AdS black holes with higher derivative corrections in ...

Eur. Phys. J. C (2020) 80:1012https://doi.org/10.1140/epjc/s10052-020-08565-9

Regular Article - Theoretical Physics

AdS black holes with higher derivative corrections in presence ofstring cloud

Tanay K. Dey1,a, Subir Mukhopadhyay2,b

1 Department of Physics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rongpo, East Sikkim, Sikkim 737136,India

2 Department of Physics, Sikkim University, 6th Mile, Gangtok 737102, India

Received: 16 July 2020 / Accepted: 18 October 2020 / Published online: 2 November 2020© The Author(s) 2020

Abstract We consider asymptotically AdS black hole solu-tions in Einstein Gauss Bonnet gravity in presence of stringclouds. As in the case of black hole solutions in Gauss Bon-net gravity, it admits three black hole solutions in presence ofstring clouds as well within a region of the parameter space.Using holography, we have studied the quark–antiquark dis-tance and binding energy in the dual gauge theory.

1 Introduction

Study of strongly coupled gauge theories remain a chal-lenge due to lack of appropriate systematic formulations andmachineries. On that score, the AdS/CFT duality has beenproved to be quite a useful route. This duality implies thatSU (Nc), N = 4 Super Yang-Mills (SYM) theory livingon the boundary of the one higher dimensional space timein the limit of strong coupling is dual to a weakly coupledgravity theory in the higher dimensional space time and viceversa [1–4]. It has been found that there is a correspondencebetween the black hole configurations in the gravity theoryand the dual boundary theory in its deconfined phase. Onthe other hand, the confined phase of the dual gauge theoryis dual to pure AdS space time configuration. The transition[5] between the pure AdS configuration and the black holeconfiguration has been interpreted as equivalent to the transi-tion between the confined and deconfined phases of the dualgauge theory living on the boundary.

A more realistic scenario is to consider a gauge theorywith fundamental quarks and baryons, which are absent inSU (Nc), N = 4 SYM theory. In order to incorporate fun-damental quarks, [6–10] consider cloud of hanging strings

a e-mails: [email protected]; [email protected] e-mails: [email protected]; [email protected] (corre-sponding author)

whose end points are tracing out the contour of the loop atthe asymptotic boundary. In the dual gauge theory living atthe boundary, the end points of the string correspond to thequark and anti-quark, whereas the string itself correspondsto the gluons.

Several works [11–17] have considered string cloudswhich amounts to considering external heavy quarks in thedual gauge theory. The Einstein theory with string cloudsadmits black hole solutions with three different radii andthe thermodynamics of them has been discussed by one ofus [12], which also showed a Hawking–Page like phase tran-sition between two black hole solutions. Such three blackhole solutions along with the phase transitions have beenshowed in [18,19] in the context of Einstein Gauss Bonnet(GB) theory in absence of the string clouds. Thermodynam-ics of black hole solutions in Einstein Gauss Bonnet theoryin presence of strings clouds has been extensively discussedin [13]. Effect of the strings clouds on the thermodynamicsof the extended phase space of Gauss Bonnet black hole hasalso been discussed in [15].

In this work we have considered the black hole solutionof Gauss Bonnet gravity in presence of string clouds anddiscussed thermodynamics as well as its effect on the dualgauge theory. In a similar model [20,21], found that thereis an upper bound for the quark–antiquark (QQ) separationfor large black hole and thus identified it with the decon-fined phase. Along the similar line, we have studied the QQdistance, as well as the binding energy for the black hole solu-tions in GB theory with string clouds. We find the screeninglength of QQ pair increases as the temperature decreases andthough for the small black hole, it can go to a larger valuecompared to that of large black hole, there remains an upperbound. Therefore one can conclude that the present modeldoes not admit a confined phase. The value of the bindingenergy is more negative in the case of the large black holecompared to that of the small black hole showing QQ bound

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state in the case of large black hole is more stable. In partic-ular, at high enough temperature, the QQ distance vanishesshowing that the QQ bound state does not exist.

The paper is structured as follows. In Sect. 2, a brief dis-cussion of the black hole solution in presence of externalstring cloud is included. We then discuss the thermodynam-ical quantities and their behaviour in Sects. 3 and 4 respec-tively. Thereafter in Sect. 5 we check the instability of QQbound state by studying the depth of the U-shape string hang-ing from boundary of the asymptotically AdS background.Finally we study QQ bound state and its binding energy atlow temperature in Sect. 6. At the end we summarize ourwork in Sect. 7.

