Holographic vector mesons from spectral functions at finite baryon or isospin density
aa r X i v : . [ h e p - t h ] M a y MPP-2007-136
Holographic vector mesons from spectral functionsat finite baryon or isospin density
Johanna Erdmenger, ∗ Matthias Kaminski, † and Felix Rust ‡ Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, 80805 M¨unchen, Germany
We consider gauge/gravity duality with flavor for the finite-temperature field theory dual of the AdS-Schwarzschild black hole background with embedded D7-brane probes. In particular, we investigate spectralfunctions at finite baryon density in the black hole phase. We determine the resonance frequencies correspond-ing to meson-mass peaks as function of the quark mass over temperature ratio. We find that these frequencieshave a minimum for a finite value of the quark mass. If the quotient of quark mass and temperature is increasedfurther, the peaks move to larger frequencies. At the same time the peaks narrow, in agreement with the for-mation of nearly stable vector meson states which exactly reproduce the meson mass spectrum found at zerotemperature. We also calculate the diffusion coefficient, which has finite value for all quark mass to temperatureratios, and exhibits a first-order phase transition. Finally we consider an isospin chemical potential and find thatthe spectral functions display a resonance peak splitting, similar to the isospin meson mass splitting observed ineffective QCD models.
PACS numbers: 11.25.Tq, 11.25.Wx, 12.38.Mh, 11.10.Wx
Contents
I. Introduction and Summary II. Holographic setup and thermodynamics
III. Spectral functions at finite baryon density
IV. Spectral functions at finite isospin density SU (2) -background gauge field 11B. Results at finite isospin density 12 V. Conclusion Acknowledgments A. Notation References ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
I. INTRODUCTION AND SUMMARY
Recently in the context of gauge/gravity duality, there hasbeen an intensive study of the phase diagram of N = 4 large N c SU ( N c ) supersymmetric Yang-Mills theory with addedfundamental degrees of freedom, by considering the AdS-Schwarzschild black hole background with added D7-braneprobes [1, 2, 3, 4, 5, 6]. There are two kinds of D7-braneprobes in the black hole background: Either they end beforereaching the black hole horizon, since the S wrapped by theD7-brane probe shrinks to zero as in [7], or they reach all theway to the black hole horizon. The first class of embeddings isusually called ‘Minkowski embeddings’, while the second isreferred to as ‘black hole embeddings’. The parameter whichparametrizes different embeddings is the temperature normal-ized quark mass m q /T , which may be given in terms of theasymptotic value χ of the embedding coordinate at the AdShorizon. The phase transition between both classes of embed-dings is of first order. The analysis of the meson spectrumshows that this phase transition corresponds to a fundamen-tal confinement/deconfinement transition at which the mesonsmelt.Particular interest has arisen in the more involved struc-ture of the phase diagram when a baryon chemical potentialis present [8]. It was argued that for non-vanishing baryondensity, there are no embeddings of Minkowski type, and allembeddings reach the black hole horizon. This is due to thefact that a finite baryon density generates strings in the dualsupergravity picture which pull the brane towards the blackhole. A chemical potential for these baryons corresponds toa vev ˜ A for the time component of the gauge field on thebrane. In the dual thermal SU ( N c ) -gauge theory a baryon iscomposed of N c quarks, such that the baryon density n B canbe directly translated into a quark density n q = n B N c . Thethermodynamic dual quantity of the quark density is the quarkchemical potential µ q . In the brane setup we use, the chemical PSfrag replacements . . . . . . . . . . . . . . . .
81 11 1 µ q / m q µ q / m q T / ¯ MT / ¯ M ˜ d = 0 ˜ d = 0 . d = 4 ˜ d = 0 . FIG. 1: The phase diagram for quarks: The quark chemical po-tential µ q divided by the quark mass is plotted versus the tempera-ture T divided by ¯ M = 2 m q / √ λ . Two different regions are dis-played: The shaded region with vanishing baryon density and theregion above the transition line with finite baryon density, in whichwe work here. The multivalued region at the lower tip of the transi-tion line is not resolved here. The curves are lines of equal baryondensity parametrized by ˜ d = 2 / n B / ( N f √ λT ) . The critical den-sity ˜ d ∗ = 0 . , at which the first order phase transition betweentwo black hole phases disappears, is shown as short-dashed line closeto the transition line. It virtually coincides with the short-dashed linefor ˜ d = 0 . . potential is determined by the choice of quark density and bythe embedding parameter χ .Very recently, however, it was found that for a vanishingbaryon number density, there may indeed be Minkowski em-beddings if a constant vev ˜ A is present, which does not de-pend on the holographic coordinate [9, 10, 11, 12, 13]. Thephase diagram found there is sketched in figure 1. In the greyshaded region, the baryon density vanishes ( n B = 0 ) but tem-perature, quark mass and chemical potential can be nonzero.This low temperature region only supports Minkowski em-beddings with the brane ending before reaching the horizon.In contrast, the unshaded region supports black hole embed-dings with the branes ending on the black hole horizon. In thisregime the baryon density does not vanish ( n B = 0 ). In thispaper we exclusively explore the latter region. At the lower tipof the line separating n B = 0 from n B = 0 in figure 1, thereexists also a small region of multivalued embeddings, whichare thermodynamically unstable [10].In the black hole phase considered here, there is a funda-mental phase transition between different black hole embed-dings [8]. This is a first order transition, which occurs in aregion of the phase diagram close to the separation line be-tween the two regions with vanishing (gray shaded) and non-vanishing (unshaded) baryon density. This transition disap-pears above a critical value for the baryon density n B givenby ˜ d ∗ = 0 . , ˜ d = 2 / n B / ( N f √ λT ) . (1.1)In this paper we make use of the methods developed in thecontext of AdS/CFT applied to hydrodynamics, for instance[14, 15, 16], in order to determine the spectral function atfinite temperature and finite baryon density. For vanishing chemical potential, a similar analysis of the spectral functionshas been performed in [17]. It was found that the spectrumis discrete at large quark mass, or equivalently at low temper-ature. At low quark mass, a quasiparticle structure is seenwhich displays the broadening decay width of the mesons.As the mass decreases or temperature rises, the mesons arerendered unstable as the resonance frequencies develop imag-inary parts. Modes corresponding to such frequencies arecalled quasinormal. These excitations are then dissipated inthe plasma. – Note that for this case, there are also latticegauge theory results [18].In this paper we study the differences in the spectral func-tions with and without chemical potential. Relating our workto the phase diagram shown in [10, figure 2] (and reproducedhere in figure 1), we here consider the