Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

8
Research Article Study of Confinement/Deconfinement Transition in AdS/QCD with Generalized Warp Factors Shobhit Sachan Department of Physics, Banaras Hindu University, Varanasi 221005, India Correspondence should be addressed to Shobhit Sachan; [email protected] Received 27 June 2014; Accepted 13 August 2014; Published 21 August 2014 Academic Editor: George Siopsis Copyright © 2014 Shobhit Sachan. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP 3 . We study analytical solutions of charged black holes and thermally charged AdS with generalized warped factors in Einstein- Maxwell-Dilaton system. We calculate Euclidean action for charged AdS and thermally charged AdS. e actions in both backgrounds are regularized by the method of background subtraction. e study of phase transition between charged black hole and thermally charged AdS gives an insight into the confinement/deconfinement transition. e plots of grand potential versus temperature and chemical potential versus transition temperature are obtained. 1. Introduction Strongly interacting systems are always a challenge to our analytical knowledge. Quantum chromodynamics is such a theory, which cannot be solved analytically in low energy regime. ere are two methods to solve QCD; one is “lattice QCD” [1] and the other is “AdS/QCD.” e formulation of lattice QCD is based on discretization of spacetime and it requires high performance computing. On the other hand, AdS/QCD is analytic approach and is motivated by the gauge/gravity duality [2–5]. Some properties of QCD like the- ories motivated by gauge/gravity duality such as confinement and chiral symmetry breaking have been studied extensively in [6–21] and spectrum of mesons and baryons is studied in [22, 23]. ere are two approaches from where one can construct QCD like theories. ese approaches are known as top-down and bottom-up approaches. In top-down approach, one starts from stringy D brane configurations and constructs models for QCD [24, 25] while in bottom-up approach, one starts from QCD and attempts to construct its five-dimensional gravity dual. ese five-dimensional dual models can be gen- eralized to study various properties of QCD. e bottom-up approach is divided into two categories, hard wall [7] and soſt wall models [8]. In hard wall model, one imposes a cutoff at IR boundary. e IR cutoff in hard wall model is inverse of the QCD scale. e hard wall model describes many properties of QCD such as form factors, effective coupling constants, chiral symmetry breaking, and correlation functions but fails to accommodate Regge trajectory of meson masses. e problem of mass spectra can be removed by introduction of a dilaton field. is model is known as soſt wall model of AdS/QCD and the IR boundary in this model is shiſted to infinity. e transition between confining and deconfining phase is studied by Hawking-Page transition in bulk spacetime [26]. e high temperature phase is charged AdS black hole while low temperature phase is thermally charged AdS geometry. e confinement/deconfinement is studied in hard wall and soſt wall models in [14] and models with chemical potential are studied in [19–21, 27]. In the charged black hole solutions, charge of black hole is related to chemical potential of the quarks. e dual gauge theory defining the deconfining phase is AdS black hole while the confining phase is defined by thermally charged AdS solutions [21]. e UV divergences in these actions are removed by subtraction of action of thermal AdS [19, 21, 28]. e study of gauge/gravity duality provides a relation between the gravity theories in the AdS spacetime and con- formal field theories on the boundary of the AdS spacetime. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 543526, 7 pages http://dx.doi.org/10.1155/2014/543526

Transcript of Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Page 1: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Research ArticleStudy of ConfinementDeconfinement Transition inAdSQCD with Generalized Warp Factors

Shobhit Sachan

Department of Physics Banaras Hindu University Varanasi 221005 India

Correspondence should be addressed to Shobhit Sachan shobhitsachangmailcom

Received 27 June 2014 Accepted 13 August 2014 Published 21 August 2014

Academic Editor George Siopsis

Copyright copy 2014 Shobhit Sachan This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study analytical solutions of charged black holes and thermally charged AdS with generalized warped factors in Einstein-Maxwell-Dilaton system We calculate Euclidean action for charged AdS and thermally charged AdS The actions in bothbackgrounds are regularized by the method of background subtraction The study of phase transition between charged black holeand thermally charged AdS gives an insight into the confinementdeconfinement transition The plots of grand potential versustemperature and chemical potential versus transition temperature are obtained

1 Introduction

Strongly interacting systems are always a challenge to ouranalytical knowledge Quantum chromodynamics is such atheory which cannot be solved analytically in low energyregime There are two methods to solve QCD one is ldquolatticeQCDrdquo [1] and the other is ldquoAdSQCDrdquo The formulation oflattice QCD is based on discretization of spacetime and itrequires high performance computing On the other handAdSQCD is analytic approach and is motivated by thegaugegravity duality [2ndash5] Someproperties ofQCD like the-ories motivated by gaugegravity duality such as confinementand chiral symmetry breaking have been studied extensivelyin [6ndash21] and spectrum of mesons and baryons is studied in[22 23]

There are two approaches from where one can constructQCD like theoriesThese approaches are known as top-downand bottom-up approaches In top-down approach one startsfrom stringy D brane configurations and constructs modelsfor QCD [24 25] while in bottom-up approach one startsfrom QCD and attempts to construct its five-dimensionalgravity dualThese five-dimensional dual models can be gen-eralized to study various properties of QCD The bottom-upapproach is divided into two categories hard wall [7] and softwall models [8] In hard wall model one imposes a cutoff at

IR boundaryThe IR cutoff in hardwallmodel is inverse of theQCD scale The hard wall model describes many propertiesof QCD such as form factors effective coupling constantschiral symmetry breaking and correlation functions but failsto accommodate Regge trajectory of meson masses Theproblem of mass spectra can be removed by introduction ofa dilaton field This model is known as soft wall model ofAdSQCD and the IR boundary in this model is shifted toinfinity

The transition between confining and deconfining phaseis studied byHawking-Page transition in bulk spacetime [26]The high temperature phase is charged AdS black hole whilelow temperature phase is thermally charged AdS geometryThe confinementdeconfinement is studied in hard wall andsoft wall models in [14] and models with chemical potentialare studied in [19ndash21 27]

In the charged black hole solutions charge of black holeis related to chemical potential of the quarks The dual gaugetheory defining the deconfining phase is AdS black holewhile the confining phase is defined by thermally chargedAdS solutions [21] The UV divergences in these actions areremoved by subtraction of action of thermal AdS [19 21 28]

The study of gaugegravity duality provides a relationbetween the gravity theories in the AdS spacetime and con-formal field theories on the boundary of the AdS spacetime

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 543526 7 pageshttpdxdoiorg1011552014543526

2 Advances in High Energy Physics

In recent years a large number of generalized geometries arestudied which gives a dual scale invariant gauge theory Oneof the metrics representing such a geometry is given by

1198891199042= 1199032120572(1199032119911119891 (119903) 119889119905

2+

1

1199032119891 (119903)1198891199032+ 11990321198892)

where 120572 = minus120579

119889

(1)

In thismetric the constants 119911 and 120579 are dynamical and hyper-scaling violation exponents respectively This metric givesAdS solutions for 120579 = 0 and 119911 = 1 The scale transformations119905 rarr 120582

119911 119903 rarr 120582

minus1119903 119909119894rarr 120582119909

119894lead to 1198891199042

119889+2rarr 1205821205791198891198891199042

119889+2

Thus the transformations retain the spatially homogeneousand covariant nature but the distance scales as powers of120582 for nonzero values of 120579 The noninvariance of distance inreference of AdSCFT correspondence leads to violation ofhyperscaling in dual field theory In other words it is shownin [29] that if the hyperscaling violation exponent is includedin the metric the entropy scales as 119879(119889minus120579)119911 If hyperscaling isnot taken into account the entropy scales as 119879119889119911 [29ndash31]

In this paper we study the effect of warping on confine-mentdeconfinement transition in the simplest case by takingthe value of 119911 = 1 In this context we have only single chargein AdS black hole [32] We first study the holographic modelof QCDwith a dilaton potential in Einstein-Maxwell-Dilatonsystem The form of potential is taken to be exponentialwith some free parameters These parameters are fixed byvarious boundary conditions We calculate Euclidean actionsof charged AdS black hole and thermally charged AdSThese actions are regularized by subtraction of thermal AdSaction The Hawking-Page transition is studied and plotsbetween grand potential versus temperature and transitiontemperature versus chemical potential are given The plotsof transition temperature (119879

119888) versus chemical potential 120583 in

Figure 2 are plotted for different values of 120572 The warp factor120572 depends on hyperscaling violation exponent thereforethe transition temperature varies with hyperscaling violationexponent The transition temperature becomes independentof warp factor (or hyperscaling violation exponent) forcertain value of 120583 This point where all curves meet may bea signature of second-order transition

This paper is organized in six sections In Section 2we briefly summarized the calculations for 119911 = 1 andsingle gauge field In Sections 3 and 4 grand potentialsare calculated for charged AdS black hole and thermallycharged AdS Section 5 is devoted to the study of confine-mentdeconfinement transition and in Section 6 we concludeour analysis

