QCD condensates and holographic Wilson loops for asymptotically AdS spaces
QQCD condensates and holographic Wilson loops for asymptotically AdS spaces
R. Carcasses Quevedo a,b , ∗ J. L. Goity c,d , † and R. C. Trinchero a,b ‡ a Instituto Balseiro, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Argentina. b CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina.. c Department of Physics, Hampton University, Hampton, VA 23668, USA and d Thomas Jefferson National Accelerator Facility,Newport News, VA 23606, USA.
The minimization of the Nambu-Goto action for a surface whose contour defines a circular Wilsonloop of radius a placed at a finite value of the coordinate orthogonal to the boundary is considered.This is done for asymptotically AdS spaces. The condensates of even dimension n = 2 through 10 arecalculated in terms of the coefficient of a n in the expansion of the on-shell subtracted Nambu-Gotoaction for small a . The subtraction employed is such that it presents no conflict with conformalinvariance in the AdS case and need not introduce an additional infrared scale for the case ofconfining geometries. It is shown that the UV value of the condensates is universal in the sense thatthey only depends on the first coefficients of the difference with the AdS case. PACS numbers: 11.15-q, 11.15-Tk, 11.25-Tq, 12.38.Aw, 12.38.Lg
I. INTRODUCTION
The relation between large N gauge theories and stringtheory [1] together with the AdS/CFT correspondence[2–5] have opened new insights into strongly interact-ing gauge theories. The application of these ideas toQCD has received significant attention since those break-throughs. From the phenomenological point of view, theso called AdS/QCD approach has produced very inter-esting results in spite of the strong assumptions involvedin its formulation [6–11]. It seems important to furtherproceed investigating these ideas and refining the currentunderstanding of a possible QCD gravity dual.As is well known the vacuum of pure gauge QCD isthe simplest setting that presents key non-perturbativeeffects of QCD. In this regard, the gluon condensate G ≡ g π (cid:104) F aµν F µνa (cid:105) plays an important role. The exis-tence of a non-vanishing G was early on identified [12].It has important manifestations in hadron phenomenol-ogy [12, 13], and there are indications of its non-vanishingfrom lattice QCD [14–16]. The gluon condensate can beobtained from the vacuum expectation value of a smallWilson loop. In the holographic approach, such an ex-pectation value is obtained by minimizing the Nambu-Goto (NG) action for a loop lying in the boundary space[17, 18]. This is known to work in the strictly AdS case,i.e. for a conformal boundary field theory. In this workwe assume that this procedure also works in the non-conformal-QCD case provided an adequate 5-dimensionalbackground metric is chosen.The features and results of this work are summarizedas follows: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] • The NG action for a circular loop of radius a lyingat a given value of the coordinate orthogonal tothe boundary of an asymptotically AdS space isconsidered.• The minimization of this action leads to equationsof motion, whose solution is approximated by apower series in a .• The on-shell NG action is subtracted following theprocedure in ref. [17]. More precisely, an extensionof this procedure is proposed for the case underconsideration, where the base of the loop is at afinite value of the radial coordinate and a naturalinfrared limit is considered for the case of confiningtheories.• The gluon condensates of even dimension n = 2 through 10 are obtained from the coefficients of theexpansion in powers of a of the subtracted on-shellNG action S subNG , the last four ones assuming theabsence of the condensate of dimension .• It is shown that the UV value of these condensatesis universal in the sense that for a condensate of agiven dimension, its value does not depend on thevalue of the warp factor’s higher order coefficients.The paper is organized as follows. Section II definesthe problem to be considered, including the NG actionfor the circular loop and the asymptotically AdS back-ground metric. Section III deals with the subtraction ofthe on-shell NG action. Section IV gives some modelindependent results, which clarify the relation betweencondensates and the expansion coefficients of the warpfactor. Section V deals with the approximate solution ofthe equations of motion and the evaluation of the on-shellNG action as a power series in a . Section VI gives the re-sults for the gluon condensates, showing the above men- a r X i v : . [ h e p - ph ] N ov tioned universality. Section VII includes some concludingremarks. In addition four appendices are included. II. NG ACTION FOR A CIRCULAR LOOP ONAN ASYMPTOTICALLY
AdS
SPACE