region of black holeembeddings (unshaded region) with nonvanishing quark den-sity. We find that at low temperature to quark mass ratio,the spectrum is asymptotically discrete and coincides with thezero-temperature supersymmetric meson mass formula foundin [19], which in our coordinates reads M = L ∞ R p n + 1)( n + 2) . (1.2)In [19], L ∞ denotes the asymptotic separation of the D3- andD7-branes and n counts the nodes of the embedding fluctua-tions. Here we are considering s-wave modes in the Kaluza-Klein expansion of the D7 brane probe wrapping S , so theangular quantum number l is zero. We connect the structure ofthe spectra found to the phase diagram in figure 1: The mesonmass behavior described above occurs close to the Minkowskiregion of the phase diagram, where temperature effects aresubdominant. Moreover, as a function of decreasing temper-ature to quark mass ratio, the quasiparticle peaks behave dif-ferently with and without finite quark density. As discussedin [17], at vanishing density n B = 0 the peak maxima movetowards smaller frequencies as a function of increasing quarkmass. Here, in the case of finite quark density, we observea similar behavior at small quark mass. However, keepingthe temperature fixed as we increase the quark mass further,the peaks turn around at a value m turn q and move to larger andlarger frequencies as the associated mesons become more sta-ble. Note that a turning point behavior was also observed forvanishing quark density in the context of quasinormal modesfor scalar modes of melting mesons [6].Our spectra also show that for given quark mass and tem-perature, lower n meson excitations can be nearly stable in theplasma, while higher n excitations remain unstable. At van-ishing baryon density, the formation of resonance peaks forhigher excitations has also been observed in [20]. We discussthe distinct behavior of resonance peaks in section III B 2, in-cluding a comparison of the observed turning points at finitebaryon density with previous results.We also calculate the quark diffusion constant D and showthat at finite density, it exhibits the first-order fundamentalphase transition up to the critical density given by ˜ d ∗ =0 . . For very large values of the density, the diffusionconstant asymptotes to D · T = 1 / (2 π ) . This reflects the factthat in this case, the free quarks outnumber the quarks boundin mesons.As a second point we consider the case of an isospin chem-ical potential, on which previous work in the holographic con-text has appeared in [3, 21]. In this case, two coincident D7-brane probes are considered. In particular we extend the re-sults of our previous paper [22], in which we calculated the re-tarded Green function and diffusion coefficient at finite SU (2) isospin chemical potential for the flat embedding m q = 0 . Inthis previous work we also restricted to the case of constantvev for the non-Abelian gauge field A , where is the flavorand the Lorentz index. This means that we chose A to beindependent of the AdS radial direction. In this case we founda non-analytic frequency dependence of the Green functionsand the diffusion coefficient. Here we extend this work to thecase of non-vanishing quark mass, leading to non-trivial D7embeddings, and to the case of radially varying gauge fieldcomponent A . We find that spectral functions quantitativelydeviate from the baryonic background case. Additionally, asplitting of quasi-particle resonances is observed, which de-pends on the magnitude of the chemical potential.This paper is organized as follows. In the following sec-tion II, we introduce the gravity background, field and braneconfiguration, used for the subsequent calculations. We alsosketch the method to obtain retarded real-time correlators ofthermal field theories from supergravity calculations. In sec-tion III we discuss the spectral functions and diffusion behav-ior of fundamental matter at finite baryon density. For matterwith isospin chemical potential, the same analysis is carriedout in section IV. The results are briefly summarized in sec-tion V. II. HOLOGRAPHIC SETUP AND THERMODYNAMICSA. Background and brane configuration
We consider asymptotically
AdS × S space-time whicharises as the near horizon limit of a stack of N c coincidenttest D3-branes. More precisely, our background is an AdS blackhole, which is the geometry dual to a field theory at finite tem-perature (see e.g. [23]). We make use of the coordinates of [8]to write this background in Minkowski signature as d s = 12 (cid:16) ̺R (cid:17) (cid:18) − f ˜ f d t + ˜ f d x (cid:19) + (cid:18) R̺ (cid:19) (cid:0) d ̺ + ̺ dΩ (cid:1) , (2.1)with the metric dΩ of the unit -sphere, wheretest f ( ̺ ) = 1 − ̺ H ̺ , ˜ f ( ̺ ) = 1 + ̺ H ̺ ,R = 4 πg s N c α ′ , ̺ H = T πR . (2.2)Here R is the AdS radius, g s is the string coupling constant, T the temperature, N c the number of colors. In the followingsome equations may be written more conviniently in terms ofthe dimensionless radial coordinate ρ = ̺/̺ H , which covers a range from ρ = 1 at the event horizon to ρ → ∞ , representingthe boundary of AdS space.Into this ten-dimensional space-time we embed N f coin-ciding D7-branes, hosting flavor gauge fields A µ . The em-bedding we choose lets the D7-branes extend in all directionsof AdS space and, in the limit ρ → ∞ , wraps an S on the S . It is convenient to write the D7-brane action in coordi-nates where d ̺ + ̺ dΩ = d ̺ + ̺ (d θ + cos θ d φ + sin θ dΩ ) , (2.3)with ≤ θ < π/ . From the viewpoint of ten dimen-sional Cartesian AdS × S , θ is the angle between the sub-space spanned by the 4,5,6,7-directions, into which the D7-branes extend perpendicular to the D3-branes, and the sub-space spanned by the 8,9-directions, which are transverse toall branes.Due to the symmetries of this background, the embeddingsdepend only on the radial coordinate ρ . Defining χ ≡ cos θ ,the embeddings of the D7-branes are parametrized by thefunctions χ ( ρ ) . They describe the location of the D7-branesin , -directions. Due to our choice of the gauge field fluc-tuations in the next subsection, the remaining three-sphere inthis metric will not play a prominent role.The metric induced on the D7-brane probe is then given by d s = 12 (cid:16) ̺R (cid:17) (cid:18) − f ˜ f d t + ˜ f d x (cid:19) + (cid:18) R̺ (cid:19) − χ + ̺ χ ′ − χ d ̺ + R (1 − χ )dΩ . (2.4)Here and in what follows we use a prime to denote a derivativewith respect to ̺ (resp. to ρ in dimensionless equations). Thesymbol √− g denotes the square root of the determinant of theinduced metric on the D7-brane, which is given by √− g = ̺ f ˜ f − χ ) q − χ + ̺ χ ′ . (2.5)The table below gives an overview of the indices we use torefer to certain directions and subspaces. AdS S coord. names x x x x ̺ – µ , ν , . . .indices i , j , . . . ̺ The background geometry described so far is dual to ther-mal N = 4 supersymmetric SU ( N c ) Yang-Mills theory with N f additional N = 2 hypermultiplets. These hypermulti-plets arise from the lowest excitations of the strings stretchingbetween the D7-branes and the background-generating D3-branes. The particles represented by the fundamental fields ofthe N = 2 hypermultiplets model the quarks in our system.Their mass m q is given by the asymptotic value of the sepa-ration of the D3- and D7-branes. In the coordinates used herewe