2 Gravitational Solution of EMD Theory

In this section the solution Einstein-Maxwell-Dilaton systemwith hyperscaling violation [32] is givenWe take the solutionobtained in [32] and take dynamical exponent 119911 = 1 Taking119911 = 1 limits the number of gauge fields in solution to one

We begin with well-known Einstein-Maxwell-Dilaton actionwith exponential potential (119881 = 119881

0119890120574120601) given by

119878 = minusint119889119889+2

119909radic119892

times [1

21205812(119877 minus

1

2(120597120601)2

+ 1198810119890120574120601) minus

1

411989221198652119890120582120601]

(2)

where 1205812 = 8120587119866 119892 is coupling constant of dimension 119889 + 2and 120582 120574 and 119881

0are parameters of the model which will be

fixed laterThe equations of motion for gravitational part of action

(2) can be written as

1

21205812119866120583] minus

1

2(minus

119892120583]

2(120597120601)2

+ 120597120583120601120597]120601) minus

1

2119892120583]1198810119890120574120601 = 119879

120583]

(3)

where

119866120583] = 119877

120583] minus1

2119892120583]119877 119879

120583] =1

21198922119890120582120601119865120588

120583119865120588] minus

1

41198652119892120583]

(4)

The equations of motion (3) can be written in modified formas

119877120583] +

1198810119890120574120601

119889119892120583] =

1

2120597120583120601120597]120601 +

1205812

1198922119890120582120601119865120588

120583119865120588] minus

1

21198891198652119892120583]

(5)

The equation of motion for scalar field is

1

radic119892120597120583(radic119892119892

120583]120597]120601) = minus

120597 (1198810119890120574120601)

120597120601+1

2

1205812

11989221205821198901205821206011198652 (6)

and for gauge field is

1

radic119892120597120583(radic119892119890120582120601119865120583]) = 0 (7)

Let us consider the ansatz for our metric (with 119911 = 1)scalar field and gauge field which are given as

1198891199042= 1199032120572(1199032119891 (119903) 119889119905

2+

1

1199032119891 (119903)1198891199032+ 11990321198892)

120601 = 120601 (119903) 119865119903119905

= 0

(8)

We consider 119865120583] as the only function of 119903 and the rest of the

components are equal to zeroUsing our ansatz the solution forMaxwellrsquos equations can

be written as

119865119903119905= 119890minus120582120601

119903120572(2minus119889)minus119889

120588 (9)

where 120588 is integration constant and is to be related to chargeof the black hole later

On solving 119905119905 and 119903119903 components of Einsteinrsquos equationswe determine the scalar field which is given as

119890120601= 1198901206010119903radic2119889120572(120572+1)

= 1198901206010119903120577 (10)

Advances in High Energy Physics 3

The exponent on 119903 shows that to get well-defined solutionswe must have 120572(120572 + 1) ge 0

Using equations of motion the metric function is givenby

119891 (119903) = 1 minus119898

119903119889(1+120572)+1+

1198762

1199032119889(1+120572) (11)

where 119898 is related to the mass of the black hole and 119876 isrelated to 120588 by the following relation

1198762= minus

1205812

1198922

119890minus1205821206010

119889 (1 + 120572) (minus1 + 119889 minus 2120572 + 119889120572 + 120577120582)1205882 (12)

The parameter 120574 appearing in exponential of potential is fixedby using the fact that the a constant term (independent of119903) appears in metric Equating the powers of 119903 to zero theparameter 120574 can be fixed as minus2120572120577 The constant term isequated to unity to get the value119881

0 which is given by relation

1198810= 119889 (1 + 120572) (1 + 119889 + 119889120572) 119890

minus1205741206010 (13)

and using equation of motion for scalar field the value ofparameter 120582 is fixed as minus120574 The solution for field strength (9)becomes

119865119903119905= 119894119876119903

minus119889(120572+1) (14)

where we have defined 119876 = (119892120581)119876radic119889(1 + 120572)(119889120572 + 119889 minus 1)

119890minus12058212060102 The solution of gauge field 119860

119905is given by

119860119905(119903) =

119894119876

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 119862 (15)

where 119862 is a constant and related to boundary value of 119860119905

which is chemical potential of the system

3 AdS Black Hole

In this section we consider the black hole solution forthe warped geometry The solution of 119860

119905with appropriate

boundary conditions leads us to the solution of charged AdSblack hole Using the solution of 119860

119905in (15) we apply the

condition that at the boundary (119903 rarr infin) the value of 119860119905is

119894120583 where 120583 is chemical potential of the black hole and 119894 is dueto the consideration of Euclidean signature The boundaryvalue gives us the constant 119862 = 119894120583 and the solution of 119860

119905has

the form

119860119905(119903) = 119894 (120583 minus

119876

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

) (16)

At horizon (119903119867) 119860119905= 0 leads us to the relation between 120583

and 119876 which is given by

119876 =119889 (120572 + 1) minus 1

119903minus119889(120572+1)+1

119867

120583

997904rArr 119876 =120581

119892120583radic

119889 (1 + 120572) minus 1

119889 (1 + 120572)11989012058212060102119903119889(120572+1)minus1

119867

(17)

The radius of horizon for charged black hole solution isobtained by equating the metric function 119891(119903

119867) = 0 This

leads to the equation for 119903119867 which is given as

1199032119889(1+120572)

119867minus 119898119903119889(1+120572)minus1

119867+ 1198762= 0 (18)

and the Hawking temperature of the black hole is given by

119879 =1

4120587(119889120572 + 119889 + 1) 119903

119867(1 minus 119876

2 119889120572 + 119889 minus 1

119889120572 + 119889 + 1119903minus2119889(1+120572)

119867)

=1

4120587(119889120572 + 119889 + 1) 119903

119867

times (1 minus 1205832 1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

1

1199032119867

)

(19)

Now redefining some variables for simplicity

1198631=119889120572 + 119889 + 1

4120587 119863

2=1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

(20)

Using these redefinitions in (19) and solving the quadraticequation we get positive value of horizon radius as

119903119867=

119879 + radic1198792 + 41205832119863211198632

21198631

(21)

Using equation of motion action (2) can be written as

119878AdSBH

=1

119889int119889119889+2

119909radic119892[119881

1205812+

1

211989221198901205821206011198652]

=1

119889119881119889120573int119889119903radic119892[

119881

1205812+

1

211989221198901205821206011198652]

(22)

where 119881119889is 119889-dimensional volume and 120573 is inverse of black

hole temperature On substituting various values in the aboveaction it simplifies to

119878AdSBH

=1

119889119881119889120573119889 (1 + 120572)

1205812int119889119903119903

119889(120572+1)+2120572

times [(119889120572 + 119889 + 1) 119903minus2120572

minus1198762(119889120572 + 119889 minus 1) 119903

minus2119889(120572+1)minus2120572]

=1

119889119881119889120573119889 (1 + 120572)

1205812[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

(23)

where we take 119903max rarr infin at the end of the calculationsThe above action is singular at 119903max rarr infin Therefore

to regularize this action we subtract thermal AdS from thisaction The metric for thermal AdS is given by

1198891199042= 1199032120572(11990321198891199052+1

11990321198891199032+ 11990321198892) (24)

4 Advances in High Energy Physics

and action for thermal AdS with time periodicity 1205731is given

by equation

119878tAdS

= minus1198811198891205731

(1 + 120572)

1205812(119903max)

119889(120572+1)+1

(25)

Thus the regularized action for AdS black hole is given by

119878AdSBH

= lim119903maxrarrinfin

119881119889120573(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

119891 (119903max 119898 = 119876 = 0))

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus119881119889120573(1 + 120572)

1205812((119903119867)119889(120572+1)+1

+ 1198762(119903119867)minus119889(120572+1)+1

)

(26)

The factor (119891(119903max)119891(119903max 119898 = 119876 = 0))12 in front of the

last term in the first expression is inserted to match theEuclidean time periodicity at 119903 = 119903max where both thesolutions coincidewith each otherThe singular term (powersof 119903 with positive values) of AdS black hole solution iscancelled with the term in thermal AdS solution and we getthe regularized action By using thermodynamical relationΩ(120583 119879) = 119879119878on-shell we write the regularized grand potentialfor AdS black hole as

ΩAdSBH

= minus119881119889

1 + 120572

1205812((119903119867)119889(120572+1)+1

+ (119903119867)minus119889(120572+1)+1

1198762)

(27)

4 Thermally Charged AdS

This section is devoted to the study of thermally charged AdSsolution [21] The thermally charged AdS is also asymptot-ically AdS but does not have a horizon Due to absence ofhorizon we choose a lower cutoff for thermally charged AdSas 119903IR and integrate from 119903IR to infin The metric function forthermally charged AdS is given by

