Thermal width of heavy quarkonia from an AdS/QCD model
TThermal width of heavy quarkonia from an AdS/QCD model
Nelson R. F. Braga ∗ and Luiz F. Ferreira † Instituto de F´ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, RJ 21941-972 – Brazil
Abstract
We estimate the thermal width of a heavy quark anti-quark pair inside a strongly coupled plasmausing a holographic AdS/QCD model. The imaginary part of the quark potential that producesthe thermal width appears in the gravity dual from quantum fluctuations of the string world sheetin the vicinity of the horizon. The results, obtained using a soft wall background that involves aninfrared mass scale, are consistent with previous analisys where the mass scale was introduced byaveraging over quark anti-quark states. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] O c t . INTRODUCTION Gauge string duality[1–3] provides an important tool to calculate properties of gaugetheores at strong coupling. One quantity of particular interest is the static rectangularWilson loop, that provides the potential energy between an infinitely heavy quark anti-quark pair. In refs. [4, 5] it was proposed that a Wilson loop for a gauge theory (at largenumber of colors and with extended supersymmetry) is dual to a string worldsheet in anti-deSitter space whose boundary is the loop.Static Wilson loops can also be calculated for gauge theories at finite temperature. Theenergy obtained this way can be taken as the heavy quark potential at finite tempera-ture. This potential has in general an imaginary part associated with the thermal decay,as discussed for example in [6–8]. The holographic description of Wilson loops in the finitetemperature case was developed in [9, 10]. In this case the dual geometry is an anti-de sitterblack hole.More recently, the presence of an imaginary part in the quark anti-quark potential wasinvestigated using gauge string duality in, for example, refs. [11–16]. In particular, in refs.[12, 14] the imaginary part of the quark anti-quark potential is used to calculate the thermalwidth of a quarkonium state. The state is represented by a static string in a black holeAdS space with end-points fixed on the boundary. The imaginary part of the potentialcomes from fluctuations of the string near the horizon. The thermal width is calculated asthe expectation value of the imaginary part of the potential in a state of the quarkonium.In this approach of [14] the dual geometry does not contain any dimensionfull parameter(mass scale). The geometry is just a black hole AdS (Poincar´e) space, that is dual to aconformal gauge theory. The mass scale of the quarks enters into the calculation of thethermal width through the introduction of a wave function representing a massive quarksubject to a coulomb like potential.Here we present an alternative holographic approach to determine the thermal width of aquarkonium state. The motivation is that AdS/QCD models, like hard wall [17–19] and softwall [20] and improved holographic QCD [21–23], provide a nice phenomenological descrip-tion of quark anti-quark interaction and other hadronic properties like the mass spectra. Thepotential at zero temperature is linearly confining while at finite temperature they exihibita confinement/deconfinement phase transition both in the hard wall[24, 25] and in the soft2all with positive exponential factor[26]. We will show here that such a phenomenologicaldescrition of quark anti-quark interaction provides also a tool for calculating the thermalwidth.We will consider a quark anti-quark pair in the soft wall model background, that involvesan infrared energy scale associated with the mass. The thermal width will be calculated byjust averaging over the lengths of all possible string worldsheet configurations that generateimaginary contributions to the potential. For completeness, we mention that meson widthshave been calculated in the holographic D7 brane model framework in [27].