write [17] m q √ λT = ¯ MT = lim ρ →∞ ρ χ ( ρ ) = m, (2.6)where we introduced the dimensionless scaled quark mass m .In addition to the parameters incorporated so far, we aimfor a description of the system at finite chemical potential µ and baryon density n B . In field theory, a chemical poten-tial is given by a nondynamical time component of the gaugefield. In the gravity dual, this is obtained by introducing a ρ -dependent gauge field component ¯ A ( ρ ) on the D7 braneprobe. For now we consider a baryon chemical potentialwhich is obtained from the U (1) subgroup of the flavor sym-metry group. The sum over flavors then yields a factor of N f in front of the DBI action written down below.The value of the chemical potential µ in the dual field the-ory is then given by µ = lim ρ →∞ ¯ A ( ρ ) = ̺ H πα ′ ˜ µ, (2.7)where we introduced the dimensionless quantity ˜ µ for conve-nience. We apply the same normalization to the gauge fieldand distinguish the dimensionful quantity ¯ A from the dimen-sionless ˜ A = ¯ A (2 πα ′ ) /̺ H .The action for the probe branes’ embedding function andgauge fields on the branes is S DBI = − N f T D7 Z d ξ q | det( g + ˜ F ) | . (2.8)Here g is the induced metric (2.4) on the brane, ˜ F is the fieldstrength tensor of the gauge fields on the brane and ξ are thebranes’ worldvolume coordinates. T D7 is the brane tensionand the factor N f arises from the trace over the generators ofthe symmetry group under consideration. For finite baryondensity, this factor will be different from that at finite isospindensity.In [8], the dynamics of this system of branes and gaugefields was analyzed in view of describing phase transitions atfinite baryon density. Here we use these results as a startingpoint which gives the background configuration of the braneembedding and the gauge field values at finite baryon density.To examine vector meson spectra, we will then investigate thedynamics of fluctuations in this gauge field background.In the coordinates introduced above, the action S DBI for theembedding χ ( ρ ) and the gauge fields’ field strength F is ob-tained by inserting the induced metric and the field strengthtensor into (2.8). As in [8], we get S DBI = − N f T D7 ̺ H Z d ξ ρ f ˜ f (1 − χ ) × s − χ + ρ χ ′ − ff (1 − χ ) ˜ F ρ , (2.9)where ˜ F ρ = ∂ ρ ˜ A is the field strength on the brane. ˜ A depends solely on ρ . According to [8], the equations of motion for the back-ground fields are obtained after Legendre transforming theaction (2.9). Varying this Legendre transformed action withrespect to the field χ gives the equation of motion for the em-beddings χ ( ρ ) , ∂ ρ " ρ f ˜ f (1 − χ ) χ ′ p − χ + ρ χ ′ s d ρ ˜ f (1 − χ ) = − ρ f ˜ f χ p − χ + ρ χ ′ s d ρ ˜ f (1 − χ ) × " − χ ) + 2 ρ χ ′ −
24 ˜ d − χ + ρ χ ′ ρ ˜ f (1 − χ ) + 8 ˜ d . (2.10)The dimensionless quantity ˜ d is a constant of motion. It isrelated to the baryon number density n B by [8] n B = 12 / N f √ λT ˜ d. (2.11)Below, equation (2.10) will be solved numerically for differ-ent initial values χ and ˜ d . The boundary conditions used are χ ( ρ = 1) = χ , ∂ ρ χ ( ρ ) (cid:12)(cid:12)(cid:12) ρ =1 = 0 . (2.12)The quark mass m is determined by χ . It is zero for χ = 0 and tends to infinity for χ → . Figure 2 shows the de-pendence of the scaled quark mass m = 2 m q / √ λT on thestarting value χ for different values of the baryon densityparametrized by ˜ d ∝ n B . In general, a small (large) χ isequivalent to a small (large) quark mass. For χ < . , χ can be viewed as being proportional to the large quark masses.At larger χ for vanishing ˜ d = 0 , the quark mass reaches a fi-nite value. In contrast, at finite baryon density, if χ is closeto , the mass rapidly increases when increasing χ further.In embeddings with a phase transition, there exist more thanone embedding for one specific mass value. In a small regimeclose to χ = 1 , there are more than one possible value of χ for a given m . So in this small region, χ is not proportionalto m q .The equation of motion for the background gauge field ˜ A is ∂ ρ ˜ A = 2 ˜ d f p − χ + ρ χ ′ q ˜ f (1 − χ )[ ρ ˜ f (1 − χ ) + 8 ˜ d ] . (2.13)Integrating both sides of the equation of motion from ρ H tosome ρ , and respecting the boundary condition ˜ A ( ρ = 1) =0 [8], we obtain the full background gauge field ˜ A ( ρ ) = 2 ˜ d ρ Z ρ H d ρ f p − χ + ρ χ ′ q ˜ f (1 − χ )[ ρ ˜ f (1 − χ ) + 8 ˜ d ] . (2.14)Recall that the chemical potential of the field theory is givenby lim ρ →∞ ˜ A ( ρ ) and thus can be obtained from the formulaabove. Examples for the functional behavior of A ( ρ ) are PSfrag replacements m χ ˜ d = 0˜ d = 0 . d = 0 . d = 0 . FIG. 2: The dependence of the scaled quark mass m = 2 m q / √ λT on the horizon value χ = lim ρ → χ of the embedding. shown in figure 3. Note that at a given baryon density n B = 0 there exists a minimal chemical potential which is reached inthe limit of massles quarks.The asymptotic form of the fields χ ( ρ ) and A ( ρ ) can befound from the equations of motion in the boundary limit ρ →∞ , ¯ A = µ − ˜ dρ ̺ H πα ′ + · · · , (2.15) χ = mρ + cρ + · · · . (2.16)Here µ is the chemical potential, m is the dimensionless quarkmass parameter given in (2.6), c is related to the quark conden-sate (but irrelevant in this work) and ˜ d is related to the baryonnumber density as stated in (2.11). See also figure 3 for thisasymptotic behavior. The ρ -coordinate runs from the horizonvalue ρ = 1 to the boundary at ρ = ∞ . In most of this range,the gauge field is almost constant and reaches its asymptoticvalue, the chemical potential µ , at ρ → ∞ . Only near thehorizon the field drops rapidly to zero. For small χ → ,the curves asymptote to the lowest (red) curve. So there isa minimal chemical potential for fixed baryon density in thissetup. At small baryon density ( ˜ d ≪ . ) the embed-dings resemble the Minkowski and black hole embeddingsknown from the case without a chemical potential. Only athin spike always reaches down to the horizon.In the setup described in this section we restrict ourselvesto the regime of so called ‘black hole embeddings’ which arethose embeddings ending on the horizon of the black hole, op-posed to ‘Minkowski embeddings’ , which would reach ρ = 0 without touching the horizon. The black hole embeddings weuse for this work (see figure 3) are not capable of describingmatter in all possible phases. In fact we are able to cover theregime of fixed n B > and thus examine thermal systems inthe canonical ensemble at finite baryon density. For a detaileddiscussion of this aspect see [9, 10]. PSfrag replacements χ ρrL ˜ A ˜ A / − L . . . . .
81 11 . . PSfrag replacements χ ρrL ˜ A ˜ A / − L . . . . .
81 11 . . PSfrag replacements χ ρrL ˜ A ˜ A / − L . . . . . . .
52 2 3 4 5
PSfrag replacements χ ρrL ˜ A ˜ A / − L . . . . . . .
52 2 3 4 5
PSfrag replacements χρ rL ˜ A ˜ A / − L . . . . . . . PSfrag replacements χρ rL ˜ A ˜ A / − L . . . . . . . FIG. 3: The three figures of the left column show the embeddingfunction χ versus the radial coordinate ρ , the corresponding back-ground gauge fields ˜ A and the distance L = ρ χ between the D3 andthe D7-branes at ˜ d = 10 − / . L is plotted versus r , given by ρ = r + L . In the right column, the same three quantities are depictedfor ˜ d = 0 . . The five curves in each plot correspond to parametriza-tions of the quark mass to temperature ratio with χ = χ (1) =0 , . , . , . (all solid) and . (dashed) from bottom up.These correspond to scaled quark masses m = 2 m q /T √ λ =0 , . , . , . , . in the left plot and to m =0 , . , . , . , . on the right. The curves onthe left exhibit µ ≈ − . Only the upper most curve on theleft at χ = 0 . develops a large chemical potential of µ =0 . . In the right column curves correspond to chemical po-tential values µ = 0 . , . , . , . , . frombottom up. B. Holographic spectral functions