1198911(119903) = 1 +

1198762

1

1199032119889(120572+1) (28)

where 1198761is charge associated with thermally charged AdS

This metric function also satisfies Einstein-Maxwell equa-tions This geometry is simply obtained by putting 119898 = 0

in solution of AdS black hole The charge 1198761in this case

is different from that of AdS black hole due to differentboundary conditions

The field strength tensor for thermally charged AdS isgiven by the same equation as that for AdS black hole casebut now 119876 is replaced by 119876

1 Th expression is written as

1198651119903119905

= 1198941198761119903minus119889(120572+1)

where1198761=119892

1205811198761radic119889 (120572 + 1) (119889120572 + 119889 minus 1)119890

minus12058212060102

(29)

From this field strength the gauge field can be calculated as

1198601119905(119903) =

1198941198761

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 1198622 (30)

Again at 119903 rarr infin we have 1198601119905(infin) = 119862

2= 119894120583 but at 119903 = 119903IR

we apply Dirichlet boundary condition1198601119905(119903IR) = 119894120585120583 where

120585 is a constant to determined Thus at 119903IR

1198601119905(119903IR) = 119894120585120583 = 119894120583 minus

1198941198761

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

997904rArr 1198761=120581

119892120583 (1 minus 120585)radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR

(31)

Using the same procedure as done for AdS black hole wecompute regularized action for thermally chargedAdS whichis written as

119878tcAdS

= lim119903maxrarrinfin

1198811198891205731

(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

minus(1198911(119903max)

1198911(119903max 1198761 = 0)

)

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus1198811198891205731

(1 + 120572)

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

)

(32)

and grand potential for thermally charged AdS is given byequation

ΩtcAdS

= minus119881119889

1 + 120572

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

) (33)

where 1198761is function of chemical potential 120583 Using ther-

modynamical relation 119873 = minus120597Ω120597120583 we calculate the quarknumber for thermally chargedAdS which is given by relation

119873 = 2(120572 + 1)

1198891198922120583(1 minus 120585)

2(119889 (120572 + 1) minus 1) 119890

1205821206010119903119889(120572+1)minus1

IR (34)

As shown in [21] one has to useDirichlet boundary conditioninstead of Neumann we get free energy from Legendretransformation of grand potential and 120583119873120573

1is equal to

boundary action 119878tcAdS Calculating the boundary action wecan determine unknown parameter 120585 The boundary actionof thermally charged AdS is given by

119878tcAdS119887

=1

119889int120597119872

119889119889+1

119909radic119892(119889+1)120578120590119860120588119865120583120590119892120588120583119890120582120601

=1

119889119892211988111988912057311205831198761119890120582120601

(35)

where unit vector 120578119903 = (0 minusradic1198911(119903)119903120572minus1

0 0 ) and 119892119889+1 =119903(120572+1)(119889+1)

radic1198911(119903) Comparing (34) and (35) we evaluated

Advances in High Energy Physics 5

20

15

10

05

0

minus05

minus10

times1010

ΔΩV

3(M

eV)

T (MeV)0 20 40 60 80 100 120 140

10

100

200

400

Value of 120583

(a) For 120572 = 01

Value of 12058310

100

200

400

0 50 100 150

T (MeV)ΔΩV

3(M

eV)

4

3

2

1

0

minus1

minus2

minus3

times1012

(b) For 120572 = 03

Figure 1 Grand potential versus temperature at various values of 120583 for constant 120572

constant 120585 = 12 Thus charge 1198761of thermally charged AdS

is given by

1198761=

120581

2119892120583radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR (36)

5 ConfinementDeconfinement Transition

Now we study the transition from AdS black hole phase tothermally charged AdS To study this we take the differencebetween the actions of AdS black hole and thermally chargedAdS geometries with appropriate periodicity matching Thedifference in grand potentials is proportional to difference inactions The difference in actions is given by

Δ119878 = lim119903maxrarrinfin

1

119889119881119889120573119889 (1 + 120572)

1205812

times [119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

1198911(119903max)

)

12

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

= 1198811198891205731

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(37)

The factor (119891(119903max)1198911(119903max))12 in front of second term

comes from periodicity matching of AdS black hole andthermally charged AdS geometries Using this expression wecalculated grand potential which is given as

ΔΩ = 119881119889

1

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(38)

If the value of ΔΩ is less than zero the dominant geom-etry is AdS black hole and vice versa The sign of ΔΩgoverns the nature of the stability of phase and confine-mentdeconfinement transition The value of 119903IR is set to be323MeVwhich is calculated from themass of lightestmesons[14] We have plotted grand potential difference (ΔΩ) versustemperature (119879) for various values of warp factor 120572which aregiven in Figure 1

The relation describing the five-dimensional gravitationalconstant and that of the five-dimensional gauge couplingconstant are evaluated by the application of AdSCFT toQCD These constants relates the colour gauge group (119873

119888)

and number of flavours (119873119891) as

1

21205812=1198732

119888

81205872

1

21198922=119873119888119873119891

81205872 (39)

In our study we have used119873119891= 2 and119873

119888= 3

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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GravityJournal of

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AstrophysicsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

Page 2: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

2 Advances in High Energy Physics

In recent years a large number of generalized geometries arestudied which gives a dual scale invariant gauge theory Oneof the metrics representing such a geometry is given by

1198891199042= 1199032120572(1199032119911119891 (119903) 119889119905

2+

1

1199032119891 (119903)1198891199032+ 11990321198892)

where 120572 = minus120579

119889

(1)

In thismetric the constants 119911 and 120579 are dynamical and hyper-scaling violation exponents respectively This metric givesAdS solutions for 120579 = 0 and 119911 = 1 The scale transformations119905 rarr 120582

119911 119903 rarr 120582

minus1119903 119909119894rarr 120582119909

119894lead to 1198891199042

119889+2rarr 1205821205791198891198891199042

119889+2

Thus the transformations retain the spatially homogeneousand covariant nature but the distance scales as powers of120582 for nonzero values of 120579 The noninvariance of distance inreference of AdSCFT correspondence leads to violation ofhyperscaling in dual field theory In other words it is shownin [29] that if the hyperscaling violation exponent is includedin the metric the entropy scales as 119879(119889minus120579)119911 If hyperscaling isnot taken into account the entropy scales as 119879119889119911 [29ndash31]

In this paper we study the effect of warping on confine-mentdeconfinement transition in the simplest case by takingthe value of 119911 = 1 In this context we have only single chargein AdS black hole [32] We first study the holographic modelof QCDwith a dilaton potential in Einstein-Maxwell-Dilatonsystem The form of potential is taken to be exponentialwith some free parameters These parameters are fixed byvarious boundary conditions We calculate Euclidean actionsof charged AdS black hole and thermally charged AdSThese actions are regularized by subtraction of thermal AdSaction The Hawking-Page transition is studied and plotsbetween grand potential versus temperature and transitiontemperature versus chemical potential are given The plotsof transition temperature (119879

119888) versus chemical potential 120583 in

Figure 2 are plotted for different values of 120572 The warp factor120572 depends on hyperscaling violation exponent thereforethe transition temperature varies with hyperscaling violationexponent The transition temperature becomes independentof warp factor (or hyperscaling violation exponent) forcertain value of 120583 This point where all curves meet may bea signature of second-order transition

This paper is organized in six sections In Section 2we briefly summarized the calculations for 119911 = 1 andsingle gauge field In Sections 3 and 4 grand potentialsare calculated for charged AdS black hole and thermallycharged AdS Section 5 is devoted to the study of confine-mentdeconfinement transition and in Section 6 we concludeour analysis

2 Gravitational Solution of EMD Theory

In this section the solution Einstein-Maxwell-Dilaton systemwith hyperscaling violation [32] is givenWe take the solutionobtained in [32] and take dynamical exponent 119911 = 1 Taking119911 = 1 limits the number of gauge fields in solution to one

We begin with well-known Einstein-Maxwell-Dilaton actionwith exponential potential (119881 = 119881

0119890120574120601) given by

119878 = minusint119889119889+2

119909radic119892

times [1

21205812(119877 minus

1

2(120597120601)2

+ 1198810119890120574120601) minus

1

411989221198652119890120582120601]

(2)

where 1205812 = 8120587119866 119892 is coupling constant of dimension 119889 + 2and 120582 120574 and 119881

0are parameters of the model which will be

fixed laterThe equations of motion for gravitational part of action

(2) can be written as

1

21205812119866120583] minus

1

2(minus

119892120583]

2(120597120601)2

+ 120597120583120601120597]120601) minus

1

2119892120583]1198810119890120574120601 = 119879

120583]