2 Black hole solution

We begin with a brief review of black hole solutions [13–17]of Einstein Gauss Bonnet theory, which are asymptoticallyAdS in presence of string clouds. The string clouds corre-spond to introducing strings in the background. They areextended from the boundary to the horizon of the black holeor the center of the space time. The higher derivative termscan be thought of as leading to subleading correction in the‘t Hooft coupling.

The (4 + 1) dimensional gravitational action along withGauss Bonnet terms is given by

S = 1

16πG5

∫dx4+1√−g(R − 2� + αLGB) + Sm, (1)

where G5 represents the gravitational constant. R is the Ricciscalar and � is the cosmological constant. We write gμν asthe space time metric tensor and g represents its determinant.The term

LGB = R2 + Rμνρσ Rμνρσ − 4RμνR

μν,

is the higher order correction term due to the quantum fieldsrenormalization. This term is called Gauss–Bonnet correc-tion in the Lagrangian. The α is the coefficient of the Gauss–Bonnet counterpart which plays a crucial role in this theory.

In order to incorporate the string clouds we have added afurther term, Sm . Since Sm represents a large number of thestrings its contribution can be written as,

Sm = −1

2

∑i

Ti∫

d2ξ√−hhβγ ∂βX

μ∂γ Xνgμν. (2)

where we have Integrated over the world-sheet, hβγ is theworld-sheet metric and β, γ correspond to the world sheetcoordinates. Sm is the contribution of a large number ofstrings. We denote the tension of i’th string by Ti .

We will assume that the strings are uniformly distributedover the three spatial directions and write the density as

a(x) = T∑i

δ3i (x − Xi ), with a > 0. (3)

With this assumption, as shown in [13–17] the action (1)admits a black hole solution. The metric tensor of this solu-tion is,

ds2 = −gtt (r)dt2 + grr (r)dr

2 + r2gi j dxi dx j . (4)

Here gi j is the metric on the (4 − 1) dimensional boundaryand

gtt (r) = 1+ r2

4α

(1 −

√1 + 32αM

r4 − 8α

l2+ 16aα

3r3

)= 1

grr,

(5)

where l corresponds to the AdS radius and related to cosmo-logical constant via the relation, � = − 6

l2. M is the constant

of integration, which represents the mass of the solution.The solution is singular at r = 0, which is cloaked by

a horizon located at r+, which can be obtained by settinggtt (r+) = 0,

1 + r2+4α

(1 −

√1 + 32αM

r4+− 8α

l2+ 16aα

3r3+

)= 0. (6)

To get the real solution of the above equation, the discrimi-nant should remain positive which implies following condi-tion,

r4bh

(1 − 8α

l2

)+ 16α

(2M + arbh

3

)= 0, (7)

to be imposed on the maximum value of the horizon radius.We can trade the ADM mass of the black hole solution forthe radius of the horizon using the following relation,

M = 3r4+ + 3l2r2+ − 2al2r+ + 6l2α

12l2. (8)

This solution is asymptotically AdS and it is necessary toanalyse the stability of this solution towards a decay into thepure AdS. We will consider a thermodynamic analysis of itin the next section. As we will see, for a given temperature itadmits three different kinds of black holes along the line of[18] and we will study the relative stability of these solutions.

3 Thermodynamics

Black holes can be considered as thermodynamic systemsand various thermodynamical quantities associated to blackhole maintain a similar type of thermodynamical law as ther-modynamic systems follow. In the present section we willconsider the solutions and their different thermodynamic

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aspects. In what follows, we set the term related to gravi-tational constant, 16πG5 = 1 and also consider unit volumeof the 3-sphere.

• Temperature: To begin with the black hole temperatureis given by,

T = 1

4π

dgttdr

|r=r+ = 6r3+ + 3l2r+ − al2

6πl2(r2+ + 4α), (9)

This gives rise to a cubic equation in r+ and for a giventemperature, generically, we can expect three solutionsfor r+.

• Entropy:Assuming that the first law of thermodynamicsis satisfied by the black hole, we obtain the followingexpression for the entropy,

S =∫

T−1dM = π

(r3+3

+ 4αr+)

. (10)

• Specific heat: Specific heat can easily be obtained byusing the formula C = T dS

dT and is given by,

C= ∂M

∂T=

π

(r2+ + 4α

)2[6r3+ + 3l2r+ − al2

]

6r2+(r2++12α

)+2al2r+−3l2

(r2+−4α

) .