II. HOLOGRAPHIC DESCRIPTION OF QUARK ANTI-QUARK POTENTIAL
Following the standard gauge/gravity prescription[4, 5] the expectation value of a staticWilson loop W(C) in a strongly coupled gauge theory that has a gravity dual is representedby the generating functional Z str of a static string in the bulk of the dual space. Theintersection of the string worldsheet with the boundary of the space is the loop C. In thesemi-classical gravity approximation we have Z str ∼ e iS str , (1)where S str is the classical Nambu-Goto action S str = S NG = 12 πα (cid:48) (cid:90) dσdτ (cid:113) det ( G µν ∂ a X µ ∂ b X ν ) , (2)where X µ ( τ, σ ) are the worldsheet embedding coordinates, µ, ν = 0 , , ..., a, b = σ, τ ;1 / πα (cid:48) is the string tension and G µν the spacetime metric that we consider to be of Euclideanform.A systematic analysis of static strings representing Wilson loops was presented in ref.[28], assuming metrics of the general form ds = G ( z ) dt + G (cid:126)x(cid:126)x ( z ) d(cid:126)x + G zz ( z ) dz , (3)where (cid:126)x denotes the usual spatial boundary coordinates while z is the radial direction. Forour case of interest the boundary is assumed to be at z →
0. Choosing the world sheetcoordinates σ = x and τ = t and assuming translation invariance along t , the string actionwith endpoints fixed at x = ± L/ S NG = T πα (cid:48) (cid:90) L/ − L/ dx (cid:113) M ( z ( x ))( z (cid:48) ( x )) + V ( z ( x )) (4)3here M ( z ) ≡ G G zz (5) V ( z ) ≡ G G xx (6)The string profile z ( x ) can be determined by considering expression (4) as representing an“action integral” for the evolution in coordinate x. The corresponding lagrangian density is L ( z, z (cid:48) ) = 12 πα (cid:48) (cid:113) M ( z ) z (cid:48) + V ( z ) (7)with conjugate momentum: p = ∂ L ∂z (cid:48) = 12 πα (cid:48) M ( z ) z (cid:48) (cid:113) M ( z ) z (cid:48) + V ( z ) , (8)and Hamiltonian H ( z, p ) = p · z (cid:48) − L ( z, z (cid:48) ( z, p )) = 12 πα (cid:48) − V ( z ) (cid:113) M ( z ) z (cid:48) + V ( z ) = 1(2 πα (cid:48) ) − V ( z ) L . (9)This quantity is a constant of motion, for evolution in x , that can be conveniently evaluatedat the maximum value of coordinate z : z ∗ = z (0) where z (cid:48) (0) = 0 leading to H ( z ∗ ,
0) = − πα (cid:48) (cid:113) V ( z ∗ ) . (10)So, one can express the Lagrangian as L = V ( z )2 πα (cid:48) (cid:113) V ( z ∗ ) , (11)and get the differential equation for the string profile: dzdx = ± (cid:113) V ( z ) (cid:113) M ( z ) (cid:113) V ( z ) − V ( z ∗ ) (cid:113) V ( z ∗ ) . (12)The distance between the infinitely massive ’quarks’ is then: L = (cid:90) dx = (cid:90) (cid:32) dzdx (cid:33) − dz = 2 (cid:90) z ∗ (cid:113) M ( z ) (cid:113) V ( z ) (cid:113) M ( z ∗ ) (cid:113) V ( z ) − V ( z ∗ ) dz . (13)The on shell action of the static string takes the form: S on shellNG = T πα (cid:48) (cid:90) z ∗ (cid:113) M ( z ) (cid:113) V ( z ) V ( z ) (cid:113) V ( z ) − V ( z ∗ ) dz . (14)4he real part of the potential is obtained as the limit: Re V Q ¯ Q = lim T →∞ S on shellNG T = T πα (cid:48) (cid:90) z ∗ (cid:113) M ( z ) (cid:113) V ( z ) V ( z ) (cid:113) V ( z ) − V ( z ∗ ) dz . (15)This expression is singular and is regularized by the subtraction of the quark masses: m Q = 