Spectral functions contain information about the quasipar-ticle spectrum of a given theory. Recently, methods were de-veloped to compute spectral functions from the holographicduals of strongly coupled finite temperature gauge theories.In this work we extend these results to investigate the quasi-particle spectrum corresponding to vector mesons in the limitof vanishing spatial momentum. Therefore, we analyze theholographic dual to spectral functions for thermal N = 4 supersymmetric SU ( N c ) Yang-Mills theory with N f funda-mental degrees of freedom (quarks) at finite baryon densityand finite chemical potential. We compute the spectral densi-ties for the flavor current J , which is dual to the fluctuations A of the flavor gauge field on the supergravity side.Within field theory, the spectral function R ( ω, q ) of someoperator J ( x ) is defined via the imaginary part of the retardedGreen function G R as follows R ( ω, q ) = − G R ( ω, q ) , (2.17)where Energy ω and spatial momentum q may be written ina four vector ~k = ( ω, q ) and the Green function G R may bewritten as G R ( ω, q ) = − i Z d x e i ~k~x θ ( x ) h [ J ( ~x ) , J (0)] i (2.18)One may find singularities of G R ( ω, q ) in the lower half ofthe complex ω -plane, including hydrodynamic poles of theretarded real-time Green function. Consider for example G R = 1 ω − ω + i Γ . (2.19)These poles emerge as peaks in the spectral densities, R = 2 Γ( ω − ω ) + Γ , (2.20)located at ω with a width given by Γ . These peaks are inter-preted as quasi-particles if their lifetime / Γ is considerablylong, i.e. if Γ ≪ ω .In this paper we use the gauge/gravity duality prescriptionof [14] for calculating Green functions in Minkowski space-time. For further reference, we outline this prescription brieflyin the subsequent. Starting out from a classical supergravityaction S cl for the gauge field A , according to [14] we extractthe function B ( ρ ) (containing metric factors and the metricdeterminant) in front of the kinetic term ( ∂ ρ A ) , S cl = Z d ρ d x B ( ρ ) ( ∂ ρ A ) + . . . . (2.21)Then we perform a Fourier transformation and solve the lin-earized equations of motion for the fields A in momentumspace. The solutions in general are functions of all five co-ordinates in Anti-de Sitter space. Near the boundary we mayseparate the radial behavior from the boundary dynamics bywriting A ( ρ, ~k ) = f ( ρ, ~k ) A bdy ( ~k ) , (2.22)where A bdy ( ~k ) is the value of the supergravity field at theboundary of AdS depending only on the four flat boundarycoordinates. Thus by definition we have f ( ρ, ~k ) | ρ →∞ = 1 .Then the retarded thermal Green function is given by G R ( ω, q ) = 2 B ( ρ ) f ( ρ, − ~k ) ∂ ρ f ( ρ, ~k ) (cid:12)(cid:12)(cid:12) ρ →∞ . (2.23)The thermal correlators obtained in this way display hydro-dynamic properties, such as poles located at complex frequen-cies. They are used to compute the spectral densities (2.17).We are going to compute the functions A ( ρ, k ) numerically inthe limit of vanishing spatial momentum q → . The func-tions f ( ρ, ~k ) are then obtained by dividing out the boundaryvalue A bdy ( ~k ) = lim ρ →∞ A ( ρ, ~k ) . Numerically we obtain theboundary value by computing the solution at a fixed large ρ . III. SPECTRAL FUNCTIONS AT FINITE BARYONDENSITYA. Baryon diffusion
In this section we calculate the baryon diffusion coefficientand its dependence on the baryon density. As discussed in[10], the baryon density affects the location and the pres-ence of the fundamental phase transition between two blackhole embeddings observed in [8]. This first order transition ispresent only very close to the separation line between the re-gions of zero and non-zero baryon density shown in figure 1.We show that this fundamental phase transition may alsobe seen in the diffusion coefficient for quark diffusion. Inorder to compute the diffusion using holography, we use themembrane paradigm approach developed in [24] and extendedin [17]. This method allows to compute various transportcoefficients in Dp/Dq-brane setups from the metric coeffi-cients. The resulting formula for our background is the sameas in [17], D = √− gg √− g g (cid:12)(cid:12)(cid:12)(cid:12) ρ =1 Z d ρ − g g √− g . (3.1)The dependence of D on the baryon density and on thequark mass originates from the dependence of the embedding χ on these variables. The results for D are shown in figure 4.The thick solid line shows the diffusion constant at vanishingbaryon density found in in [17], which reaches D = 0 at thefundamental phase transition. Increasing the baryon density,the diffusion coefficient curve is lifted up for small tempera-tures, still showing a phase transition up to the critical density ˜ d ∗ = 0 . . This is the same value as found in [8] in thecontext of the phase transition of the quark condensate.The diffusion coefficient never vanishes for finite density.Both in the limit of T / ¯ M → and T / ¯ M → ∞ , D · T con-verges to / π for all densities, i.e. to the same value as forvanishing baryon density, as given for instance in [24] for R-charge diffusion. At the phase transition, the diffusion con-stant develops a nonzero minimum at finite baryon density.Furthermore, the location of the first order phase transitionmoves to lower values of T / ¯ M while we increase ˜ d towardsits critical value.In order to give a physical explanation for this behavior,we focus on the case without baryon density first. We seethat the diffusion coefficient vanishes at the temperature of thefundamental deconfinement transition. This is simply due tothe fact that at and below this temperature, all charge carriersare bound into mesons not carrying any baryon number.For non-zero baryon density however, there is a fixed num-ber of charge carriers (free quarks) present at any finite tem-perature. This implies that the diffusion coefficient never van-ishes. Switching on a very small baryon density, even be-low the phase transition, where most of the quarks are boundinto mesons, by definition there will still be a finite amountof free quarks. By increasing the baryon density, we increasethe amount free quarks, which at some point outnumber thequarks bound in mesons. Therefore in the large density limit (cid:144) PSfrag replacements . . . . . . . . . . . . . . . . . . . . . . . .
02 0 . . . .
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45 0 . . . . . . . . . . . . .
95 11 . . . . . . . . . . . . . . .
25 2 . .
75 33 . . . . . . . . . . . . . . . . . . . . . . m q / (2 πα ′ ) M mes b = 0 b = 1 Lorentzian b = 1 Euclidean b = 1 . b = 1 R = 3 m q = 2 . m q = 2 . R = 3 exactapprox. m q = 1 R = 3 m q = 2 . /TR = 3 qa = δM/b - δM ∂ δM∂q = α E w ( ρ ) bρχ ( ρ ) A ( ρ ) ρL wR ( w , R ( w , − R D T T ¯ M (cid:144) PSfrag replacements . . . . . . . . . . . . . . . . . . . . .
005 0 . .
015 0 . .
025 0 . .