(11)

• Helmholtz free energy: Considering the energy of theblack hole E to be equal to its mass we get the followingform of the Helmholtz free energy, F = E−T S for theseblack holes.

F = E − T S

= − 3r6+ − 3l2r4+ + 108αr4+ + 4al2r3+ + 18αl2r2+ − 72α2l2

36l2(r2+ + 4α).

(12)

• Landau function: In order to study the transitionsbetween different phases, we have constructed the Lan-dau function computed around the critical point, whereradius of the horizon plays the role of the order parame-ter. The construction goes as follows; first we consider afunction G as a power series in the horizon radius r+,

G(T, r+)=α0r0++α1r

1++α2r2++α3r

3+ + α4r4++· · · ,

(13)

where αi are functions of temperature T and are chosenso as to satisfy the following conditions:

• Up to a certain temperature Tmin , G will have only oneminimum at r+ = r+1 with r+1 ≥ 0.

• Once the temperature is above Tmin another minimumappears at r+3. Minima at r+ = r+1 and r+ = r+3 mustbe separated by a maximum at r+ = r+2. We demandthat at high temperature, the second minimum is globallystable. In order to achieve this, we can consider only up tothe quartic power of the order parameter and neglect allthe terms, which are higher order in the order parameter.Therefore, the Landau function can be written as;

G(T, r+) = α0 + α1r1+ + α2r

2+ + α3r3+ + α4r

4+. (14)

• In order to determine these five αi we impose the follow-ing five conditions.

– At the three extreme points the function should satisfy∂G∂r+ |r1,r2,r3 = 0,

– Expression of the temperature can be obtained fromthe condition of extrema of this function.

– Finally we get back the expression of free energy ofthe system by substituting temperature in the Landaufunction.

From the above five conditions we can compute the constants(α0, α1, α2, α3, α4). The Landau function, when written interms of temperature and the order parameter, reduces to thefollowing expression,

G = 3r4+ − 4πl2r3+T+3l2r2+−48παl2r+T−2al2r++6αl2

12l2.

(15)

An analysis of these thermodynamic quantities of theblack hole and their dependence on various parameters ofthe model reveals a complex thermodynamic phase structure.This will enable us to study the issue of thermodynamic sta-bility of the various solution admitted by the present model.

4 Phases of black hole

In order to explore the phase structures of the solutions, wewill examine the behaviour of the various thermodynamicquantities with respect to variation of different parameters.Similar discussion of thermodynamics of Einstein GaussBonnet black hole in presence of string clouds has alsoappeared in [13,15]. For the sake of simplicity of notation,we scale a, α and the horizon radius with appropriate powersof l to render them dimensionless as follows:

a = a

l, α = α

l2and r = r+

l. (16)

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(a) (b)

Fig. 1 Both the graphs are for α = 0. a Temperature T vs. scaled horizon radius r and b string cloud density a vs. temperature π T

We have introduced the temperature for the black holesolutions in (9), which can be expressed as

T = 6r3 + 3r − a

6πl(r2 + 4α). (17)

It shows that for a fixed temperature, the scaled horizon radiusr satisfies a cubic equation. If the cubic equation admits threereal and positive solutions there are three black holes or it willbe only one black hole solution.

The discriminant of the cubic equation is given by,

� = −[

16α(π T )4 + 2a

3(π T )3 +

(432α2 − 36α − 1

4

)

(π T )2 − 9a

(1

6− 4α

)(π T ) + 3

4a2 + 1

2

]. (18)

In the above equation and in the following plots we have setl = 1. In the region of the parameter space where � > 0,it admits three real solutions and otherwise there is only onereal solution. First we have plotted the region for � > 0 forα = 0 in Fig. 1.

In the left subfigure of Fig. 1, we consider a set of valuesof a, the string cloud density and plotted temperature againstr , the radius of the horizon scaled appropriately. We observethat for a = 0(red curve), there are two black hole solutionfor any temperature greater than a minimum temperature.However once a gets some positive value but below a criticalvalue a = ac = 1√

6= .408 (for details see [12]) there are

certain range of temperature where three black hole solutionsexist. Outside this range only one black hole solution exist.Once a is above its critical value (purple curve), for all valuesof temperature we get one black hole solution.