12 πα (cid:48) (cid:90) ∞ (cid:113) M ( z ) dz . (16)The regularized form of the real part of the potential is: Re V regQ ¯ Q = 1 πα (cid:48) (cid:90) z ∗ (cid:113) M ( z ) (cid:113) V ( z ) V ( z ) (cid:113) V ( z ) − V ( z ∗ ) dz − πα (cid:48) (cid:90) ∞ (cid:113) M ( z ) dz . (17)Now we discuss the imaginary part of the potential. We follow at this point ref. [14] whereone calculates the fluctuations of the string that cross the horizon leading to imaginarycontributions to the energy. The calculations of this reference were performed using theradial coordinate U = R /z . Here, with the purpose of simplifying the description of thestring profile in the soft wall background, we use coordinate z . In order to consider the samekind of fluctuations of the metric world sheet and find a result that can be compared to thisreference, we consider for the fluctuations of the string profile the coordinate U .To extract the imaginary part of the quark anti-quark potential one considers the effectof thermal worldsheet fluctuations about the classical configurations U = U c ( x ) = R /z c .Fluctuation of the form U ( x ) = U c ( x ) → U ( x ) = U c ( x ) + δU ( x ) , (18)produce negative contributions to the root square that appears in the Nambu Goto stringaction of eq. (2) near x = 0 and generate an imaginary part in effective string action. Consid-ering the long wavelength limit the fluctuations δU ( x ) at each string point are independentfunctions. The condition of fixed endpoits is: δU ( ± L/
2) = 0.The string partition function that takes into account the fluctuations is then a functionalintegral over the contributions coming from S NG ( U c ( x )+ δU ( x )). One discretizes the interval − L/ < x < L/ N points located at coordinates x j = j ∆ x ( j = − N, − N +1 , ..., N ) with ∆ x ≡ L/ (2 N ). The continuum limit N → ∞ is taken at the end of calculation.Then, Z str becomes Z str ∼ lim N →∞ (cid:90) d [ δU ( x − N )] ...d [ δU ( x N )] exp T ∆ x πα (cid:48) (cid:88) j (cid:113) M ( U j )( U (cid:48) j ) + V ( U j ) , (19)5here U j ≡ U ( x j ) and U (cid:48) j ≡ U (cid:48) ( x j ). The thermal fluctuations are more important around x = 0, where U = U ∗ and the string is closer to the horizon. Thus, it is reasonable to expand U c ( x j ) around x = 0 and keep only terms up to second order in x j . Given that U (cid:48) c (0) = 0one has: U c ( x j ) ≈ U ∗ + x j U (cid:48)(cid:48) c (0) . (20)The corresponding expansion for the relevant quantities V ( U ) and M ( U ), keeping only theterm up to second order in the monomial x mj δU n (that means m + n ≤
2) reads V ( U j ) ≈ V ∗ + δU V (cid:48)∗ + U (cid:48)(cid:48) c (0) V (cid:48)∗ x j δU V (cid:48)(cid:48)∗ M ( U ) ≈ M ( U ∗ ) , (21)where V ∗ ≡ V ( U ∗ ), V (cid:48)∗ ≡ V (cid:48) ( U ∗ ), etc. So, one can approximate the exponent in Eq.(19) as S NGj = T δx πα (cid:48) (cid:113) C x j + C , (22)with C = U (cid:48)(cid:48) c (0)2 [2 M ∗ U (cid:48)(cid:48) c (0) + V (cid:48)∗ ] ; C = V ∗ + δU V (cid:48)∗ + δU V (cid:48)(cid:48)∗ . (23)If the function in the square root of eq.(22) is negative then S jj contributes to an imaginarypart in the potential. The relevant region of the fluctuations is the one between the valuesof δU that lead to a vanishing argument in the square root in the action (22). So, one canisolate the j-th contribution I j ≡ (cid:90) δU jmax δU jmin d ( δU j ) exp (cid:20) i T ∆ x πα (cid:48) (cid:113) C x j + C (cid:21) , (24)where δU jmin , δU jmax are the roots of C x j + C in δU .The integral in eq.(24) can be evaluated using the saddle point method in the classicalgravity approximation where α (cid:48) (cid:28)