(3)

where

119866120583] = 119877

120583] minus1

2119892120583]119877 119879

120583] =1

21198922119890120582120601119865120588

120583119865120588] minus

1

41198652119892120583]

(4)

The equations of motion (3) can be written in modified formas

119877120583] +

1198810119890120574120601

119889119892120583] =

1

2120597120583120601120597]120601 +

1205812

1198922119890120582120601119865120588

120583119865120588] minus

1

21198891198652119892120583]

(5)

The equation of motion for scalar field is

1

radic119892120597120583(radic119892119892

120583]120597]120601) = minus

120597 (1198810119890120574120601)

120597120601+1

2

1205812

11989221205821198901205821206011198652 (6)

and for gauge field is

1

radic119892120597120583(radic119892119890120582120601119865120583]) = 0 (7)

Let us consider the ansatz for our metric (with 119911 = 1)scalar field and gauge field which are given as

1198891199042= 1199032120572(1199032119891 (119903) 119889119905

2+

1

1199032119891 (119903)1198891199032+ 11990321198892)

120601 = 120601 (119903) 119865119903119905

= 0

(8)

We consider 119865120583] as the only function of 119903 and the rest of the

components are equal to zeroUsing our ansatz the solution forMaxwellrsquos equations can

be written as

119865119903119905= 119890minus120582120601

119903120572(2minus119889)minus119889

120588 (9)

where 120588 is integration constant and is to be related to chargeof the black hole later

On solving 119905119905 and 119903119903 components of Einsteinrsquos equationswe determine the scalar field which is given as

119890120601= 1198901206010119903radic2119889120572(120572+1)

= 1198901206010119903120577 (10)

Advances in High Energy Physics 3

The exponent on 119903 shows that to get well-defined solutionswe must have 120572(120572 + 1) ge 0

Using equations of motion the metric function is givenby

119891 (119903) = 1 minus119898

119903119889(1+120572)+1+

1198762

1199032119889(1+120572) (11)

where 119898 is related to the mass of the black hole and 119876 isrelated to 120588 by the following relation

1198762= minus

1205812

1198922

119890minus1205821206010

119889 (1 + 120572) (minus1 + 119889 minus 2120572 + 119889120572 + 120577120582)1205882 (12)

The parameter 120574 appearing in exponential of potential is fixedby using the fact that the a constant term (independent of119903) appears in metric Equating the powers of 119903 to zero theparameter 120574 can be fixed as minus2120572120577 The constant term isequated to unity to get the value119881

0 which is given by relation

1198810= 119889 (1 + 120572) (1 + 119889 + 119889120572) 119890

minus1205741206010 (13)

and using equation of motion for scalar field the value ofparameter 120582 is fixed as minus120574 The solution for field strength (9)becomes

119865119903119905= 119894119876119903

minus119889(120572+1) (14)

where we have defined 119876 = (119892120581)119876radic119889(1 + 120572)(119889120572 + 119889 minus 1)

119890minus12058212060102 The solution of gauge field 119860

119905is given by

119860119905(119903) =

119894119876

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 119862 (15)

where 119862 is a constant and related to boundary value of 119860119905

which is chemical potential of the system

3 AdS Black Hole

In this section we consider the black hole solution forthe warped geometry The solution of 119860

119905with appropriate

boundary conditions leads us to the solution of charged AdSblack hole Using the solution of 119860

119905in (15) we apply the

condition that at the boundary (119903 rarr infin) the value of 119860119905is

119894120583 where 120583 is chemical potential of the black hole and 119894 is dueto the consideration of Euclidean signature The boundaryvalue gives us the constant 119862 = 119894120583 and the solution of 119860

119905has

the form

119860119905(119903) = 119894 (120583 minus

119876

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

) (16)

At horizon (119903119867) 119860119905= 0 leads us to the relation between 120583

and 119876 which is given by

119876 =119889 (120572 + 1) minus 1

119903minus119889(120572+1)+1

119867

120583

997904rArr 119876 =120581

119892120583radic

119889 (1 + 120572) minus 1

119889 (1 + 120572)11989012058212060102119903119889(120572+1)minus1

119867

(17)

The radius of horizon for charged black hole solution isobtained by equating the metric function 119891(119903

119867) = 0 This

leads to the equation for 119903119867 which is given as

1199032119889(1+120572)

119867minus 119898119903119889(1+120572)minus1

119867+ 1198762= 0 (18)

and the Hawking temperature of the black hole is given by

119879 =1

4120587(119889120572 + 119889 + 1) 119903

119867(1 minus 119876

2 119889120572 + 119889 minus 1

119889120572 + 119889 + 1119903minus2119889(1+120572)

119867)

=1

4120587(119889120572 + 119889 + 1) 119903

119867

times (1 minus 1205832 1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

1

1199032119867

)

(19)

Now redefining some variables for simplicity

1198631=119889120572 + 119889 + 1

4120587 119863

2=1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

(20)

Using these redefinitions in (19) and solving the quadraticequation we get positive value of horizon radius as

119903119867=

119879 + radic1198792 + 41205832119863211198632

21198631

(21)

Using equation of motion action (2) can be written as

119878AdSBH

=1

119889int119889119889+2

119909radic119892[119881

1205812+

1

211989221198901205821206011198652]

=1

119889119881119889120573int119889119903radic119892[

119881

1205812+

1

211989221198901205821206011198652]

(22)

where 119881119889is 119889-dimensional volume and 120573 is inverse of black

hole temperature On substituting various values in the aboveaction it simplifies to

119878AdSBH

=1

119889119881119889120573119889 (1 + 120572)

1205812int119889119903119903

119889(120572+1)+2120572

times [(119889120572 + 119889 + 1) 119903minus2120572

minus1198762(119889120572 + 119889 minus 1) 119903

minus2119889(120572+1)minus2120572]

=1

119889119881119889120573119889 (1 + 120572)

1205812[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

(23)

where we take 119903max rarr infin at the end of the calculationsThe above action is singular at 119903max rarr infin Therefore

to regularize this action we subtract thermal AdS from thisaction The metric for thermal AdS is given by

1198891199042= 1199032120572(11990321198891199052+1

11990321198891199032+ 11990321198892) (24)

4 Advances in High Energy Physics

and action for thermal AdS with time periodicity 1205731is given

by equation

119878tAdS

= minus1198811198891205731

(1 + 120572)

1205812(119903max)

119889(120572+1)+1

(25)

Thus the regularized action for AdS black hole is given by

119878AdSBH

= lim119903maxrarrinfin

119881119889120573(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

119891 (119903max 119898 = 119876 = 0))

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus119881119889120573(1 + 120572)

1205812((119903119867)119889(120572+1)+1

+ 1198762(119903119867)minus119889(120572+1)+1

)

(26)

The factor (119891(119903max)119891(119903max 119898 = 119876 = 0))12 in front of the

last term in the first expression is inserted to match theEuclidean time periodicity at 119903 = 119903max where both thesolutions coincidewith each otherThe singular term (powersof 119903 with positive values) of AdS black hole solution iscancelled with the term in thermal AdS solution and we getthe regularized action By using thermodynamical relationΩ(120583 119879) = 119879119878on-shell we write the regularized grand potentialfor AdS black hole as

ΩAdSBH

= minus119881119889

1 + 120572

1205812((119903119867)119889(120572+1)+1

+ (119903119867)minus119889(120572+1)+1

1198762)

(27)

4 Thermally Charged AdS

This section is devoted to the study of thermally charged AdSsolution [21] The thermally charged AdS is also asymptot-ically AdS but does not have a horizon Due to absence ofhorizon we choose a lower cutoff for thermally charged AdSas 119903IR and integrate from 119903IR to infin The metric function forthermally charged AdS is given by

1198911(119903) = 1 +

1198762

1

1199032119889(120572+1) (28)

where 1198761is charge associated with thermally charged AdS

This metric function also satisfies Einstein-Maxwell equa-tions This geometry is simply obtained by putting 119898 = 0

in solution of AdS black hole The charge 1198761in this case

is different from that of AdS black hole due to differentboundary conditions

The field strength tensor for thermally charged AdS isgiven by the same equation as that for AdS black hole casebut now 119876 is replaced by 119876

1 Th expression is written as

1198651119903119905

= 1198941198761119903minus119889(120572+1)

where1198761=119892

1205811198761radic119889 (120572 + 1) (119889120572 + 119889 minus 1)119890

minus12058212060102

(29)

From this field strength the gauge field can be calculated as

1198601119905(119903) =

1198941198761

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 1198622 (30)

Again at 119903 rarr infin we have 1198601119905(infin) = 119862

2= 119894120583 but at 119903 = 119903IR

we apply Dirichlet boundary condition1198601119905(119903IR) = 119894120585120583 where