In right subfigure of Fig. 1 we plot string cloud density aagainst π T . For a given value of string cloud density 0 < a <

ac there are two values of temperatures T1 and T2, such thatthe three black hole solutions exist in the region between

T1 and T2. For a = 0, π T1 = √2 and T2 is extended up

to infinity and for T >√

2/π , we get only two black holesolutions, which cease to exist as we decrease the temperaturebelow

√2/π .

As we turn on the string clouds, its density a becomesnon-zero and T2 becomes finite. As the density of the stringclouds a increases more and more, the upper bound for threesolutions T2 keeps on decreasing. On the other hand, thelower bound T1 also decreases but at a much slower rate.Finally, when a reaches a critical value, a = ac, T1 andT2 meet, indicating the merger of all the three black holesolutions. For a > ac, there exists only a single black holesolution.

Once we turn on the higher derivative correction α, theregion of the three dimensional parameter space correspond-ing to T , a and α, which admits three black hole solutionsare shown in the Fig. 2. As α increases, the upper bound ofthe temperature for the existence of three black hole solutionT2 becomes finite. For example, for α = .001 and a = 0, T2

becomes approximately 3/π . T2 keeps on decreasing withincrease of α. The critical value of a denoted by ac, wherethe two bounds T1 and T2 meet, keeps on decreasing as α

increases and vanishes at around αc = .01386. For α > αc

there will be only a single black hole solution.In the Fig. 3 we have plotted temperature with respect to r

for two specific values of string cloud density a and Gauss–Bonnet coupling α. It shows that at a particular temperature,the horizon radii of these three black hole solutions are ofdifferent sizes and we call them small, medium and largedepending on their horizon radii.

For T < T1 only a single black hole exists, whose radiusdecreases as the temperature decreases. This single blackhole persists even at zero temperature, where the radiusreaches its minimum value, which can be expanded in apower series of a,

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Fig. 2 Temperature vs string cloud density with higher derivative cor-rection

r0 = a

3+ O(a2). (19)

Having established the different black hole solutions fromthe study of their temperatures, we will enquire about theirthermodynamic stability in the respective domains of theparameter space. For this purpose, we will examine the threedifferent thermodynamic quantities: the specific heat, theHelmholtz free energy and the Landau function, that we haveintroduced in the earlier section.

We begin with the specific heat. The expression for thespecific heat of the black hole solutions in terms of the threedimensionless parameters turns out to be;

C = πl3(4α + r2)2[6r3 + 3r − a]6r2(r2 + 12α) − 3(r2 − 4α) + 2ar

. (20)

We have plotted the specific heat vs. temperature in Fig. 4balong with a plot of the horizon radii vs. temperature in Fig. 4aand we have chosen α = 0, a = 0.1. In these figures, the reddashed line represents the small black hole which exists upto T < T2 and as one can observe from Fig. 4b, its specificheat remains positive (though small) throughout the rangewhere it exists. The blue line represents the medium blackhole, which exists between T1 and T2 and the Fig. 4b showsthat the specific heat is negative leading to thermodynamicinstability of the solution. The green, dotted line representsthe large black hole. It starts off its existence from T = T1 andpersists for the entire range of temperature greater than T1.Once temperature is above T2 it represents the single blackhole. As one can observe in Fig. 4b, it has a positive specificheat, showing it is thermodynamically stable. The behaviourof specific heat at the temperature, where unstable mediumblack hole meets the black holes with large horizon radius,indicates that the medium black hole decays at T1 through afirst order phase transition.

Similar diagrams has been plotted in Fig. 5a, b for α =.004, a = 0.2. As discussed earlier, due to turning on ofthe higher derivative correction, T2 has been reduced andapproached T1 resulting the range of the temperature for theexistence of the medium black hole smaller. The specific heatshows qualitatively similar features.

Next, we analyse the Helmholtz free energy, which can bewritten as;

F = − l2[3r6 − 3r4 + 108αr4 + 4ar3 + 18αr2 − 72α2]36(r2 + 4α)

.

(21)

In the Fig. 6a, we have plotted the free energy vs. thetemperature for α = 0, a = 0.1. Since for our choice ofparameters, the free energy of the small black hole is alwaysless than that of AdS configuration, for small temperature

(a) (b)

Fig. 3 T vs. r plots. a a = 0.1, α = 0.001 and b a = 0.1, α = 0.01

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(a) (b)

Fig. 4 a r vs π T plot b C vs. π T with α = 0, a = 0.1 for black holes with horizon radii small (red), medium (blue) and large(green)

(a) (b)

Fig. 5 a r vs π T plot b C vs. π T with α = 0.004, a = 0.2 for black holes with horizon radii small (red), medium (blue) and large(green)

the small black hole remains thermodynamically stable tillT < T1, which is consistent with the analysis of the specificheat. As T reaches T1, the other two black holes appear. Atany temperature, a comparison of free energies shows theblack hole with intermediate horizon radius is not thermody-namically favoured. For T > Tc, a critical temperature whichis less than T2, the small black hole has a free energy greaterthan that of the large black hole and therefore, the large blackhole continues to be the thermodynamically favoured config-uration for the range of temperature T > Tc.