1. The exponent has a stationary point when the functioninside the root square of eq. (24) D ( δU j ) ≡ C x j + C ( δU j ) , (25)assumes an extremal value. This happens for δU = − V (cid:48)∗ V (cid:48)(cid:48)∗ . (26)6equiring that the square root has an imaginary part implies that D ( δU j ) < → − x c < x j < x c , where x c = (cid:118)(cid:117)(cid:117)(cid:116) C (cid:34) V (cid:48) ∗ V (cid:48)(cid:48)∗ (cid:35) . (27)We take x c = 0 if the square root in Eq.(27) is not real. Under these conditions, we canapproximate D ( δU ) by D ( − V (cid:48)∗ V (cid:48)(cid:48)∗ ) in eq.(24) I j ∼ exp (cid:34) i T ∆ x πα (cid:48) (cid:115) C x j + V ∗ − V (cid:48)∗ V (cid:48)(cid:48)∗ (cid:35) . (28)The total contribution to the imaginary part comes from superposing the individualterms: Π j I j . The result is[14] Im V Q ¯ Q = − πα (cid:48) (cid:90) | x |
1. Then, we can rewrite the expressions (34),(35)and (36) as LT = 2 π h √ − h (cid:90) dy y (cid:113) (1 − y h ) (cid:114) e √ γ h ( y − (1 − y h ) − (1 − h ) y ) , (37)8 e V Q ¯ Q T = − h + 1 h (cid:90) dy y e √ γ h y √ − h y (cid:114) (1 − h y ) − e √ γ h y y (1 − h ) − (38)and Im V Q ¯ Q T = − π √ e √ γ h h [ (2 − √ γ h (1 − h )( − cz ∗ + (4( √ γ h ) + 6 √ γ h )(1 − h ) + 12)) − (1 − h )2(2 − √ γ h (1 − h )) ] (39) III. THERMAL WIDTH IN THE SOFT WALL MODELA. Review of the black hole case
For the sake of comparison, let us first briefly review how the thermal width was calculatedin ref. [14]. In this reference the background describing the quark gluon plasma is just anAdS black hole in Poincare coordinates. This geometry is dual to a gauge theory, with nomass scale. The strategy in this reference was to get the thermal width from the expectationvalue of the imaginary part of the potential in a state of a quark anti-quark pair in non-relativistic approximation Γ Q ¯ Q = −(cid:104) ψ | ImV Q ¯ Q ( L, T ) | ψ (cid:105) , (40)where (cid:104) (cid:126)r | ψ (cid:105) = 1 √ πa / e − r/a , (41)is the ground-state wave function of a particle in a Coulomb-like potential of the form V ( L ) = − D/L and a = 2 / ( m Q D ) is the Bohr radius. In this way, a mass scale wasintroduced in the problem through the parameter a that is related to heavy quark masss m Q = 4 . GeV . It is interesting to mention that an expression similar to eq.(41) was obtainedin ref.[30] from a two point correlator obtained holographically.The thermal width is then given byΓ T = − a T ) (cid:90) d ( LT ) ( LT ) e − LTa T Im V Q ¯ Q ( L, T ) T (42)This integral is performed in the interval of LT where the string solutions with “U-shaped” profile exist. As discussed in refs. [9, 10] these solutions exist in the black hole9 .1 0.2 0.3 0.4 0.50.00000.00050.00100.0015 Ta _ Γ T FIG. 1. Thermal width obtained in [14] using an AdS black hole metric.
AdS space for LT ≤ ( LT ) max ≈ . LT : ( LT ) min ≈ .