120585 is a constant to determined Thus at 119903IR

1198601119905(119903IR) = 119894120585120583 = 119894120583 minus

1198941198761

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

997904rArr 1198761=120581

119892120583 (1 minus 120585)radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR

(31)

Using the same procedure as done for AdS black hole wecompute regularized action for thermally chargedAdS whichis written as

119878tcAdS

= lim119903maxrarrinfin

1198811198891205731

(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

minus(1198911(119903max)

1198911(119903max 1198761 = 0)

)

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus1198811198891205731

(1 + 120572)

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

)

(32)

and grand potential for thermally charged AdS is given byequation

ΩtcAdS

= minus119881119889

1 + 120572

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

) (33)

where 1198761is function of chemical potential 120583 Using ther-

modynamical relation 119873 = minus120597Ω120597120583 we calculate the quarknumber for thermally chargedAdS which is given by relation

119873 = 2(120572 + 1)

1198891198922120583(1 minus 120585)

2(119889 (120572 + 1) minus 1) 119890

1205821206010119903119889(120572+1)minus1

IR (34)

As shown in [21] one has to useDirichlet boundary conditioninstead of Neumann we get free energy from Legendretransformation of grand potential and 120583119873120573

1is equal to

boundary action 119878tcAdS Calculating the boundary action wecan determine unknown parameter 120585 The boundary actionof thermally charged AdS is given by

119878tcAdS119887

=1

119889int120597119872

119889119889+1

119909radic119892(119889+1)120578120590119860120588119865120583120590119892120588120583119890120582120601

=1

119889119892211988111988912057311205831198761119890120582120601

(35)

where unit vector 120578119903 = (0 minusradic1198911(119903)119903120572minus1

0 0 ) and 119892119889+1 =119903(120572+1)(119889+1)

radic1198911(119903) Comparing (34) and (35) we evaluated

Advances in High Energy Physics 5

20

15

10

05

0

minus05

minus10

times1010

ΔΩV

3(M

eV)

T (MeV)0 20 40 60 80 100 120 140

10

100

200

400

Value of 120583

(a) For 120572 = 01

Value of 12058310

100

200

400

0 50 100 150

T (MeV)ΔΩV

3(M

eV)

4

3

2

1

0

minus1

minus2

minus3

times1012

(b) For 120572 = 03

Figure 1 Grand potential versus temperature at various values of 120583 for constant 120572

constant 120585 = 12 Thus charge 1198761of thermally charged AdS

is given by

1198761=

120581

2119892120583radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR (36)

5 ConfinementDeconfinement Transition

Now we study the transition from AdS black hole phase tothermally charged AdS To study this we take the differencebetween the actions of AdS black hole and thermally chargedAdS geometries with appropriate periodicity matching Thedifference in grand potentials is proportional to difference inactions The difference in actions is given by

Δ119878 = lim119903maxrarrinfin

1

119889119881119889120573119889 (1 + 120572)

1205812

times [119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

1198911(119903max)

)

12

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

= 1198811198891205731

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(37)

The factor (119891(119903max)1198911(119903max))12 in front of second term

comes from periodicity matching of AdS black hole andthermally charged AdS geometries Using this expression wecalculated grand potential which is given as

ΔΩ = 119881119889

1

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(38)

If the value of ΔΩ is less than zero the dominant geom-etry is AdS black hole and vice versa The sign of ΔΩgoverns the nature of the stability of phase and confine-mentdeconfinement transition The value of 119903IR is set to be323MeVwhich is calculated from themass of lightestmesons[14] We have plotted grand potential difference (ΔΩ) versustemperature (119879) for various values of warp factor 120572which aregiven in Figure 1

The relation describing the five-dimensional gravitationalconstant and that of the five-dimensional gauge couplingconstant are evaluated by the application of AdSCFT toQCD These constants relates the colour gauge group (119873

119888)

and number of flavours (119873119891) as

1

21205812=1198732

119888

81205872

1

21198922=119873119888119873119891

81205872 (39)

In our study we have used119873119891= 2 and119873

119888= 3

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

Page 3: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Advances in High Energy Physics 3

The exponent on 119903 shows that to get well-defined solutionswe must have 120572(120572 + 1) ge 0

Using equations of motion the metric function is givenby

119891 (119903) = 1 minus119898

119903119889(1+120572)+1+

1198762

1199032119889(1+120572) (11)

where 119898 is related to the mass of the black hole and 119876 isrelated to 120588 by the following relation

1198762= minus

1205812

1198922

119890minus1205821206010

119889 (1 + 120572) (minus1 + 119889 minus 2120572 + 119889120572 + 120577120582)1205882 (12)

The parameter 120574 appearing in exponential of potential is fixedby using the fact that the a constant term (independent of119903) appears in metric Equating the powers of 119903 to zero theparameter 120574 can be fixed as minus2120572120577 The constant term isequated to unity to get the value119881

0 which is given by relation

1198810= 119889 (1 + 120572) (1 + 119889 + 119889120572) 119890

minus1205741206010 (13)

and using equation of motion for scalar field the value ofparameter 120582 is fixed as minus120574 The solution for field strength (9)becomes

119865119903119905= 119894119876119903

minus119889(120572+1) (14)

where we have defined 119876 = (119892120581)119876radic119889(1 + 120572)(119889120572 + 119889 minus 1)

119890minus12058212060102 The solution of gauge field 119860

119905is given by

119860119905(119903) =

119894119876

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 119862 (15)

where 119862 is a constant and related to boundary value of 119860119905

which is chemical potential of the system

3 AdS Black Hole

In this section we consider the black hole solution forthe warped geometry The solution of 119860

119905with appropriate

boundary conditions leads us to the solution of charged AdSblack hole Using the solution of 119860

119905in (15) we apply the

condition that at the boundary (119903 rarr infin) the value of 119860119905is

119894120583 where 120583 is chemical potential of the black hole and 119894 is dueto the consideration of Euclidean signature The boundaryvalue gives us the constant 119862 = 119894120583 and the solution of 119860

119905has

the form

119860119905(119903) = 119894 (120583 minus

119876

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

) (16)

At horizon (119903119867) 119860119905= 0 leads us to the relation between 120583

and 119876 which is given by

119876 =119889 (120572 + 1) minus 1

119903minus119889(120572+1)+1

119867

120583

997904rArr 119876 =120581

119892120583radic

119889 (1 + 120572) minus 1

119889 (1 + 120572)11989012058212060102119903119889(120572+1)minus1

119867

(17)

The radius of horizon for charged black hole solution isobtained by equating the metric function 119891(119903

119867) = 0 This

leads to the equation for 119903119867 which is given as

1199032119889(1+120572)

119867minus 119898119903119889(1+120572)minus1

119867+ 1198762= 0 (18)

and the Hawking temperature of the black hole is given by

119879 =1

4120587(119889120572 + 119889 + 1) 119903

119867(1 minus 119876

2 119889120572 + 119889 minus 1

119889120572 + 119889 + 1119903minus2119889(1+120572)

119867)

=1

4120587(119889120572 + 119889 + 1) 119903

119867

times (1 minus 1205832 1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

1

1199032119867

)

(19)

Now redefining some variables for simplicity

1198631=119889120572 + 119889 + 1

4120587 119863

2=1205812

1198922

(119889120572 + 119889 minus 1)2

119889 (1 + 120572) (119889120572 + 119889 + 1)1198901205821206010

(20)

Using these redefinitions in (19) and solving the quadraticequation we get positive value of horizon radius as

119903119867=

119879 + radic1198792 + 41205832119863211198632

21198631

(21)

Using equation of motion action (2) can be written as

119878AdSBH

=1

119889int119889119889+2

119909radic119892[119881

1205812+

1

211989221198901205821206011198652]

=1

119889119881119889120573int119889119903radic119892[

119881

1205812+

1

211989221198901205821206011198652]

(22)

where 119881119889is 119889-dimensional volume and 120573 is inverse of black

hole temperature On substituting various values in the aboveaction it simplifies to

119878AdSBH

=1

119889119881119889120573119889 (1 + 120572)

1205812int119889119903119903

119889(120572+1)+2120572

times [(119889120572 + 119889 + 1) 119903minus2120572

minus1198762(119889120572 + 119889 minus 1) 119903

minus2119889(120572+1)minus2120572]

=1

119889119881119889120573119889 (1 + 120572)

1205812[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

(23)

where we take 119903max rarr infin at the end of the calculationsThe above action is singular at 119903max rarr infin Therefore

to regularize this action we subtract thermal AdS from thisaction The metric for thermal AdS is given by

1198891199042= 1199032120572(11990321198891199052+1

11990321198891199032+ 11990321198892) (24)

4 Advances in High Energy Physics

and action for thermal AdS with time periodicity 1205731is given

by equation

119878tAdS

= minus1198811198891205731

(1 + 120572)