In the Fig. 6b, we have plotted the free energy vs. thetemperature for α = .001, a = 0.2. One noticeable differ-ence that happens here is that the free energy for the smallblack hole is positive up to a certain temperature and then itbecomes negative. On the other hand the free energy for themedium and large black holes is negative for any tempera-ture. Except these differences other behaviour are same asprevious case.

In the Fig. 7, we have plotted the free energy vs. the scaledhorizon radius for two different sets: (a) a = 0.1, α = 0.001and (b) a = 0.1, α = 0.01. As one can observe from theseplots at r = 0 the free energy is positive. As the radius of theblack hole increases, the free energy decreases and becomesminimum. As we increase r further, the free energy startsincreasing, becomes positive and reaches a maximum valueat a certain radius. For further increase of the horizon radius,the free energy becomes negative and keeps on decreasing.We can associate again free energy of r = 0 as free energy ofAdS configuration, the local minima of the free energy as thesmall black hole free energy and the negative free energy forthe large r is related to the large black hole. The maxima inthe free energy corresponds to the unstable black hole. Fromthe consideration of free energy, AdS configuration is not astable configuration and small and large black hole can bestable configuration only.

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(a) (b)

Fig. 6 Free energy is plotted against temperature: black holes with horizon radii small, medium and large are plotted in red, blue and green. aα = 0, a = 0.1, b α = 0.004, a = 0.2

0.03

(a) (b)

Fig. 7 Free energy is plotted against horizon radius: a a = 0.1 and α = 0.001 and plot b is for a = 0.1, α = 0.01 and l = 1

We consider the Landau function which is constructedaround the critical point to investigate about stability of dif-ferent solutions and is given by:

G = 3l2r4−4πl3r3T+3l2r2−2l2ar−48πl3αr T+6αl2

12.

(22)

We consider four different temperatures and have plottedLandau function with respect to scaled horizon in Fig. 8awith a = 0.1 and α = 0.001. The temperatures we con-sider are T0 = 0.42 (red curve), T1 = 0.44 (green curve),Tc = 0.4545 (blue curve) and Tc1 = 0.4568 (pink curve)such that T0 < T1 < Tc < Tc1. For temperature T0, only oneminimum of Landau function at finite r exists, that corre-sponds to the black hole solution with small horizon radius.With rise of temperature two more extrema nucleates at tem-perature T1. The maxima of the Landau function correspondsto unstable black hole and minima corresponds to the largeblack hole. As we increase the temperature further at temper-ature Tc the Landau function associated with the black holes

having smallest and largest radii become equal. Above Tc , theLandau function for large black hole becomes lower than thatof the small black hole indicating stability of the small blackhole at low temperature and of the large black hole at hightemperature. It also indicates that there will be a transitionbetween the small and large black hole at T = Tc similarto Hawking–Page phase transition. In Fig. 8b, we plot theLandau function against scaled horizon radius for a = 0.1and α = 0.01. We observed that there is only one minimaat finite r corresponds to existence of one stable black holesolution from which one may conclude that pure AdS is nota stable configuration at any temperature.

5 Quark–antiquark distance

As pointed out in the introduction, it is interesting to examinethe phase of the dual model with reference to the confine-ment. For that purpose, we consider a string configurationhaving its endpoints on the boundary and elongated in the

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0.03

(a) (b)

Fig. 8 Landau function vs. r for four different temperatures: T0 = 0.42 (red) , T1 = 0.44 (green) , Tc = 0.4545 (blue) Tc1 = 0.4568 (pink) ;T0 < T1 < Tc < Tc1 a a = 0.1 and α = 0.001, b is for a = 0.1, α = 0.01 and l = 1

bulk. There may be two such configurations: a straight stringfrom boundary to horizon or a U-shaped string where thestring is hanging from the boundary, with its tip located atu0, but not touching the horizon.