035 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m q / (2 πα ′ ) M mes b = 0 b = 1 Lorentzian b = 1 Euclidean b = 1 . b = 1 R = 3 m q = 2 . m q = 2 . R = 3 exactapprox. m q = 1 R = 3 m q = 2 . /TR = 3 qa = δM/b - δM ∂ δM∂q = α E w ( ρ ) bρχ ( ρ ) A ( ρ ) ρL wR ( w , R ( w , − R D T T ¯ M FIG. 4: The diffusion coefficient times temperature is plottedagainst the mass-scaled temperature for diverse baryon densitiesparametrized by ˜ d = 0 . (uppermost line in upper plot, not visible inlower plot), . , (long-dashed), . (thin solid), . (long-short-dashed), . (short-dashed) and (thick solid). The fi-nite baryon density lifts the curves at small temperatures. Thereforethe diffusion constant never vanishes but is only minimized near thephase transition. The lower plot zooms into the region of the tran-sition. The phase transition vanishes above a critical value ˜ d ∗ =0 . . The position of the transition shifts to smaller T / ¯ M , as ˜ d is increased towards its critical value. the diffusion coefficient approaches D = 1 / (2 πT ) for allvalues of T / ¯ M , because only a negligible fraction of thequarks is still bound in this limit.Note that as discussed in [8, 9, 10] there exists a regionin the ( n B , T ) phase diagram at small n B and T where theembeddings are unstable. In figure 4, this corresponds to theregion just below the phase transition at small baryon density.This instability disappears for large n B . B. Vector mesons in the black hole phase
1. Application of calculation method
We now compute the spectral functions of flavor currentsat finite baryon density n B , chemical potential µ and tem-perature in the ‘black hole phase’. As black hole phase theauthors of [9] denote the phase of matter which has nonzerobaryon density. Compared to the limit of vanishing chemicalpotential treated in [17], we discover a qualitatively differentbehavior of the finite temperature oscillations correspondingto vector meson resonances. To obtain the spectral functions, we compute the correla-tions of flavor gauge field fluctuations A µ about the back-ground given by (2.9), denoting the full gauge field by ˆ A µ ( ρ, ~x ) = δ µ ˜ A ( ρ ) + A µ ( ~x, ρ ) . (3.2)According to section II A, the background field has a non-vanishing time component, which depends solely on ρ . Thefluctuations in turn are gauged to have non-vanishing compo-nents along the Minkowski coordinates ~x only and only de-pend on these coordinates and on ρ . Additionally they areassumed to be small, so that it suffices to consider their lin-earized equations of motion. At this point we neglect the fluc-tuation of the scalar and pseudoscalar modes and their cou-pling to the vector fluctuations. In fact there is no such cou-pling in the limit of vanishing spatial momentum, which werestrict to below.The resulting equations of motion are obtained from the ac-tion (2.8), where we introduce small fluctuations A by setting ˜ F µν → ˆ F µν = 2 ∂ [ µ ˆ A ν ] with ˆ A = ˜ A + A . The backgroundgauge field ˜ A is given by (2.13). The fluctuations now propa-gate on a background G given by G ≡ g + ˜ F , (3.3)and their dynamics is determined by the Lagrangian L = p | det( G + F ) | , (3.4)with the fluctuation field strength F µν = 2 ∂ [ µ A ν ] . Since thefluctuations and their derivatives are chosen to be small, weconsider their equations of motion only up to linear order, asderived from the part of the Lagrangian L which is quadraticin the fields and their derivatives. Denoting this part by L ,we get L = − p | det G |× (cid:18) G µα G βγ F αβ F γµ − G µα G βγ F µα F βγ (cid:19) . (3.5)Here and below we use upper indices on G to denote elementsof G − . The equations of motion for the components of A are ∂ ν "p | det G |× (cid:16) G µν G σγ − G µσ G νγ − G [ νσ ] G γµ (cid:17) ∂ [ γ A µ ] . (3.6)The terms of the corresponding on-shell action at the ρ -boundaries are (with ρ as an index for the coordinate ρ , notsummed) S on-shellD7 = ̺ H π R N f T D7 Z d x p | det G |× (cid:16) (cid:0) G (cid:1) A ∂ ρ A − G G ik A i ∂ ρ A k − A G tr( G − F ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ B ̺ H . (3.7)Note that on the boundary ρ B at ρ → ∞ , the background ma-trix G reduces to the induced D7-brane metric g . Therefore,the analytic expression for boundary contributions to the on-shell action is identical to the one found in [17]. There, the co-ordinates in Minkowski directions were chosen such that thefluctuation four vector ~k exhibits only one non vanishing spa-tial component, e. g. in x -direction as ~k = ( ω, q, , . Thenthe action was expressed in terms of the gauge invariant fieldcomponent combinations E x = ωA x + qA , E y,z = ωA y,z . (3.8)In the case of vanishing spatial momentum q → , the Greenfunctions for the different components coincide and werecomputed as [17] G R = G Rxx = G Ryy = G Rzz = N f N c T ρ →∞ (cid:18) ρ ∂ ρ E ( ρ ) E ( ρ ) (cid:19) , (3.9)where the E ( ρ ) in the denominator divides out the boundaryvalue of the field in the limit of large ρ , as discussed after(2.23). The indices on the Green function denote the compo-nents of the operators in the correlation function, in our caseall off-diagonal correlations (as G yz , for example) vanish.In our case of finite baryon density, new features arisethrough the modified embedding and gauge field background,which enter the equations of motion (3.6) for the field fluctu-ations. To apply the prescription to calculate the Green func-tion, we Fourier transform the fields as A µ ( ρ, ~x ) = Z d k (2 π ) e i~k~x A µ ( ρ, ~k ) . (3.10)As above, we are free to choose our coordinate system togive us a momentum vector of the fluctuation with nonvanish-ing spatial momentum only in x -direction, ~k = ( ω, q, , .For simplicity we restrict ourselves to vanishing spatial mo-mentum q = 0 . In this case the equations of motion fortransversal fluctuations E y,z match those for longitudinal fluc-tuations E x . For a more detailed discussion see [17]. As anexample consider the equation of motion obtained from (3.6)with σ = 2 , determining E y = ωA , E ′′ + ∂ ρ [ p | det G | G G ] p | det G | G G E ′ − G G ̺ H ω E = E ′′ + 8 w ˜ ff − χ + ρ χ ′ ρ (1 − χ ) E + ∂ ρ ln ρ f (cid:0) − χ (cid:1) q − χ + ρ χ ′ − f (1 − χ )˜ f ( ∂ ρ ˜ A ) E ′ . (3.11)The symbol w denotes the dimensionless frequency w = ω/ (2 πT ) , and we made use of the dimensionless radial co-ordinate ρ .In order to numerically integrate this equation, we deter-mine local solutions of that equation near the horizon ρ = 1 . These can be used to compute initial values in order to inte-grate (3.11) forward towards the boundary. The equation ofmotion (3.11) has coefficients which are singular at the hori-zon. According to standard methods [25], the local solutionof this equation behaves as ( ρ − ρ H ) β , where β is a so-called‘index’ of the differential equation. We compute the possibleindices to be β = ± i w . (3.12)Only the negative one will be retained in the following, sinceit casts the solutions into the physically relevant incomingwaves at the horizon and therefore satisfies the incoming waveboundary condition. The solution E can be split into two fac-tors, which are ( ρ − − i w and some function F ( ρ ) , which isregular at the horizon. The first coefficients of a series expan-sion of F ( ρ ) can be found recursively as described in [15, 16].At the horizon the local solution then reads E ( ρ ) = ( ρ − − i w F ( ρ )= ( ρ − − i w (cid:20) i w ρ −
1) + · · · (cid:21) . (3.13)So, F ( ρ ) asymptotically assumes values F ( ρ = 1) = 1 , ∂ ρ F ( ρ ) (cid:12)(cid:12)(cid:12) ρ =1 = i w . (3.14)For the calculation of numbers, we have to specify thebaryon density ˜ d and the mass parameter χ ∼ m q /T to ob-tain the embeddings χ used in (3.11). Then we obtain a so-lution for a given frequency w using initial values (3.13) and(3.14) in the equation of motion (3.11). This eventually givesus the numerical solutions for E ( ρ ) .Spectral functions are then obtained by combining (3.9) and(2.17), R ( ω,
0) = − N f N c T ρ →∞ (cid:18) ρ ∂ ρ E ( ρ ) E ( ρ ) (cid:19) . (3.15)
2. Results for spectral functions