1205812(119903max)

119889(120572+1)+1

(25)

Thus the regularized action for AdS black hole is given by

119878AdSBH

= lim119903maxrarrinfin

119881119889120573(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

119891 (119903max 119898 = 119876 = 0))

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus119881119889120573(1 + 120572)

1205812((119903119867)119889(120572+1)+1

+ 1198762(119903119867)minus119889(120572+1)+1

)

(26)

The factor (119891(119903max)119891(119903max 119898 = 119876 = 0))12 in front of the

last term in the first expression is inserted to match theEuclidean time periodicity at 119903 = 119903max where both thesolutions coincidewith each otherThe singular term (powersof 119903 with positive values) of AdS black hole solution iscancelled with the term in thermal AdS solution and we getthe regularized action By using thermodynamical relationΩ(120583 119879) = 119879119878on-shell we write the regularized grand potentialfor AdS black hole as

ΩAdSBH

= minus119881119889

1 + 120572

1205812((119903119867)119889(120572+1)+1

+ (119903119867)minus119889(120572+1)+1

1198762)

(27)

4 Thermally Charged AdS

This section is devoted to the study of thermally charged AdSsolution [21] The thermally charged AdS is also asymptot-ically AdS but does not have a horizon Due to absence ofhorizon we choose a lower cutoff for thermally charged AdSas 119903IR and integrate from 119903IR to infin The metric function forthermally charged AdS is given by

1198911(119903) = 1 +

1198762

1

1199032119889(120572+1) (28)

where 1198761is charge associated with thermally charged AdS

This metric function also satisfies Einstein-Maxwell equa-tions This geometry is simply obtained by putting 119898 = 0

in solution of AdS black hole The charge 1198761in this case

is different from that of AdS black hole due to differentboundary conditions

The field strength tensor for thermally charged AdS isgiven by the same equation as that for AdS black hole casebut now 119876 is replaced by 119876

1 Th expression is written as

1198651119903119905

= 1198941198761119903minus119889(120572+1)

where1198761=119892

1205811198761radic119889 (120572 + 1) (119889120572 + 119889 minus 1)119890

minus12058212060102

(29)

From this field strength the gauge field can be calculated as

1198601119905(119903) =

1198941198761

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 1198622 (30)

Again at 119903 rarr infin we have 1198601119905(infin) = 119862

2= 119894120583 but at 119903 = 119903IR

we apply Dirichlet boundary condition1198601119905(119903IR) = 119894120585120583 where

120585 is a constant to determined Thus at 119903IR

1198601119905(119903IR) = 119894120585120583 = 119894120583 minus

1198941198761

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

997904rArr 1198761=120581

119892120583 (1 minus 120585)radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR

(31)

Using the same procedure as done for AdS black hole wecompute regularized action for thermally chargedAdS whichis written as

119878tcAdS

= lim119903maxrarrinfin

1198811198891205731

(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

minus(1198911(119903max)

1198911(119903max 1198761 = 0)

)

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus1198811198891205731

(1 + 120572)

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

)

(32)

and grand potential for thermally charged AdS is given byequation

ΩtcAdS

= minus119881119889

1 + 120572

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

) (33)

where 1198761is function of chemical potential 120583 Using ther-

modynamical relation 119873 = minus120597Ω120597120583 we calculate the quarknumber for thermally chargedAdS which is given by relation

119873 = 2(120572 + 1)

1198891198922120583(1 minus 120585)

2(119889 (120572 + 1) minus 1) 119890

1205821206010119903119889(120572+1)minus1

IR (34)

As shown in [21] one has to useDirichlet boundary conditioninstead of Neumann we get free energy from Legendretransformation of grand potential and 120583119873120573

1is equal to

boundary action 119878tcAdS Calculating the boundary action wecan determine unknown parameter 120585 The boundary actionof thermally charged AdS is given by

119878tcAdS119887

=1

119889int120597119872

119889119889+1

119909radic119892(119889+1)120578120590119860120588119865120583120590119892120588120583119890120582120601

=1

119889119892211988111988912057311205831198761119890120582120601

(35)

where unit vector 120578119903 = (0 minusradic1198911(119903)119903120572minus1

0 0 ) and 119892119889+1 =119903(120572+1)(119889+1)

radic1198911(119903) Comparing (34) and (35) we evaluated

Advances in High Energy Physics 5

20

15

10

05

0

minus05

minus10

times1010

ΔΩV

3(M

eV)

T (MeV)0 20 40 60 80 100 120 140

10

100

200

400

Value of 120583

(a) For 120572 = 01

Value of 12058310

100

200

400

0 50 100 150

T (MeV)ΔΩV

3(M

eV)

4

3

2

1

0

minus1

minus2

minus3

times1012

(b) For 120572 = 03

Figure 1 Grand potential versus temperature at various values of 120583 for constant 120572

constant 120585 = 12 Thus charge 1198761of thermally charged AdS

is given by

1198761=

120581

2119892120583radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR (36)

5 ConfinementDeconfinement Transition

Now we study the transition from AdS black hole phase tothermally charged AdS To study this we take the differencebetween the actions of AdS black hole and thermally chargedAdS geometries with appropriate periodicity matching Thedifference in grand potentials is proportional to difference inactions The difference in actions is given by

Δ119878 = lim119903maxrarrinfin

1

119889119881119889120573119889 (1 + 120572)

1205812

times [119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

1198911(119903max)

)

12

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

= 1198811198891205731

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(37)

The factor (119891(119903max)1198911(119903max))12 in front of second term

comes from periodicity matching of AdS black hole andthermally charged AdS geometries Using this expression wecalculated grand potential which is given as

ΔΩ = 119881119889

1

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(38)

If the value of ΔΩ is less than zero the dominant geom-etry is AdS black hole and vice versa The sign of ΔΩgoverns the nature of the stability of phase and confine-mentdeconfinement transition The value of 119903IR is set to be323MeVwhich is calculated from themass of lightestmesons[14] We have plotted grand potential difference (ΔΩ) versustemperature (119879) for various values of warp factor 120572which aregiven in Figure 1

The relation describing the five-dimensional gravitationalconstant and that of the five-dimensional gauge couplingconstant are evaluated by the application of AdSCFT toQCD These constants relates the colour gauge group (119873

119888)

and number of flavours (119873119891) as

1

21205812=1198732

119888

81205872

1

21198922=119873119888119873119891

81205872 (39)

In our study we have used119873119891= 2 and119873

119888= 3

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Page 4: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

4 Advances in High Energy Physics

and action for thermal AdS with time periodicity 1205731is given

by equation

119878tAdS

= minus1198811198891205731

(1 + 120572)

1205812(119903max)

119889(120572+1)+1

(25)

Thus the regularized action for AdS black hole is given by

119878AdSBH

= lim119903maxrarrinfin

119881119889120573(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

119891 (119903max 119898 = 119876 = 0))

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus119881119889120573(1 + 120572)

1205812((119903119867)119889(120572+1)+1

+ 1198762(119903119867)minus119889(120572+1)+1

)

(26)

The factor (119891(119903max)119891(119903max 119898 = 119876 = 0))12 in front of the

last term in the first expression is inserted to match theEuclidean time periodicity at 119903 = 119903max where both thesolutions coincidewith each otherThe singular term (powersof 119903 with positive values) of AdS black hole solution iscancelled with the term in thermal AdS solution and we getthe regularized action By using thermodynamical relationΩ(120583 119879) = 119879119878on-shell we write the regularized grand potentialfor AdS black hole as

ΩAdSBH

= minus119881119889

1 + 120572

1205812((119903119867)119889(120572+1)+1

+ (119903119867)minus119889(120572+1)+1

1198762)

(27)

4 Thermally Charged AdS

This section is devoted to the study of thermally charged AdSsolution [21] The thermally charged AdS is also asymptot-ically AdS but does not have a horizon Due to absence ofhorizon we choose a lower cutoff for thermally charged AdSas 119903IR and integrate from 119903IR to infin The metric function forthermally charged AdS is given by

1198911(119903) = 1 +

1198762

1

1199032119889(120572+1) (28)

where 1198761is charge associated with thermally charged AdS

This metric function also satisfies Einstein-Maxwell equa-tions This geometry is simply obtained by putting 119898 = 0

in solution of AdS black hole The charge 1198761in this case

is different from that of AdS black hole due to differentboundary conditions

The field strength tensor for thermally charged AdS isgiven by the same equation as that for AdS black hole casebut now 119876 is replaced by 119876

1 Th expression is written as

1198651119903119905

= 1198941198761119903minus119889(120572+1)

where1198761=119892

1205811198761radic119889 (120572 + 1) (119889120572 + 119889 minus 1)119890

minus12058212060102

(29)

