. For χ₀ .0.5, we nearly observe proportionality. For vanishingd˜= 0, we only can model quarks withm≤1.3, heavier quarks at vanishing density are described by embeddings of Minkowski type, which we do not discuss here. (The trained eye can see that there is a maximum ofm= 1.3in figure 3.3 beforemdrops

FIGURE3.3: Dependence of the quark mass parameter m on the initial valueχ0 of the embedding.

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

χ⁰

m ^˜_d= 0

d˜= 0.05 d˜= 0.5 d˜= 1 d˜= 2

to smaller values towardsχ0 = 1. This is reflecting the existence of a phase transition to Minkowski embeddings atd˜= 0).

The equation of motion for the background gauge fieldA˜is given by

∂_ρA˜₀ = 2 ˜d fp

1−χ²+ρ²χ⁰² r

f(1˜ −χ²)h

ρ⁶f˜³(1−χ²)³+ 8 ˜d²i

. (3.46)

Integrating both sides of the equation of motion fromρ_H= 1to someρ, and respecting the boundary condition A˜₀(ρ = 1) = 0 [58], we obtain the full background gauge field

A˜₀(ρ) = 2 ˜d

ρ

Z

1

dρ fp

1−χ²+ρ²χ⁰² r

f(1˜ −χ²)h

ρ⁶f˜³(1−χ²)³+ 8 ˜d²i

. (3.47)

Examples for the functional behavior ofA˜0(ρ)are shown in figure 3.4. While there is a significant slope of A(ρ)˜ near the horizon at ρ = 1, the gauge field tends to a constant at largeρ. From(3.35)we recall that this value is the chemical potential of the field theory. We will henceforth computeµ˜by evaluating the formula above for largeρ. Note that at any finite baryon density d˜∝nB 6= 0there exists a minimal chemical potential which is reached in the limit of massless quarks.

Im Dokument In-medium effects in the holographic quark-gluon plasma (Seite 72-77)