We now discuss the resulting spectral functions at finitebaryon density, and observe crucial qualitative differencescompared to the case of vanishing baryon density. In fig-ures 5 to 8, some examples for the spectral function at fixedbaryon density n B ∝ ˜ d are shown. To emphasize the reso-nance peaks, in some plots we subtract the quantity R = N f N c T π w , (3.16)around which the spectral functions oscillate, cf. figure 9.The graphs are obtained for a value of ˜ d above ˜ d ∗ (given by(1.1)), where the fundamental phase transition does not occur.The different curves in these plots show the spectral functionsfor different quark masses, corresponding to different posi-tions on the solid blue line in the phase diagram shown in fig-ure 1. Regardless whether we chose ˜ d to be below or above thecritical value of ˜ d , we observe the following behavior of the - - - PSfrag replacements w R ( w , ) − R ˜ d = 0 . χ = 0 . χ = 0 . χ = 0 . χ = 0 . FIG. 5: The finite temperature part of the spectral function R − R (inunits of N f N c T / ) at finite baryon density ˜ d . The maximum growsand shifts to smaller frequencies as χ is increased towards χ =0 . , but then turns around to approach larger frequency values. - PSfrag replacements w R ( w , ) − R ˜ d = 0 . χ = 0 . χ = 0 . χ = 0 . FIG. 6: The finite temperature part of the spectral function R − R (in units of N f N c T / ) at finite baryon density ˜ d . In the regimeof χ shown here, the peak shifts to larger frequency values withincreasing χ . spectral functions with respect to changes in the quark massto temperature ratio.Increasing the quark mass from zero to small finite valuesresults in more and more pronounced peaks of the spectralfunctions. This eventually leads to the formation of resonancepeaks in the spectrum. At small masses, though, there are nonarrow peaks. Only some maxima in the spectral functionsare visible. At the same time as these maxima evolve intoresonances with increasing quark mass, their position changesand moves to lower freqencies w , see figure 5. This behaviorwas also observed for the case of vanishing baryon density in[17].However, further increasing the quark mass leads to a cru-cial difference to the case of vanishing baryon density. Abovea value m turn of the quark mass, parametrized by χ turn , thepeaks change their direction of motion and move to larger val-ues of w , see figure 6. Still the maxima evolve into more andmore distinct peaks. - - PSfrag replacements w R ( w , ) − R ˜ d = 0 . χ = 0 . n = 0 n = 1 n = 2 n = 3 FIG. 7: The finite temperature part R − R of the spectral func-tion (in units of N f N c T / ) at finite baryon density ˜ d . The oscilla-tion peaks narrow and get more pronounced compared to smaller χ .Dashed vertical lines show the meson mass spectrum given by equa-tion (3.17). Eventually at very large quark masses, given by χ closerand closer to 1, the positions of the peaks asymptotically reachexactly those frequencies which correspond to the masses ofthe vector mesons at zero temperature [19]. In our coordi-nates, these masses are given by M = L ∞ R p n + 1)( n + 2) , (3.17)where n labels the Kaluza-Klein modes arising from the D7-brane wrapping S , and L ∞ is the radial distance in (8 , -direction between the stack of D3-branes and the D7, evalu-ated at the AdS -boundary, L ∞ = lim ̺ →∞ ̺χ ( ̺ ) . (3.18)The formation of a line-like spectrum can be interpreted asthe evolution of highly unstable quasi-particle excitations inthe plasma into quark bound states, finally turning into nearlystable vector mesons, cf. figures 7 and 8.We now consider the turning behavior of the resonancepeaks shown in figures 5 and 6. There are two different sce-narios, depending on whether the quark mass is small or large.First, when the quark mass is very small m q ≪ T , we arein the regime of the phase diagram corresponding to the righthalf of figure 1. In this regime the influence of the Minkowskiphase is negligible, as we are deeply inside the black holephase. We therefore observe only broad structures in the spec-tral functions, instead of peaks.Second, when the quark mass is very large, m q ≫ T , orequivalently the temperature is very small, the quarks behavejust as they would at zero temperature, forming a line-likespectrum. This regime corresponds to the left side of the phasediagram in figure 1, where all curves of constant ˜ d asymptoteto the Minkowski phase.The turning of the resonance peaks is associated to being inthe first or in the second regime. At χ turn the two regimes areconnected and none of them is dominant.0 PSfrag replacements w R ( w , ) ˜ d = 0 . χ = 0 . n = 0 n = 1 n = 2 n = 3 FIG. 8: The spectral function R (in units of N f N c T / ) at finitebaryon density ˜ d . At large χ , as here, the peaks approach the dasheddrawn line spectrum given by (3.17). The turning behavior is best understood by following a lineof constant density ˜ d in the phase diagram of figure 1. Con-sider for instance the solid blue line in figure 1, starting atlarge temperatures/small masses on the right of the plot. First,we are deep in the unshaded region ( n B = 0 ), far inside theblack hole phase. Moving along to lower T / ¯ M , the solid blueline in figure 1 rapidly bends upwards, and asymptotes to boththe line corresponding to the onset of the fundamental phasetransition, as well as to the separation line between black holeand Minkowski phase (gray region).This may be interpreted as the quarks joining in boundstates. Increasing the mass further, quarks form almost sta-ble mesons, which give rise to resonance peaks at larger fre-quency if the quark mass is increased. The confined and de-confined phase are coexistent asymptotically for T / ¯ M → .We also observe a dependence of χ turn on the baryon den-sity. As the baryon density is increased from zero, the valueof χ turn decreases.Figures 8 and 9 show that higher n excitations from theKaluza-Klein tower are less stable. While the first resonancepeaks in this plot are very narrow, the following peaks show abroadening with decreasing amplitude.This broadening of the resonances is due to the behaviourof the quasinormal modes of the fluctuations, which corre-spond to the poles of the correlators in the complex ω plane,as described in the example (2.19) and sketched in figure 10.The location of the resonance peaks on the real frequency axiscorresponds to the real part of the quasinormal modes. It is aknown fact that the the quasinormal modes develop a largerreal and imaginary part at higher n . So the sharp resonancesat low w , which correspond to quasiparticles of long lifetime,originate from poles whith small imaginary part. For higherexcitations in n at larger w , the resonances broaden and getdamped due to larger imaginary parts of the correspondingquasi normal modes.For increasing mass we described above that the peaks ofthe spectral functions first move to smaller frequencies un-til they reach the turning point m turn . Further increasing themass leads to the peaks moving to larger frequencies, asymp- PSfrag replacements w R R R ( w , ˜ d = 0 . χ = 0 . FIG. 9: The thermal spectral function R (in units of N f N c T / )compared to the zero temperature result R .FIG. 10: Qualitative relation between the location of the poles inthe complex frequency plane and the shape of the spectral functionson the real ω axis. The function plotted here is an example for theimaginary part of a correlator. Its value on the real ω axis representsthe spectral function. The poles in the right plot are closer to the realaxis and therefore there is more structure in the spectral function. totically approaching the line spectrum. This behavior can betranslated into a movement of the quasinormal modes in thecomplex plane. It would be interesting to compare our resultsto a direct calculation of the quasinormal modes of vector fluc-tuations in analogy to [6].In [6] the quasinormal modes are considered for scalar fluc-tuations exclusively, at vanishing baryon density. The au-thors observe that starting from the massless case, the realpart of the quasinormal frequencies increases with the quarkmass first, and then turns around to decrease. This behavioragrees with the peak movement for scalar spectral functionsobserved in [17, figure 9] (above the fundamental phase tran-sition, χ ≤ . ) where the scalar meson resonances moveto higher frequency first, turn around and move to smaller fre-quency increasing the mass further. These results do not con-tradict the present work since we consider vector modes ex-clusively. The vector meson spectra considered in [17] at van-ishing baryon density only show peaks moving to smaller fre-quency as the quark mass is increased. Note that the authorsthere continue to consider black hole embeddings below thefundamental phase transition which are only metastable, theMinkowski embeddings being thermodynamically favored.At small baryon density and small quark mass our spectraare virtually coincident with those of [17]. In our case, atfinite baryon density, black hole embeddings are favored for1all values of the mass over temperature ratio. At small val-ues of T / ¯ M in the phase diagram of figure 1, we are veryclose to the Minkowski regime, temperature effects are small,and the meson mass is proportional to the quark mass as inthe supersymmetric case. Therefore, the peaks in the spectralfunction move to the right (higher frequencies) as function ofincreasing quark mass.The turning point in the location of the peaks is a con-sequence of the transition between two regimes, i.e. thetemperature-dominated