Meson Life Time in the Anisotropic Quark-Gluon Plasma
PPrepared for submission to JHEP
Meson Life Time in the Anisotropic Quark-GluonPlasma
Mohammad Ali-Akbari a,b
Davood Allahbakhshi b a Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran. b School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395-5531, Tehran, Iran
E-mail: [email protected] , [email protected] Abstract:
In the hot (an)isotropic plasma the meson life time τ is defined as a timescale after which the meson dissociates. According to the gauge/gravity duality, this timecan be identified with the inverse of the imaginary part of the frequency of the quasinormalmodes, ω I , in the (an)isotropic black hole background. In the high temperature limit, wenumerically show that at fixed temperature(entropy density) the life time of the mesonsdecreases(increases) as the anisotropy parameter raises. For general case, at fixed tem-perature we introduce a polynomial function for ω I and observe that the meson life timedecreases. Moreover, we realize that ( s/T ) , where s and T are entropy density and tem-perature of the plasma respectively, can be expressed as a function of anisotropy parameterover temperature. Interestingly, this function is a Pad´e approximant . a r X i v : . [ h e p - t h ] A ug ontents Contents1 Introduction
A new phase of quantum chromodynamics, quark-gluon plasma (QGP), is produced atrelativistic heavy ion collider (RHIC) or these days at large hardon collider (LHC) bycolliding two heavy nuclei such as gold (Au) or lead (Pb), relativistically. Experimentalobservations imply that the plasma is strongly coupled [1] and hence the perturbativecalculation is not trustworthy. Therefore non-perturbative methods such as gauge/gravityduality may be applied to explain various properties of the plasma.The gauge/gravity duality claims that for certain strongly coupled gauge theories thedynamics of the quantum fields can be described by the dynamics of the classical fieldsliving in a higher dimensional space-time [2]. In particular, N = 4 super Yang-Mills theory(SYM) in the limit of large colors N and large but finite t’Hooft coupling λ , which isexpected to behave in a similar way with the strongly coupled QGP, is dual to type IIbsupergravity on AdS × S background [3]. Similarly a thermal SYM theory corresponds tothe supergravity in an AdS-Shwarzschild background where the temperature of the SYMtheory is identified with the Hawking temperature of AdS black hole [4]. Moreover Mateosand Trancanelli have introduced an interesting generalization of this duality to the thermaland spatially anisotropic SYM theory [5, 6].In order to add matter (quark) in the fundamental representation of the correspondinggauge group, one needs to introduce a D-brane into the background in the probe limit[7]. The probe limit means that D-brane does not back-react the geometry. Then theasymptotic shape of the brane gives the mass and condensation of the matter field. Inaddition, the shape of the brane can be classified into two types, one is the Mikowskiembedding (ME) and the other is black hole embedding (BE). While the ME does not seethe horizon, the BE crosses it. Various aspects of these embeddings have been studied inthe literature, for instance see [8].The results reported in [9] show that the mesons living in the QGP can be describedby quasinormal modes. They are considered as certain small fluctuations around the BE– 1 –ith a complex frequency. Therefore, they are unstable modes where the imaginary partof their frequencies is identified with the inverse of the meson life time. The question wewould like to answer in this paper is how the anisotropy affects the mass of the meson andits life time. The background we are interested in is an anisotropic solution of the IIb supergravityequations of motion. This solution in the string frame is given by [6] ds = −F B u − dt + u − ( dx + dy ) + H u − dz + F − u − du + e φ d Ω ,d Ω = dθ + sin θd Ω + cos θdϕ ,χ = az, φ = φ ( u ) , (2.1)where a is a constant. χ and φ are axion and dilaton fields, respectively. H , F and B depend only on the radial direction, u . In terms of the dilaton field, they are H = e − φ , (2.2a) F = e − φ (cid:104) a e φ (4 u + u φ (cid:48) ) + 16 φ (cid:48) (cid:105) φ (cid:48) + uφ (cid:48)(cid:48) ) , (2.2b) B (cid:48) B = 124 + 10 uφ (cid:48) (cid:0) φ (cid:48) − uφ (cid:48) + 20 uφ (cid:48)(cid:48) (cid:1) , (2.2c)In order to find the solution one needs to solve the equation