From this field strength the gauge field can be calculated as

1198601119905(119903) =

1198941198761

1 minus 119889 (120572 + 1)119903minus119889(120572+1)+1

+ 1198622 (30)

Again at 119903 rarr infin we have 1198601119905(infin) = 119862

2= 119894120583 but at 119903 = 119903IR

we apply Dirichlet boundary condition1198601119905(119903IR) = 119894120585120583 where

120585 is a constant to determined Thus at 119903IR

1198601119905(119903IR) = 119894120585120583 = 119894120583 minus

1198941198761

119889 (120572 + 1) minus 1119903minus119889(120572+1)+1

997904rArr 1198761=120581

119892120583 (1 minus 120585)radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR

(31)

Using the same procedure as done for AdS black hole wecompute regularized action for thermally chargedAdS whichis written as

119878tcAdS

= lim119903maxrarrinfin

1198811198891205731

(1 + 120572)

1205812

times

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

minus(1198911(119903max)

1198911(119903max 1198761 = 0)

)

12

119903119889(120572+1)+110038161003816100381610038161003816

119903max

0

= minus1198811198891205731

(1 + 120572)

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

)

(32)

and grand potential for thermally charged AdS is given byequation

ΩtcAdS

= minus119881119889

1 + 120572

1205812((119903IR)

119889(120572+1)+1

+ 1198762

1(119903IR)minus119889(120572+1)+1

) (33)

where 1198761is function of chemical potential 120583 Using ther-

modynamical relation 119873 = minus120597Ω120597120583 we calculate the quarknumber for thermally chargedAdS which is given by relation

119873 = 2(120572 + 1)

1198891198922120583(1 minus 120585)

2(119889 (120572 + 1) minus 1) 119890

1205821206010119903119889(120572+1)minus1

IR (34)

As shown in [21] one has to useDirichlet boundary conditioninstead of Neumann we get free energy from Legendretransformation of grand potential and 120583119873120573

1is equal to

boundary action 119878tcAdS Calculating the boundary action wecan determine unknown parameter 120585 The boundary actionof thermally charged AdS is given by

119878tcAdS119887

=1

119889int120597119872

119889119889+1

119909radic119892(119889+1)120578120590119860120588119865120583120590119892120588120583119890120582120601

=1

119889119892211988111988912057311205831198761119890120582120601

(35)

where unit vector 120578119903 = (0 minusradic1198911(119903)119903120572minus1

0 0 ) and 119892119889+1 =119903(120572+1)(119889+1)

radic1198911(119903) Comparing (34) and (35) we evaluated

Advances in High Energy Physics 5

20

15

10

05

0

minus05

minus10

times1010

ΔΩV

3(M

eV)

T (MeV)0 20 40 60 80 100 120 140

10

100

200

400

Value of 120583

(a) For 120572 = 01

Value of 12058310

100

200

400

0 50 100 150

T (MeV)ΔΩV

3(M

eV)

4

3

2

1

0

minus1

minus2

minus3

times1012

(b) For 120572 = 03

Figure 1 Grand potential versus temperature at various values of 120583 for constant 120572

constant 120585 = 12 Thus charge 1198761of thermally charged AdS

is given by

1198761=

120581

2119892120583radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR (36)

5 ConfinementDeconfinement Transition

Now we study the transition from AdS black hole phase tothermally charged AdS To study this we take the differencebetween the actions of AdS black hole and thermally chargedAdS geometries with appropriate periodicity matching Thedifference in grand potentials is proportional to difference inactions The difference in actions is given by

Δ119878 = lim119903maxrarrinfin

1

119889119881119889120573119889 (1 + 120572)

1205812

times [119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

1198911(119903max)

)

12

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

= 1198811198891205731

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(37)

The factor (119891(119903max)1198911(119903max))12 in front of second term

comes from periodicity matching of AdS black hole andthermally charged AdS geometries Using this expression wecalculated grand potential which is given as

ΔΩ = 119881119889

1

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(38)

If the value of ΔΩ is less than zero the dominant geom-etry is AdS black hole and vice versa The sign of ΔΩgoverns the nature of the stability of phase and confine-mentdeconfinement transition The value of 119903IR is set to be323MeVwhich is calculated from themass of lightestmesons[14] We have plotted grand potential difference (ΔΩ) versustemperature (119879) for various values of warp factor 120572which aregiven in Figure 1

The relation describing the five-dimensional gravitationalconstant and that of the five-dimensional gauge couplingconstant are evaluated by the application of AdSCFT toQCD These constants relates the colour gauge group (119873

119888)

and number of flavours (119873119891) as

1

21205812=1198732

119888

81205872

1

21198922=119873119888119873119891

81205872 (39)

In our study we have used119873119891= 2 and119873

119888= 3

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Page 5: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Advances in High Energy Physics 5

20

15

10

05

0

minus05

minus10

times1010

ΔΩV

3(M

eV)

T (MeV)0 20 40 60 80 100 120 140

10

100

200

400

Value of 120583

(a) For 120572 = 01

Value of 12058310

100

200

400

0 50 100 150

T (MeV)ΔΩV

3(M

eV)

4

3

2

1

0

minus1

minus2

minus3

times1012

(b) For 120572 = 03

Figure 1 Grand potential versus temperature at various values of 120583 for constant 120572

constant 120585 = 12 Thus charge 1198761of thermally charged AdS

is given by

1198761=

120581

2119892120583radic

119889 (120572 + 1) minus 1

119889 (120572 + 1)11989012058212060102119903119889(120572+1)minus1

IR (36)

5 ConfinementDeconfinement Transition

Now we study the transition from AdS black hole phase tothermally charged AdS To study this we take the differencebetween the actions of AdS black hole and thermally chargedAdS geometries with appropriate periodicity matching Thedifference in grand potentials is proportional to difference inactions The difference in actions is given by

Δ119878 = lim119903maxrarrinfin

1

119889119881119889120573119889 (1 + 120572)

1205812

times [119903119889(120572+1)+1

+ 1198762119903minus119889(120572+1)minus1

]119903max

119903119867

minus(119891 (119903max)

1198911(119903max)

)

12

[119903119889(120572+1)+1

+ 1198762

1119903minus119889(120572+1)minus1

]119903max

119903IR

= 1198811198891205731

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(37)

The factor (119891(119903max)1198911(119903max))12 in front of second term

comes from periodicity matching of AdS black hole andthermally charged AdS geometries Using this expression wecalculated grand potential which is given as

ΔΩ = 119881119889

1

1205812 ((119903IR)

119889(120572+1)+1

minus (119903119867)119889(120572+1)+1

)

+ 1205832119889 (1 + 120572) minus 1 119890

1205821206010

times(1

4119903119889(120572+1)minus1

IR minus 119903119889(120572+1)minus1

119867)

(38)

If the value of ΔΩ is less than zero the dominant geom-etry is AdS black hole and vice versa The sign of ΔΩgoverns the nature of the stability of phase and confine-mentdeconfinement transition The value of 119903IR is set to be323MeVwhich is calculated from themass of lightestmesons[14] We have plotted grand potential difference (ΔΩ) versustemperature (119879) for various values of warp factor 120572which aregiven in Figure 1

The relation describing the five-dimensional gravitationalconstant and that of the five-dimensional gauge couplingconstant are evaluated by the application of AdSCFT toQCD These constants relates the colour gauge group (119873

119888)

and number of flavours (119873119891) as

1

21205812=1198732

119888

81205872

1

21198922=119873119888119873119891

81205872 (39)

In our study we have used119873119891= 2 and119873

119888= 3

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Page 6: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

6 Advances in High Energy PhysicsTc

(MeV

)

140

120

100

80

60

40

20

0

100 200 300 400 500 600

120583 (MeV)

0

01

02

05

Value of 120572

Figure 2 119879 versus 120583 for various values of 120572

The value of 120572 is considered to be equal to minus120579119889 whichis commonly used in literature mentioned in [32] We havetaken dimension for our estimation to be five that is 119889 = 3

and 1206010= 0 To get plot 119879 versus 120583 we equate ΔΩ = 0 and

these plots for various values of 120572 are given in Figure 2

6 Conclusion

In this paper we studied the thermodynamic behavior ofAdSQCD from holographic approach with generalized warpfactorThe plots of grand potential per unit volume are shownin Figure 1 Figure 1(a) shows grand potential per unit volumeversus temperature for 120572 = 01 and Figure 1(b) for 120572 =

03 These plots show that the increasing value of chemicalpotential 120583 for constant 120572 and transition temperature gotlowered but the maximum value of grand potential increaseswhich indicates the stability of thermally charged AdS atlower temperatures The entropy difference Δ119878 (entropy isdefined as 119878 = minus(120597Ω120597119879)

119881120583) is nonzero which shows that

the transition is of first order (using Ehrenfest scheme forclassification of phase transition)