one also observed in [17], and thepotential-dominated one which asymptotes to the supersym-metric spectrum.We expect the physical interpretation of the left-moving ofthe peaks in the temperature-dominated regime to be relatedto the strong dissipative effects present in this case. This isconsistent with the large baryon diffusion coefficient presentin this regime as discussed in section III A and shown in fig-ure 4. A detailed understanding of the physical picture in thisregime requires a quantitative study of the quasipaticle behav-ior which we leave to future work.In our approach it is straightforward to investigate the T → limit since black hole embeddings are thermodynamicallyfavored even near T = 0 at finite baryon density. We expectthat a right-moving of the peaks consistent with the SUSYspectrum should also be observable for Minkowski embed-dings at vanishing baryon density. However this has not beeninvestigated for vector modes neither in [6] nor in [17]. IV. SPECTRAL FUNCTIONS AT FINITE ISOSPINDENSITYA. Radially varying SU (2) -background gauge field In order to examine the case N f = 2 in the strongly cou-pled plasma, we extend our previous analysis of vector mesonspectral functions to a chemical potential with SU (2) -flavor(isospin) structure. Starting from the general action S iso = − T r T D Z d ξ q | det( g + ˆ F ) | , (4.1)we now consider field strength tensors ˆ F µν = σ a (cid:18) ∂ [ µ ˆ A aν ] + ̺ H πα ′ f abc ˆ A bµ ˆ A cν (cid:19) , (4.2)with the Pauli matrices σ a and ˆ A given by equation (3.2). Thefactor ̺ H / (2 πα ′ ) is due to the introduction of dimensionlessfields as described below (2.7). In order to obtain a finiteisospin-charge density n I and its conjugate chemical poten-tial µ I , we introduce an SU (2) -background gauge field ˜ A [22] ˜ A σ = ˜ A ( ρ ) (cid:18) − (cid:19) . (4.3)This specific choice of the 3-direction in flavor space as wellas spacetime dependence simplifies the isospin background field strength, such that we get two copies of the baryonicbackground ˜ F ρ on the diagonal of the flavor matrix, ˜ F ρ σ = (cid:18) ∂ ρ ˜ A − ∂ ρ ˜ A (cid:19) . (4.4)The action for the isospin background differs from the ac-tion (2.9) for the baryonic background only by a group the-oretical factor: The factor T r = 1 / (compare (4.1)) replacesthe baryonic factor N f in equation (2.8), which arises by sum-mation over the U (1) representations. We can thus use theembeddings χ ( ρ ) and background field solutions ˜ A ( ρ ) ofthe baryonic case of [8], listed here in section II A. As be-fore, we collect the induced metric g and the background fieldstrength ˜ F in the background tensor G = g + ˜ F .We apply the background field method in analogy to thebaryonic case examined in section III. As before, we obtainthe quadratic action by expanding the determinant and squareroot in fluctuations A aµ . The term linear in fluctuations againvanishes by the equation of motion for our background field.This leaves the quadratic action S (2) iso = ̺ H (2 π R ) T D T r ∞ Z d ρ d x p | det G |× h G µµ ′ G νν ′ (cid:16) ∂ [ µ A aν ] ∂ [ µ ′ A aν ′ ] + ̺ H (2 πα ′ ) ( ˜ A ) f ab f ab ′ A b [ µ δ ν ]0 A b ′ [ µ ′ δ ν ′ ]0 (cid:17) + ( G µµ ′ G νν ′ − G µ ′ µ G ν ′ ν ) ̺ H πα ′ ˜ A f ab ∂ [ µ ′ A aν ′ ] A b [ µ δ ν ]0 i . (4.5)Note that besides the familiar Maxwell term, two other termsappear, which are due to the non-Abelian structure. One ofthe new terms depends linearly, the other quadratically on thebackground gauge field ˜ A and both contribute nontrivially tothe dynamics. The equation of motion for gauge field fluctua-tions on the D7-brane is ∂ κ hp | det G | ( G νκ G σµ − G νσ G κµ ) ˇ F aµν i (4.6) − p | det G | ̺ H πα ′ ˜ A f ab (cid:0) G ν G σµ − G νσ G µ (cid:1) ˇ F bµν , with the modified field strength linear in fluctuations ˇ F aµν =2 ∂ [ µ A aν ] + f ab ˜ A ( δ µ A bν + δ ν A bµ ) ̺ H / (2 πα ′ ) .Integration by parts of (4.5) and application of (4.6) yieldsthe on-shell action S on-shelliso = ̺ H T r T D π R Z d x p | det G |× (cid:16) G ν G ν ′ µ − G νν ′ G µ (cid:17) A aν ′ ˇ F aµν (cid:12)(cid:12)(cid:12) ρ B ρ H . (4.7)The three flavor field equations of motion (flavor index a =1 , , ) for fluctuations in transversal Lorentz-directions α =2 , can again be written in terms of the combination E aT = qA a + ωA aα . At vanishing spatial momentum q = 0 we get E T ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G E T ′ (4.8) − G G " ( ̺ H ω ) + (cid:18) ̺ H πα ′ ˜ A (cid:19) E T + 2 i̺ H ωG G ̺ H πα ′ ˜ A E T , E T ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G E T ′ (4.9) − G G " ( ̺ H ω ) + (cid:18) ̺ H πα ′ ˜ A (cid:19) E T − i̺ H ωG G ̺ H πα ′ ˜ A E T , E T ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G E T ′ − G ( ̺ H ω ) G E T . (4.10)Note that we use the dimensionless background gaugefield ˜ A = ¯ A (2 πα ′ ) /̺ H and ̺ H = πT R . Despite thepresence of the new non-Abelian terms, at vanishing spatialmomentum the equations of motion for longitudinal fluctua-tions are the same as the transversal equations (4.8), (4.9) and(4.10), such that E = E T = E L .Note at this point that there are two essential differenceswhich distinguish this setup from the approach with a constantpotential ¯ A at vanishing mass followed in [22]. First, the in-verse metric coefficients g µν contain the embedding function χ ( ρ ) computed with varying background gauge field. Second,the background gauge field ¯ A giving rise to the chemical po-tential now depends on ρ .Two of the ordinary second order differential equa-tions (4.8), (4.9), (4.10) are coupled through their flavor struc-ture. Decoupling can be achieved by transformation to theflavor combinations [22] X = E + iE , Y = E − iE . (4.11)The equations of motion for these fields are given by X ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G X ′ (4.12) − ̺ H R G G ( w − m ) X , Y ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G Y ′ (4.13) − ̺ H R G G ( w + m ) Y , E ′′ + ∂ ρ ( p | det G | G G ) p | det G | G G E ′ − ̺ H R G G w E , (4.14) with dimensionless m = ¯ A / (2 πT ) and w = ω/ (2 πT ) . Pro-ceeding as described in section III, we determine the local so-lution of (4.12), (4.13) and (4.14) at the horizon. The indicesturn out to be β = ± i (cid:20) w ∓ ¯ A ( ρ = 1)(2 πT ) (cid:21) . (4.15)Since ¯ A ( ρ = 1) = 0 in the setup considered here, we are leftwith the same index as in (3.12) for the baryon case. There-fore, here the chemical potential does not influence the sin-gular behavior of the fluctuations at the horizon. The localsolution coincides to linear order with the baryonic solutiongiven in (3.13).Application of the recipe described in section II B yieldsthe spectral functions of flavor current correlators shown infigures 11 and 12. Note that after transforming to flavor com-binations X and Y , given in (4.11), the diagonal elementsof the propagation submatrix in flavor-transverse X, Y direc-tions vanish, G XX = G Y Y = 0 , while the off-diagonal el-ements give non-vanishing contributions. However, the com-ponent E , longitudinal in flavor space, is not influenced bythe isospin chemical potential, such that G E E is nonzero,while other combinations with E vanish (see [22] for de-tails).Introducing the chemical potential as described above fora zero-temperature AdS × S background, we obtain thegauge field correlators in analogy to [26]. The resulting spec-tral function for the field theory at zero temperature but finitechemical potential and density R , iso is given by R , iso = N c T T r π ( w ± m ∞ ) , (4.16)with the dimensionless chemical potential m ∞ =lim ρ →∞ ¯ A / (2 πT ) = µ/ (2 πT ) . Note that (4.16) is in-dependent of the temperature. This part is always subtractedwhen we consider spectral functions at finite temperature, inorder to determine the effect of finite temperature separately,as we did in the baryonic case. B. Results at finite isospin density
In figure 11 we compare typical spectral functions foundfor the isospin case (solid lines) with that found in the bary-onic case (dashed line). While the qualitative behavior of theisospin spectral functions agrees with the one of the bary-onic spectral functions, there nevertheless is a quantitative dif-ference for the components
X, Y , which are transversal tothe background in flavor space. We find that the propagatorfor flavor combinations G Y X exhibits a spectral function forwhich the zeroes as well as the peaks are shifted to higher fre-quencies, compared to the Abelian case curve. For the spec-tral function computed from G XY , the opposite is true. Itszeroes and peaks appear at lower frequencies. As seen fromfigure 12, also the quasiparticle resonances of these two dif-ferent flavor correlations show distinct behavior. The quasi-particle resonance peak in the spectral function R Y X appears3
PSfrag replacements . . .