of motion for dilaton field.Then the above equations for metric components and suitable boundary conditions willspecify the solution. For more detail see [6]. Note also that the solution also contains aself dual five-form field.The function F ( u ) in the temporal and radial components of the metric is the black-ening factor. Therefore the horizon is located at u = u h where F ( u h ) = 0 and the Hawkingtemperature is given by T = − π F (cid:48) ( u h ) (cid:112) B ( u h ). The boundary lies at u = 0 and themetric approaches AdS × S asymptotically. The coordinates of the spacetime where thegauge theory lives are ( t, x, y, z ) where there is a U (1) symmetry in the xy -plane. We call x and y the transverse directions and the longitudinal direction is z . An anisotropy isclearly seen between the transverse and longitudinal directions. The entropy density perunit volume in the xyz -directions is given by s = π N e − φ ( u h ) π u h . (2.3)In order to add the fundamental matter to the SU ( N ) gauge theory we have to intro-duce a D7-brane into the anisotropic background in the probe limit. The probe limit meansthat the D7-brane does not modify the geometry. Flavour D7-branes in this backgroundhave been studied previously, for example see [10]. In fact the open strings stretched be-tween probe D7-brane and the D3-D7 system leading to the geometry (2.1) give rise to the– 2 –atter in the fundamental representation of the gauge group. The dynamics of the openstrings is described by the DBI action S DBI = − τ (cid:90) d ξ e − φ (cid:112) det( G ab + 2 πα (cid:48) F ab ) . (2.4)The D7-brane tension is τ where τ − = (2 π ) l s g s and G ab = g MN ∂ a X M ∂ b X N where inthe large N and t’ Hooft coupling limits the D3-D7 system is replaced with g MN given by(2.1). The D7-brane is extended along t, x, y, z, u and wrapped around S ⊂ S . Althoughthe four-form and the axion fields are non-zero in the background, in such an embeddingthe Chern-Simon action has no contribution to the action. The shape of the brane is givenby the transverse directions θ and ϕ where we choose ϕ to be zero. Since we do not like tostudy the effect of the gauge field living on the brane, we also set A a to be zero. Because ofthe translational symmetry of the metric components in xyz directions and the rotationalsymmetry in Ω directions, we consider that θ depends on the radial direction and time asit is shown in (2.6). Therefore, the Lagrangian reduces to L = e − φ ( u ) cos θ ( u, t ) u √F (2.5) × (cid:113) Z H [ BF (1 + u F Z θ (cid:48) ( u, t ) ) − u Z ˙ θ ( u, t ) ] . The physical parameters we are interested in can be found from the asymptotic solutionto θ ( u ) equation of motion, θ c ( u ) = θ u + θ u + . . . [11], where m = θ πα (cid:48) is the mass ofthe fundamental matter and c = θ − θ corresponds to condensation that is proportionalto (cid:104) ¯ ψψ (cid:105) .It is well known that the small fluctuations about the shape (the equilibrium configu-ration) of the probe branes represent the low spin mesons [9]. They are classified into twotypes according to their frequencies. In the MEs the normal modes, which are the fluctua-tions with discrete real frequencies, only exist. However, in the case of the BH embeddings,the fluctuations fall into the black hole and the corresponding frequencies, the so-calledquasinormal modes, are complex. Applying the AdS/CFT corresponding, the meson willbe dissociated in the QGP after the life time, which is given by the inverse of the imaginarypart of the frequency i.e. τ ∝ ω − I [9]. In order to find the meson life time τ , let us startwith the following ansatz θ ( u, t ) = θ c ( u ) + (cid:15) e iωt ζ ( u ) , (2.6)where θ c ( u ) is a time-independent solution of the equation of motion for θ ( u, t ) resultingfrom (2.5). Substituting the above ansatz into the equation of motion for θ ( u, t ) andexpanding it up to the first order in (cid:15) , one finds a nonlinear equation for θ c ( u ) and a linearised equation for ζ ( u ). The suitable boundary conditions to solve the nonlinearequation are θ h = θ c ( u h ) and θ (cid:48) c ( u h ). The latter is fixed in terms of θ h by using theequation of motion for θ c ( u ). Therefore, we have a one parameter family of solutions forthe background profile of the brane θ c ( u ).In order to find the quasinormal modes, one needs to solve the linear equation of motionfor the ζ ( u ) by applying the following boundary conditions: modes which are ingoing at– 3 –he horizon and have zero source term at the boundary. The analytic solutions to the nearhorizon equation for ζ ( u ) are ζ ( u ) ≈ e ± i ωT Log (1 − u/u h ) , (2.7)where the +(-) sign corresponds to the ingoing(outgoing) modes. On the other hand thenear boundary equation can be analytically solved and the solution is ζ ( u ) = ζ u + ζ u + ... . (2.8)To find the quasinormal modes we have to force the source term, ζ , to equal zero orequivalently ζ (cid:48) ( u ) | u =0 = 0. Considering the field redefinition ζ ( u ) = e + i ωT Log (1 − u/u h ) ψ ( u ) , (2.9)one can see that ψ ( u ) has the regular expansion ψ ( u ) = ψ + ψ ( u − u h ) + ψ ( u − u h ) + ... , (2.10)near the horizon. Since the equation for ψ is linear, ψ can be set to 1 and the othercoefficients will be determined from the equation of motion for ψ ( u ). Interpolating betweentwo asymptotic solutions (2.7) and (2.8) is possible only by a set of discrete complex valuesof ω which can be found by some standard methods such as shooting method. We wouldlike to emphasize that the meson in its ground state, corresponding to the first quasinormalmode, is considered in this paper. Fortunately in the high temperature limit, T (cid:29) a , the anisotropic solution has been ana-lytically introduced in [6]. In this limit, up to leading order in a , the functions F , B andthe dilaton field are given by F = 1 − u u h + a ˆ F ( u ) + ..., (2.11a) B = 1 − a u h (cid:18) u u h + u + log(1 + u u h ) (cid:19) + ..., (2.11b) φ = − a u h u u h ) + ..., (2.11c)where ˆ F ( u ) = 124 u h (cid:18) u ( u h − u ) − u log 2+ (3 u h + 7 u ) log(1 + u u h ) (cid:19) , (2.12)– 4 –he temperature and the entropy density of the solution in terms of the anisotropy param-eter are T = 1 πu h + (5 log 2 − u h π a + O ( a ) , (2.13a) s = 12 N π T + N T a + O ( a ) . (2.13b)On the other hand at low temperature limit, i.e. a (cid:29) T , the entropy density is s = c ent N a / T / + ..., (2.14)where c ent ≈ . ω R,I depends on anisotropy parameter as a for any given value of thetemperature i.e. ω R,I = ω R,I ( T, m ) + α R,I ( T, m ) a , (2.15)where ω R,I ( T, m ) are frequencies of the quasinormal modes for the isotropic case i.e. a = 0.Figure 1a shows that at fixed temperature although α R is almost constant with increasingthe mass of the faundamental matter, α I decreases (a few values for ω R,I are given in table1). It is important to notice that in a region around m = T we expect a first order phasetransition between black hole and Mikowski embeddings [12] and therefore our results arenot reliable in this region. Moreover, we numerically observe that for any given value ofthe mass a raise in the anisotropy parameter will increase ω I . And, in turn, as it is clearlyseen from figure 1b, it means that the τ /τ decreases. Note that τ is the value of mesonlife time at a = 0 for each corresponding mass. As a result the mesons will melt soonerin the QGP. This somehow indicates that anisotropy parameter and temperature behavesimilarly and it is in agreement with results in [13, 14]. We observe that the decrease in τ /τ is almost the same for different masses.In the case of fixed entropy density, the behaviour of the real and imaginary parts ofthe frequency is similar to (2.15) as ω R,I = ω R,I ( sN , m ) + α R,I ( sN , m ) a , (2.16)where its coefficients have been shown in the figure 1a. Compared to the mass dependenceof α R,I ( T, m ) at fixed temperature case, a notable increase can be seen for α R,I ( s/N , m ).Opposite to that seen in the fixed temperature case, raising the anisotropy in the systemwill increase the value of the τ /τ . In this section we are going to compute the real and imaginary parts of the frequencyfor arbitrary values of the temperature and anisotropy parameter. 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At fixed entropy density: α R and − α I , normalized to 1 at m = 0, have been plotted in terms of m . In this plot α R ( s/N = 500 π ,
0) = 0 . α I ( s/N = 500 π ,
0) = − . α R ( T = 10 ,
0) = 0 . α I ( T = 10 ,
0) =0 . a , at fixed entropy density (blue points) and at fixed temperature (red squares).limited range of mass 0 < m < T when a = 0. For instance, at fixed temperature, in thezero mass case we find ω R = 6 . T + δω R , (2.17) ω I = 5 . T + δω I . The two second terms in above equations, δω R and δω I , are a consequence of the anisotropyparameter. The function for the deviations seems complicated but for example in masslesscase δω R and δω I may be approximated by the following polynomials δω R,I = T (cid:88) n =2 c n ( R,I ) ( aT ) n , (2.18) c R ( I ) = 0 . . , c R ( I ) = − . − . ,c R ( I ) = 4(0 . × − , c R ( I ) = − . . × − . Here we would like to emphasize that these functions can be applied in the range of ournumerics ( 0 . < T <