Figure 2 shows the plot between chemical potential andtransition temperature for various values of 120572 The resultsobtained here show similar qualitative behavior with variousresults obtained without warping dependence on dimensionexcept the fact that for different values of warping all plotsof Figure 2 meet at a point It means that the transition isindependent of warping on this pointWe believe that it is theonset of second order transition This is also expected fromrecent lattice data [33] It would be interesting to study massspectra of mesons in this scenario transport properties andcorrections arising due to Gauss-Bonnet gravity

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

S Sachan is supported by CSIR-Senior Research FellowshipGrant no (09013(0239)2009-EMR-I) The author wouldalso like to thank Dr Sanjay Siwach for discussing theproblem at various stages of this work

References

[1] J B Kogut and M A Stephanov The Phases of QuantumChromody-namics From Confinement to Extreme Environ-ments Cambridge University Press 2004

[2] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[3] E Witten ldquoAnti-de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti-de Sitter space thermal phase transition andconfinement in gauge theoriesrdquo Advances in Theoretical andMathematical Physics vol 2 no 3 pp 505ndash532 1998

[6] A Brandhuber N Itzhaki J Sonnenschein and S Yankielow-icz ldquoWilson loops confinement and phase transitions in large119873 gauge theories from supergravityrdquoThe Journal of High EnergyPhysics vol 1998 no 6 article 001 1998

[7] J Erlich E Katz D T Son and M A Stephanov ldquoQCD and aholographic model of hadronsrdquo Physical Review Letters vol 95no 26 Article ID 261602 2005

[8] A Karch E Katz D T Son and M A Stephanov ldquoLinearconfinement and AdSQCDrdquo Physical Review D vol 74 no 1Article ID 015005 2006

[9] C A Ballon Bayona H Boschi-Filho N R F Braga andL A Pando Zayas ldquoOn a holographic model for con-nementdeconfinementrdquo Physical Review D vol 77 Article ID046002 2008

[10] E Megıas H J Pirner and K Veschgini ldquoQCD thermodynam-ics using five-dimensional gravityrdquo Physical Review D vol 83no 5 Article ID 056003 2011

[11] K Veschgini E Megıas and H J Pirner ldquoTrouble finding theoptimal AdSQCDrdquo Physics Letters B vol 696 no 5 pp 495ndash498 2011

[12] L Da Rold and A Pomarol ldquoChiral symmetry breaking fromfive-dimensional spacesrdquoNuclear Physics B vol 721 no 1ndash3 pp79ndash97 2005

[13] A Parnachev andDA Sahakyan ldquoChiral phase transition fromstring theoryrdquo Physical Review Letters vol 97 no 11 Article ID111601 4 pages 2006

[14] C P Herzog ldquoA holographic prediction of the deconfinementtemperaturerdquo Physical Review Letters vol 98 Article ID 0916012007

[15] T Gherghetta J I Kapusta and T M Kelley ldquoChiral symmetrybreaking in the soft-wall AdSQCD modelrdquo Physical Review Dvol 79 Article ID 076003 2009

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Page 7: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Advances in High Energy Physics 7

[16] J Erdmenger N Evans I Kirsch and E JThrelfall ldquoMesons ingaugegravity dualsrdquo European Physical Journal A vol 35 no 1pp 81ndash133 2008

[17] R Cai and J P Shock ldquoHolographic confinementdeconfine-ment phase transitions of AdSQCD in curved spacesrdquo Journalof High Energy Physics vol 2007 no 8 article 095 2007

[18] O Andreev ldquoCold quark matter quadratic corrections andgaugestring dualityrdquo Physical Review D vol 81 no 8 ArticleID 087901 2010

[19] C Park D-Y Gwak B-H Lee Y Ko and S Shin ldquoSoft wallmodel in the hadronic mediumrdquo Physical Review D vol 84Article ID 046007 2011

[20] S Sachan and S Siwach ldquoThermodynamics of soft wallAdSQCD at finite chemical potentialrdquo Modern Physics LettersA vol 27 no 28 Article ID 1250163 2012

[21] B-H Lee C Park and S-J Sin ldquoA dual geometry of the hadronin dense matterrdquo Journal of High Energy Physics vol 7 article087 2009

[22] P Zhang ldquoLinear confinement for mesons and nucleons inAdSQCDrdquo Journal of High Energy Physics vol 2010 no 5article 039 2010

[23] D K Hong T Inami and H Yee ldquoBaryons in AdSQCDrdquoPhysics Letters B Nuclear Elementary Particle and High-EnergyPhysics vol 646 no 4 pp 165ndash171 2007

[24] T Sakai and S Sugimoto ldquoLow energy hadron physics inholographic QCDrdquo Progress of Theoretical Physics vol 113 pp843ndash882 2005

[25] T Sakai and S Sugimoto ldquoMore on a holographic dual of QCDrdquoProgress ofTheoretical Physics vol 114 no 5 pp 1083ndash1118 2005

[26] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 198283

[27] Y Kim B-H Lee S Nam C Park and S-J Sin ldquoDecon-finement phase transition in holographic QCD with matterrdquoPhysical Review D vol 76 Article ID 086003 2007

[28] M Cvetic S Nojiri and S D Odintsov ldquoBlack hole thermo-dynamics and negative entropy in de Sitter and anti-de SitterEinstein-Gauss-Bonnet gravityrdquo Nuclear Physics B vol 628 no1-2 pp 295ndash330 2002

[29] B S Kim ldquoHyperscaling violation a unified frame for effectiveholographic theoriesrdquo Journal of High Energy Physics vol 2012no 11 article 061 2012

[30] X Dong S Harrison S Kachru G Torroba and H WangldquoAspects of holography for theorieswith hyperscaling violationrdquoJournal of High Energy Physics vol 2012 article 41 2012

[31] J Gath J Hartong R Monteiro and N A Obers ldquoHolographicmodels for theories with hyperscaling violationrdquo Journal of HighEnergy Physics vol 2013 no 4 article 159 2013

[32] M Alishahiha E OrsquoColgain and H Yavartanoo ldquoChargedBlack Branes with Hyperscaling Violating Factorrdquo Journal ofHigh Energy Physics vol 2012 article 137 2012

[33] M Fromm J Langelage S Lottini and O Philipsen ldquoTheQCDdeconfinement transition for heavy quarks and all baryonchemical potentialsrdquo Journal of High Energy Physics vol 2012no 1 article 042 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Page 8: Study of confinement/deconfinement transition in AdS/QCD with generalized warp factors

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of


B-meson properties from lattice QCD

B-meson properties from lattice QCD

MATHEMATICAL ANALYSIS OF WARP ELONGATION IN ...

MATHEMATICAL ANALYSIS OF WARP ELONGATION IN ...

Heavy-light quarks interactions in QCD vacuum

Heavy-light quarks interactions in QCD vacuum

Nonsupersymmetric Seiberg duality, orientifold QCD, and noncritical strings

Nonsupersymmetric Seiberg duality, orientifold QCD, and noncritical strings

Two-loop QCD corrections of the massive fermion propagator

Two-loop QCD corrections of the massive fermion propagator

Radiative decay properties of charmonia in nonrelativistic QCD

Radiative decay properties of charmonia in nonrelativistic QCD

Bounds on |B from lattice QCD

Bounds on |B from lattice QCD

Chiral and deconfinement aspects of the QCD transition

Chiral and deconfinement aspects of the QCD transition

Exceptional Deconfinement in G(2) Gauge Theory - arXiv

Exceptional Deconfinement in G(2) Gauge Theory - arXiv

Lattice Hadron Structure: Applications within and beyond QCD

Lattice Hadron Structure: Applications within and beyond QCD

Numerical approach to Coulomb gauge QCD

Numerical approach to Coulomb gauge QCD

WARP FILMS AUSTRALIA - MIFF Industry

WARP FILMS AUSTRALIA - MIFF Industry

QCD jets from coherent states

QCD jets from coherent states

Synthetic Aperture Focusing using a Shear-Warp Factorization of the Viewing Transform

Synthetic Aperture Focusing using a Shear-Warp Factorization of the Viewing Transform

Warp architecture and implementation

Warp architecture and implementation

Towards the glueball spectrum from unquenched lattice QCD

Towards the glueball spectrum from unquenched lattice QCD

Low-energy Meson action from QCD : Extended Skyrme model

Low-energy Meson action from QCD : Extended Skyrme model

Hadronic correlators and condensate fluctuations in QCD vacuum

Hadronic correlators and condensate fluctuations in QCD vacuum

Divergence-Aware Warp Scheduling - CiteSeerX

Divergence-Aware Warp Scheduling - CiteSeerX

Addendum on High Density QCD with Heavy Ions - IYTE ...

Addendum on High Density QCD with Heavy Ions - IYTE ...