266 since for smaller values of LT the expression (27) for x c becomes imaginary. The quantity x c defines the interval where the coordinate xj is defined,so it must be real otherwise the approximation used would not be valid. So, we integratein the interval ( LT ) min ≤ LT ≤ ( LT ) max . This leads to a negative imaginary part for thepotential. The result for the thermal width as a function of a T is shown in figure B. Soft wall results
Now, returning to our soft wall case, the background already contains a mass scale, thesoft wall parameter √ c associated with the string tension. So, we follow a different strategy.We calculate the expectation value of the imaginary potential by integrating over the stringlengths using the semiclassical approximation:Γ Q ¯ Q = −(cid:104) ImV Q ¯ Q ( ˆ L ) (cid:105) T = − (cid:82) dL e − S NG ( L,T ) Im V Q ¯ Q ( L, T ) (cid:82) dL e − S NG ( L,T ) (43)where S NG is the Nambu-Goto action with the soft wall background.In the desconfined regime we can approximate the Nambu Goto action in a Coulombform for LT <<
1. Then, using the fact that the time integral in this Euclidean metric givesa factor of the inverse of the temperature, we can rewrite the equation in the dimensionlessform as a function of γ = T /T c Γ Q ¯ Q T ( γ ) = − (cid:82) dw e D/w ( γ ) Im V Q ¯ Q T ( γ ) (cid:82) dw e D/w ( γ ) (44)10here w = LT and D = 4 π √ λ/ Γ(1 / ≈ . h L T FIG. 2. The quark distance LT versus the parameter h = z ∗ /z h for γ = ∞ (solid line), γ = 2 . γ = 2 (dashes line) for the soft wall model. Now let us discuss what are the limits of integration in the string length L that shouldbe used in eq.(44). As it happens in the case of eq.(42) corresponding to the black holeAdS space without soft wall, in order to have a consistent procedure, the quantity x c mustbe real. This restricts our integration in the numerator of eq. (44) to a lower limit w = LT > ( LT ) min . But now, in the presence of the soft wall, the minimum value depends onthe parameter γ = T /T c as can be seen in eq. (37). The integrals in eq. (44) must also havean upper limit, since the string “U-shaped” profile considered has a maximum value of L,as discussed in refs. [9, 10]. We show in figure the maximum values of LT as a functionof h = z ∗ /z h , obtained numerically, for three different values of γ .In the case of the denominator we restrict the lower limit of integration considering thatthe quarks have a finite (large) mass, so they should lie on a D7-Brane at the position z = z D . The mass of the quark is related to the position of the D7 brane [31]: m q = R πα (cid:48) z D . (45)We can fix the mass the brane position using the mass of the bottom quark m b = 4 . GeV and choosing R /α (cid:48) = √ λ = 3 to find 1 z D = 9 . GeV (46)We use this value to find numerically that the lower limit of the normalization integral ofthe denominator is LT = 0 .
04. 11inally we estimate the thermal width using the equations (37) and (39) for differentvalues of γ to calculate the thermal width in eq. (44). We present in figure our result for thethermal width as a function of T /T c . Note that the thermal width is zero when γ = T /T c = 1because our model is confining for lower temperatures. For higher temperatures, there isa plasma. The thermal width increases in the region up to T ≈ . T c . For temperatureshigher than T ≈ . T c the thermal width decreases. This behaviour is qualitatively similarto that found in [14]. IV. CONCLUSIONS
The gauge/gravity duality is an interesting tool to study the imaginary part of the heavyquark potential in strongly coupled plasma. This imaginary part can be used to calculatethe thermal width of heavy quarkonia in such a thermal medium. In ref. [12] a method fordescribing thermal worldsheet fluctuations was developed. Then this approach was used in[14] to obtain a lower bound for the thermal width of heavy quarkonium states in AdS blackhole and also Gauss Bonnet gravity. In this previous study, the plasma is assumed to beisotropic and conformal. The same method was applied in [13, 15] to extract the imaginarypart of the heavy quark potential but in a strongly coupled anisotropic plasma.In the present work, we considered a non-conformal strongly coupled plasma. We used theAdS/QCD soft wall model that carries an infrared mass scale and introduces confinement ingauge gravity duality. This background represents a gauge theory that is a confining at lowtemperatures as has a deconfining transition at a critical temperature. Another point thatis different from the approach of [14] is that in this reference the width is calculated usinga wave function of a quarkonium state, while here we obtain the width by averaging overthe string lengths. Consistently, the result of the approach developed here is qualitativelysimilar to the one of reference[14]. Our result is also qualitatively similar to what is foundusing lattice QCD in ref. [32]. As a final remark, it is interesting to mention that ourprocedure was developed using a different radial coordinate ( z instead of U ) that is moreconvenient when working with the soft wall model. Acknowledgments:
N.B. is partially supported by CNPq and L. F. Ferreira is supported12 .0 1.2 1.4 1.6 1.80.000000.000020.000040.000060.000080.000100.00012 T T _ c Γ T FIG. 3. Thermal width in the soft wall model
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