We will denote the distance between the two endpointsby L , which is the quark–antiquark distance associated withthe bound state in the boundary theory. There is an upperbound Ls of the quark–antiquark separation L . Every L , 0 <

L < Ls corresponds to two U-shaped strings with tip u0

being nearer to and farther from the horizon. As L increases,these two configurations approach each other and they mergeat L = Ls , when the quark–antiquark distance reaches thescreening length.

Quark–antiquark distance for a bound state can beobtained by considering a probe string in the black holespacetime. The Nambu–Goto world sheet action for such anopen string is,

SNG = − 1

2πα′

∫d2σ

√−det (hβγ ), (23)

where 12πα′ is the string tension. The string tension is related

to λ, the ‘t Hooft coupling in the dual SYM gauge theory[22],

√λ = l2

α′ . (24)

The present case being a bottom-up approach, we do not havea precise description of the dual boundary theory. Neverthe-less we can expect that the λ will play an analogous role.We will assume that the above Nambu–Goto action is a validaction for our problem and corrections ensuing from cou-pling to the further bulk fields will not qualitatively modifythe essential result.

For convenience, we will use u rather than r , where r = l2u

so that u = 0 is the boundary and the horizon occurs at uh .The metric tensor of the black hole solution of the equation(4) reduces to the form as,

ds2 = f (u)

[− h(u)dt2 + dx2 + dy2 + dz2 + du2

h(u)

],

where, f (u) = l2

u2 and h(u) = u2

l2+ 1

4α(1 −

√1 − 8α + 32αMu4

l6+ 16aαu3

3l3

)(25)

and the radius of the horizon, uh , can be obtained by solvingthe equation,

h(uh) = u2h

l2

+ 1

4α

(1 −

√1 − 8α + 32αMu4

h

l6+ 16aαu3

h

3l3

)= 0.

(26)

We will consider, the end points of the open string corre-sponding to QQ pair are located at x = ± L

2 respectivelyand is elongated in the bulk space time with the symmetryaround x = 0.

With this background, we evaluate the Nambu–Gotoaction, (23). We choose: σ 0 = t, σ 1 = x (static gauge)and the metric induced on the worldsheet, given by hβγ =∂βXμ∂γ Xνgμν becomes,

ds2 = f (u)

[− h(u)dt2 +

{1 + u′2

h(u)

}dx2

]. (27)

After t integration is carried out, the Nambu-Goto actionbecomes

SNG = − T2πα′

L/2∫

−L/2

dx f (u)

√√√√h(u)

(1 + u′2

h(u)

). (28)

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(a) (b)

Fig. 9 QQ -distance vs. u0: a = .1, α = .001, T = T1 (red), T = T2 (blue); a = .2, α = .004, T = T1 (green), T = T2 (brown); black holeswith radius a small, b large

The equations of motion for the embedding coordinate u(x),ensuing from the above action is given by

u′(x) = −√h(u)

(f (u)2h(u)

f (u0)2h(u0)− 1

), (29)

implying the string is extended towards the horizon up to aturning point u = u0 and goes back to the boundary in asymmetric manner with u(0) = u0. The distance betweenthe two end points of the string can be obtained as

L =L/2∫

−L/2

dx = 2

u0∫

0

du

u′

= 2

u0∫

0

du

[h(u)

(f (u)2h(u)

f (u0)2h(u0)− 1

)]− 12

, (30)

where u0 is the upper bound on the location of the tip of thestring that is extended towards horizon.

We have plotted the quark–antiquark separation L withrespect to u0, the upper bound on the location of the tip ofthe string extended towards the horizon in Fig. 9. One canobserve from the figure, as the tip located at u0 gets closerto horizon, quark–antiquak separation increases. It becomesmaximum when u0 gets extremely near to the horizon andonce it touches the horizon, L becomes zero. This fact canbe interpreted as a breakdown of the U-shaped string con-figuration to two parallel straight string configuration whichcorresponds to unbound state of QQ pair. This happens incase of the small as well as the large black hole. It suggeststhat confined state might not be present in the dual theoryliving on the boundary, as expected in gravity analysis.

6 Quark–antiquark potential

In the present section we are interested to study free energyassociated with the heavy quark–antiquark pair of the dualtheory living on the boundary. The Wegner–Wilson loopencodes the free energy associated with a quark–antiquarkpair [22],

〈W (CL ,T )〉 ∼ exp[−i FQQ(L)T ], T → ∞, (31)

where FQQ(L) is the QQ free energy. It depends on thetemperature as the expectation value is for a thermal state.Since in the present discussion we will consider the range ofL only up to a screening distance Ls , the free energy FQQwill agree with the quark–antiquark potential VQQ in ourcase.