15 and 0 < a <
30 provided that a < T ).One can also calculate the quasinormal modes when the entropy density is kept fixed.However, it is not easy to find suitable functions for δω R,I ( s, m ) which fit our numericalresults. Instead, our data turn out to be fit with the following function s ( a, T ) = π N T (cid:18) α (cid:0) aT (cid:1) + α (cid:0) aT (cid:1) β (cid:0) aT (cid:1) (cid:19) / . (2.19)Using the expansion of the entropy density in the high temperature limit [6], α and β can be obtained in terms of α as α = 48 π α − π , β = 4 π α − π . (2.20)– 6 – T = 10) ≡ ( s/N = 500 π )mass ω R ( T, m ) = ω R ( s/N , m ) ω I ( T, m ) = ω I ( s/N , m )0 68.7739 55.19791 68.7727 55.2072 68.7694 55.23473 68.76 55.28844 68.7538 55.35375 68.7502 55.4349 Table 1 : Isotropic frequencies for fixed values of temperature and entropy density in hightemperature regime.Notice that this function gives the correct expression for the entropy density of N = 4super Yang-Milles theory ( a = 0). The value of α can be found by using the best fit forthe entropy density and is obtained as α = 1 / c ent ≈ .
205 which is in perfect agreement with (2.14).Now (2.19) clearly leads to (2.14) in the low temperature limit. The above discussion maybe generalized by considering higher order terms. As a result we suggest s ( a, T ) = π N T (cid:18) (cid:80) n +1 k =1 α k ( aT ) k (cid:80) nk =1 β k ( aT ) k (cid:19) / , (2.21)which is a [(2 n + 2) / n ] f ( a/T ) Pad´e approximant for f = ( s/T ) . In principle, all coef-ficients can be achieved in terms of α by utilizing the higher order expansion of entropydensity in terms of a [6].At fixed entropy density we found that the effect of anisotropy on the frequencies isvery small (less than 1%). In principle from (2.19), for a fixed value of the entropy densityand given a , the temperature can be found. Inserting the resultant temperature into (2.18),we obtain δω R and δω I . Although it is promising that we can achive the real and imaginaryparts of the frequecy at fixed entropy density, unfortunately the effect of anisotropy on thefrequencies (1%) is less than the error of the polynomials (2.18) (4%) and therefore theerror washes away the effect. Main aim in this paper is to understand the effect of the anisotropy on the life time ofthe mesons living in the plasma. As it was already mentioned, according to gauge/gravityduality, the life time and the mass of the meson are described by ω − I and ω R , respectively.By recalling (2.17), one can calculate the following ratios ττ = ω I (0 , T ) ω I ( a, T ) , M meson M = ω R ( a, T ) ω R (0 , T ) . (3.1)– 7 – meson (cid:144) M t (cid:144) t RHICLHC2 4 6 8 a (cid:144)
T0.850.900.951.051.101.15 (a) aT sN T (b) Figure 2 : Plot (a): The values of M meson /M meson and τ /τ has been plotted versus a/T .The dots are numerical data and solid curves are the fitted polynomials (2.18). Plot (b):The fitted function (2.19) (orange curve) and numerical data for s/T (blue dots).These ratios have been plotted in fig 2a. It was discussed in [15] that a/T forRIHC(LHC) is 5 . . (cid:40) M meson M ≈ . , ττ ≈ . , (3.2)and LHC (T = 450Mev) (cid:40) M meson M ≈ . , ττ ≈ . , (3.3)and therefore the mesons dissociate more in the presence of anisotropy. This conclusion isin agreement with the result reported in [16]. In this paper it was shown that the screeninglength as a function of the anisotropy decreases indicating that the life time of the boundstates become shorter in the anisotropic plasma. Furthermore, at RHIC(LHC) energies anincrease in the mass of the mesons occurs which is about 12(8)%. Since the QGP producedin laboratory is intrinsically anisotropic, one can not measure the mass of the meson livingin the QGP for a = 0. But, interestingly, this mass can be eliminated from our results andwe then have ( M meson ) RHIC ( M meson ) LHC ≈ .
037 (3.4)In other words the effect of anisotropy can experimentally be observed by comparing themass of the meson at RHIC and LHC. In fact at LHC energies, the meson is lighter.
Acknowledgement:
We would like to thank A. Davody and H. Ebrahim for fruitfuldiscussions.
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