51 1 2 − − XY Y X R − R w FIG. 11: The finite temperature part of spectral functions R iso − R , iso (in units of N c T T r / ) of currents dual to fields X, Y areshown versus w . The dashed line shows the baryonic chemical po-tential case, the solid curves show the spectral functions in presenceof an isospin chemical potential. Plots are generated for χ = 0 . and ˜ d = 0 . . The combinations XY and Y X split in oppositedirections from the baryonic spectral function.PSfrag replacements
01 2 4 6 8 101000200030004000
XYXY Y XY X E E E E n = 0 n = 0 n = 0 n = 1 n = 1 n = 1 R − R w FIG. 12: A comparison between the finite temperature part of thespectral functions R XY and R Y X (solid lines) in the two flavor di-rections X and Y transversal to the chemical potential is shown inunits of N c T T r / for large quark mass to temperature ratio χ =0 . and ˜ d = 0 . . The spectral function R E E along the a = 3 -flavor direction is shown as a dashed line. We observe a splitting ofthe line expected at the lowest meson mass at w = 4 . ( n = 0 ).The resonance is shifted to lower frequencies for R XY and to higherones for R Y X , while it remains in place for R E E . The secondmeson resonance peak ( n = 1 ) shows a similar behavior. So thedifferent flavor combinations propagate differently and have distinctquasiparticle resonances. at higher frequencies than expected from the vector mesonmass formula (1.2) (shown as dashed grey vertical lines in fig-ure 12). The other flavor-transversal spectral function R XY displays a resonance at lower frequency than observed in thebaryonic curve. The spectral function for the third flavor di-rection R E E behaves as E in the baryonic case.This may be viewed as a splitting of the resonance peak intothree distinct peaks with equal amplitudes. This is due to thefact that we explicitly break the symmetry in flavor space byour choice of the background field ˜ A . Decreasing the chem-ical potential reduces the distance of the two outer resonancepeaks from the one in the middle and therefore the splitting is reduced.The described behavior resembles the mass splitting ofmesons in presence of a isospin chemical potential expected tooccur in QCD [27, 28]. A linear dependence of the separationof the peaks on the chemical potential is expected. Our ob-servations confirm this behavior. Since our vector mesons areisospin triplets and we break the isospin symmetry explicitly,we see that in this respect our model is in qualitative agree-ment with effective QCD models. Note also the complemen-tary discussion of this point in [29].To conclude this section, we comment on the relation ofthe present results to those of our previous paper [22] wherewe considered a constant non-Abelian gauge field backgroundfor zero quark mass. From equation (4.15), the difference be-tween a constant non-vanishing background gauge field andthe varying one becomes clear. In [22] the field is chosento be constant in ρ and terms quadratic in the backgroundgauge field ˜ A ≪ are neglected. This implies that thesquare ( w ∓ m ) in (4.12) and (4.13) is replaced by w ∓ wm ,such that we obtain the indices β = ± w q ∓ ¯ A ( ρ =1)(2 πT ) w in-stead of (4.15). If we additionally assume w ≪ ˜ A , then the under the square root can be neglected. In this case the spec-tral function develops a non-analytic structure coming fromthe √ ω factor in the index.However in the case considered here, the background gaugefield is a non-constant function of ρ which vanishes at the hori-zon. Therefore the indices have the usual form β = ± iω from(4.15), and there is no non-analytic behavior of the spectralfunctions, at least none originating from the indices.It will also be interesting to consider isospin diffusion inthe setup of the present paper. However, in order to see non-Abelian effects in the diffusion coefficient, we need to givethe background gauge field a more general direction in fla-vor space or a dependence on further space-time coordinatesbesides ρ . In that case, we will have a non-Abelian termin the background field strength ˜ F µν = ∂ µ ˜ A aν − ∂ ν ˜ A aµ + f abc ˜ A bµ ˜ A cν ̺ H / (2 πα ′ ) in contrast to ∂ ρ ˜ A a considered here. V. CONCLUSION
Two distinct setups were examined here at non-zero chargedensity in the black hole phase. First, switching on a baryonchemical potential at non-zero baryon density, we find thatnearly stable vector mesons exist close to the transition lineto the Minkowski phase. Far from this line, at small quarkmasses, we essentially recover the picture given in the case ofvanishing chemical potential [17]. Increasing the quark massbeyond a distinct value, the plasma changes its behavior in or-der to asymptotically behave as it would at zero temperature.In the spectral functions we computed, this zero-temperature-like behavior is found in form of line-like resonances, ex-actly reproducing the zero-temperature supersymmetric vec-tor meson mass spectrum. A turning point at m = m turn ,where m = ¯ M /T , is observed: Below m turn , the resonancepeaks move to lower frequencies as function of rising quarkmass. This behavior of the system resembles the behav-4ior known of that system without a chemical potential [17].Above m turn , the resonance peaks move to higher frequenciesas function of the quark mass. This is the zero-temperature-like regime. Moreover, an examination of the diffusion coeffi-cient reveals that the phase transition separating two differentblack hole phases [8] is shifted towards smaller temperatureas the baryon density is increased.Second, we switched on a nonzero isospin density, andequivalently an isospin chemical potential arises. The spec-tral functions in this case show a qualitatively similar be-havior as those for baryonic potential. However, we ad-ditionally observe a splitting of the single resonance peakat vanishing isospin potential into three distinct resonances.This suggests that by explicitly breaking the flavor symme-try by a chemical potential, the isospin triplet states, vectormesons in our case, show a mass splitting similar to that ob-served for QCD [27]. It is an interesting task to explore thefeatures of this isospin theory in greater detail in order tocompare with available lattice data and effective QCD mod-els [30, 31, 32, 33, 34, 35, 36, 37, 38]. In most of theseapproaches, baryon and isospin chemical potential are con-sidered at the same time, which suggests another promisingextension of this work. Moreover, in the context of gravity du-als, it will be interesting to compare our results for the isospinchemical potential to the recent work [29].Alternatively, instead of giving the gauge field time compo-nent a non-vanishing vev, one may also switch on B-field com-ponents and connect the framework developed in [39, 40, 41]with the calculation of spectral functions for the dual gaugetheory. Acknowledgments
We are grateful to P. Kerner, C. Greubel, K. Landsteiner,D. Mateos, G. Policastro, A. Starinets, L. Yaffe and M. Za- germann for useful discussions and correspondence, as wellas to R. Myers for suggesting to consider a ρ -dependent A component in the isospin case.Part of this work was funded by the Cluster of Excellencefor Fundamental Physics – Origin and Structure of the Uni-verse . APPENDIX A: NOTATION
The five-dimensional
AdS
Schwarzschild black hole spacein which we work is endowed with a metric of signature ( − , + , + , + , +) , as given explicitly in (2.1). We make useof the Einstein notation to indicate sums over Lorentz indices,and additionally simply sum over non-Lorentz indices, suchas gauge group indices, whenever they occur twice in a term.To distinguish between vectors in different dimensions ofthe AdS space, we use bold symbols like q for vectors in thethe three spatial dimensions which do not live along the radial AdS coordinate.
Four-vectors which do not have componentsalong the radial
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