According to holographic principle, Nambu–Goto actionof an open string given in equation (23) is related to theWegner–Wilson loop where C, the contour of integration isidentified with open string worldsheet boundary. The on-shellstring action then is related to expectation value,

〈W (C)〉 ∼ exp[i SNG(C)], (32)

where SNG(C) is the on-shell Nambu–Goto action of thestring whose expression is given in (28). Comparing (31)and (32), we obtain

FQQ(L) ∼ − SNG(C)

T . (33)

FQQ(L), free energy for quark–antiquark can be obtainedby substituting (29) in (28). As has been discussed elabo-rately in [22], computation of SNG in general gives rise todivergences and this expression needs to be renormalised. Wefollow the same renormalisation prescription as given in [22]and obtain a renormalised expression for the quark–antiquarkfree energy, given by

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1012 Page 10 of 13 Eur. Phys. J. C (2020) 80 :1012

0.03

3.0 3.2

(a) (b)

Fig. 10 a = .05; a QQ binding energy for the large black hole for temperatures, T = 3 (green), 2 (brown), 1 (red), 0.5 (blue), b QQ bindingenergy for the small black hole for temperatures, T = 2 (brown), 1 (red), 0.5 (blue), .01 (black)

πFQQ(u0)√λ

=u0∫

0

du

⎡⎣ f (u)

√f (u)2h(u)

f (u)2h(u) − f (u0)2h(u0)− 1

u2

⎤⎦ − 1

u0. (34)

Usually quark–antiquark binding energy can be obtainedfrom U-shaped string action with action of two straightstrings from boundary to horizon subtracted from it, whichis given by [22]

πEQQ(u0)√λ

=u0∫

0

du

⎡⎣ f (u)

√f (u)2h(u)

f (u)2h(u) − f (u0)2h(u0)− 1

⎤⎦ −

uh∫

u0

f (u)du.

(35)

In what follows, in both the cases of small and large blackhole we can obtain free energy and binding energy for thequark–antiquark potential. We begin with black holes thathas been obtained without the higher derivative coupling.Some result has already appeared in [12] and here we havea more elaborate analysis. If we drop the higher derivativeterms, metric functions are [12]

f (u) = 1

u2 , h(u) = 1 −(

u

uh

)4

+ u2[

1−(

u

uh

)2]

−2a

3u3

(1 − u

uh

), (36)

where we have traded M for horizon radius uh . Here and inthe rest of the calculation, we set l = 1

We have substituted these expressions in the general for-mula for the free energy and the binding energy for the quark–antiquark pair derived in (34) and (35) respectively. Sincethe integration cannot be carried out to obtain expressions interms of known analytic functions, we have evaluated themnumerically. We obtain the quark–antiquark separation fromthe expression (30) as a function of u0, the closest approachof the string worldsheet towards the horizon. It turns out thatas u0 varies there is a maximum value of the quark–antiquarkseparation, which depends on the temperature as well.

In order to find out the thermodynamically stable quark–antiquark bound states we have plotted the binding energyversus the quark–antiquark separation for both the blackholes in the Fig. 10. One may observe that the binding energyof the bound state for large black hole is smaller than the smallblack hole background indicating that the bound states aremore stable in the background of large black hole. One canalso notice that for both the cases, after a certain value ofseparation distance the binding energy of the pair is gettingpositive value. Therefore the bound state exist up to an uppervalue of separation distance which is called screening lengthLs and that is finite. The screening length decreases with theincrease of temperature and bound state becomes more unsta-ble at higher temperature. We compare the screening lengthfor small and large black hole background and find screeninglength is larger in small black hole configuration showing itadmits larger separation, but in none of the cases, it can go toinfinity or can be interpreted as deconfined phase. In Fig. 11we have analysed the same features for different values ofstring cloud density by keeping temperature fixed. The qual-itative behaviour remains unchanged and here string clouddensity plays a role analogous to the temperature. Finally wecan conclude that deconfined state of quark and antiquark isthe stable configuration.

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(a) (b)

Fig. 11 QQ binding energy for different a values, a= .4 (green), .3 (brown), .2 (red), .1 (blue) a small black hole, T = 1, b large black hole,T = 0.4;

(a) (b)

Fig. 12 a = .05; QQ free energy vs. L: a large black hole with T = 3 (green), 2 (brown), 1 (red), 0.5 (blue), b small black hole with T = 2 (brown),1 (red), 0.5 (blue), .01 (black)

The quark–antiquark free energy obtained from (34) arenumerically evaluated for both the black holes in the samerange of temperatures and are plotted with respect to thequark–antiquark separation in Fig. 12. Comparing with thebinding energy plotted in Fig. 10, one can observe that withvariation of the temperature the free energy associated withquark–antiquark pair does not vary much, though there is areasonable difference in the binding energy, mostly arisingfrom the subtraction of the energy due to the straight strings.

Modification due to higher derivative terms can be studiedby considering black hole solution obtained from Einstein–Gauss–Bonnet action as discussed in the previous sections.Once higher derivative terms are taken into account, the met-ric is given in (4) and (5). In terms of the general form ofmetric given in (25) the metric functions become

f (u) = 1

u2 ,

h(u) = u2 + 1

4α

[1 −

√√√√1 − 8α

(1 − u4

u4h

)+ 16aα

3u3

(1− u

uh

)+32αu4

(1

4u2h

+ α

2

)],

(37)

where, as usual, we have traded mass parameter M for thehorizon radius uh .

Substituting the above expressions in the general formulafor the binding energy and the separation for the quark–antiquark pair given in (34) and (30) respectively, we haveplotted the binding energy against L . We consider four differ-ent values of α, the coefficient of higher derivative couplingin the plots for both the black holes as given in Fig. 13.

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energy with corrections

(a) (b)

Fig. 13 QQ binding energy vs. L for a = .05; T = .441; α = .001 (blue), .004 (red), .0082 (brown) a small black hole, b large black hole

(a) (b)

Fig. 14 QQ free energy vs. L for a = .05; T = .441; α = .001 (blue), .004 (red), .0082 (brown) a small black hole, b large black hole

Since, as the α varies, the critical temperature keeps onchanging, we have chosen suitable values of the temperaturesfor the two different black holes. As the plots show, with theincrease of the coefficient of the higher derivative terms thebinding energy increases making the quark–antiquark boundstate less stable. Furthermore, for a fixed amount of bindingenergy the length of the maximum separation between thequark and the antiquark decreases. This indicates, with addi-tion of subleading corrections in ‘t Hooft coupling, the QQbound states will become loosely bound favouring decon-finement. Both the small and the large black holes share thisfeature. Though we have given the plot only for a single tem-perature in each case, this feature qualitatively persists withvariation of the temperature.

At last we have studied the quark–antiquark free energyfor both the black holes. We plot the analysis in Fig. 14 and

observe that qualitative features are same as without higherderivative term.

7 Summary

We have considered solutions in the Einstein Gauss–Bonnetgravity along with an external string clouds. As happenedin the Einstein Gauss Bonnet gravity [18], the theory admitsthree different black holes in presence of string clouds. As thedensity of string cloud increases and/or the higher derivativecorrection dominates, the parameter region for which onegets the three solutions shrinks. Beyond the critical valuesof string cloud density or higher derivative correction, thetheory admits only a single solution.

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The analysis of the specific heat associated with the solu-tions implies that the medium size black hole is unstable,while the other two solutions are thermodynamically stable.From the free energy consideration we find at a higher tem-perature the thermodynamically faouvored configuration isthe black hole with largest horizon radius. We have analysedLandau function to examine possible transitions among thesesolutions.

Using holographic duality, we examine the quark–antiquark bound states. We find such bound states exist up toa distance of screening length, beyond which they get sep-arated, indicating the dual theory is in a deconfined phase.The screening length turns out to be larger for small blackhole than that in large black hole, though it is finite and can-not be extended indefinitely as happened in a confined phase.From the study of quark antiquark binding energy, we find itdecreases as the string cloud density and the higher deriva-tive correction becomes larger. This may be interpreted as,in presence of large number of quarks in the background, thequark–antiquark bound states will be loosely bound. A sim-ilar effect appears as the subleading corrections in ‘t Hooftcoupling becomes increasingly dominant.

It may be interesting to consider further extension of thismodel. In particular one can consider the charged black holesolution in presence of string clouds and examine the differ-ent phases admitted by such theory.

Data Availability Statement This manuscript has no associated dataor the data will not be deposited. [Authors’ comment: The numericalcomputations are done using “Mathematica”. The output data are plottedin the appropriate figures. The programs, if required, can be obtainedby contacting any of the authors.]

Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.

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