The distance to be considered has the following generalform, ds = e A ( z ) ( dz + η ij dx i dx j )= G µν dx µ dx ν µ, ν = 1 , · · · , d + 1 . (1)It is defined by a metric with no dependence on theboundary coordinates and preserves the boundary spacePoincare invariance. This should be the case if only vac-uum properties are considered. The form of the warpfactor A ( z ) to be considered is, A ( z ) = − ln (cid:16) zL (cid:17) + f ( z ) , (2)where f ( z ) is a dimensionless function. In this work f ( z ) is taken to be a power series in z [31], i.e., f ( z ) = (cid:88) k =1 α k z k . (3)The case f ( z ) = 0 corresponds to the AdS metric. Thisdeviation from the AdS case could be produced by a bulkgravity theory including matter fields [19]. Possible can-didates for these bulk gravity theories have been consid-ered in [20, 21].The area of a surface embedded in this space is givenby the NG action, S NG = 12 πα (cid:48) ˆ d σ √ g , (4)where g is the determinant of the induced metric on thesurface, which is given by, g ab = G µν ∂ a X µ ∂ b X ν , where X µ are the coordinates of the surface embedded inthe ambient d +1 dimensional space. The indices a, b referto coordinates on the surface. The case to be consideredis a circular loop whose contour lies at a constant value z of the coordinate z and in the i − j spatial plane. Thecoordinates on the surface are then taken to be r and φ ,the polar coordinates. Therefore the embedding can bedescribed by the following, X k = 0 (5) X = z ( r ) (6) X i = r cos φ, X j = r sin φ ( ∀ k (cid:54) = i (cid:54) = j ) , (7)with the boundary conditions, z ( a ) = z , z (cid:48) (0) = 0 , (8) which states that the contour of the circular loop of ra-dius a is located at z and that no cusps are admitted.Replacing the embedding (5) in the action (4), after atrivial integration in the angular variable, leads to thefollowing expression, S NG = 1 α (cid:48) ˆ a e A ( z ) r (cid:112) z (cid:48) dr , (9)where the prime denotes derivative respect to r . Theminimal surface is given by the solution of the followingequations of motion with the above mentioned boundaryconditions, r z (cid:48)(cid:48) ( r )1 + z (cid:48) ( r ) + z (cid:48) ( r ) − r dA ( z ) dz = 0 . (10)For the AdS case A ( z ) = − ln zL the solution is, z ( r ) = (cid:113) a + z − r , (11)which upon replacing in (4) leads to the following expres-sion for the on-shell NG action, S o.s.NG AdS = L α (cid:48) (cid:32)(cid:115) a z − (cid:33) . (12) III. THE SUBTRACTED ON-SHELL NGACTION
Replacing the solution of the previous section in theNG action leads to a divergent expression when z → ,i.e. near the UV boundary of the space. This happens inthe AdS case and also when f ( z ) (cid:54) = 0 . Therefore, a sub-traction procedure should be employed. The action re-quires regularization, where the most obvious procedureis to choose z (cid:54) = 0 , and a renormalized action is obtainedby implementing a subtraction, as it is discussed in de-tail in this section. A procedure of minimal subtraction,implemented by disregarding the /z term in S NG wasimplemented in [22].As shown in [17] for the rectangular loop, a physicallymotivated procedure is to subtract the contribution ofthe heavy "quark" mass to the action. This contributioncorresponds to the area of a cylinder with axis parallelto z , extending from z = ∞ to z = 0 , for the case of thebase of the loop located at z = 0 . It could be thoughtthat in the case considered in this paper, the base ofthe loop located at z , the area of a cylinder with axisparallel to z , extending from z = ∞ to z , should be sub-tracted. However such a procedure should be modifiedin two aspects,• First, in the AdS case, it leads to a loss of con-formal invariance, more precisely the value of S subNG would depend on the radius of the loop. Requiringindependence of the value of S subNG on the radius a ,leads to the following definition of the subtractedaction, S subNG = S NG − r ( a, z ) α (cid:48) ˆ z IR z dz e A ( z ) , (13)where z IR is an infrared scale whose motivation anddefinition is explained below. For the AdS case thefunction r ( a, z ) is fixed by conformal invarianceand given by, r AdS ( a, z ) = (cid:113) a + z , (14)leading to, S sub,AdSNG = − L α (cid:48) . (15)The radius r ( a, z ) corresponds to the radius of aloop located at the boundary whose minimal sur-face would intersect the plane z = z with a circleof radius a . Therefore the following holds, lim z → r ( a, z ) = a . (16)For non-AdS cases the same procedure couldbe employed, using the corresponding value r Non − AdS ( a, z ) . However it should be noted thatin the non-AdS case using the AdS r given in (14),also leads to a finite value for the S subNG and presentsno-conflict with conformal invariance. This lastprocedure will be employed below.• Second, confining warp factors are such that e A ( z ) presents a global minimum for a finite value for thisfactor[23, 24]. Let z m denote the location of thisminimum in the coordinate z . In these conditions,integrating e A ( z ) between z and ∞ would lead toa divergent result. Introducing a infrared integra-tion limit z IR as in (13) eliminates this divergenceat the cost of introducing this ad-hoc infrared cut-off. In this respect the following remarks are im-portant, – The result for the coefficients of a n , n > in S subNG do not depend on z IR . This fact is shownin section VI, and is due to property (16). – On the other hand these confining warp fac-tors already have a natural infrared scale.This is given by the location z m of the globalminimum. This is a natural candidate to beidentified with z IR . In this respect it is worthnoting that for z < z m the minimal surfacecould never exceed the value z m , otherwise itwould not be minimal [32]. In what followsthe choice z IR = z m is made. It is emphasizedthat other choices are by no means excluded.Different choices produce different coefficientsfor the perimeter in S subNG . Figure 1: Substraction scheme.
It is noted that this subtraction is non-vanishing even ifthe loop is located at a finite value of the coordinate z .Figure 1 illustrates the proposed subtraction procedure,for the case of a minimal surface bounded by a closedcontour C . A. Convergence of the subtracted NG action in theUV limit
The considered warp factors diverge in the UV, theleading singularity is, A ( z ) ∼ − ln (cid:16) zL (cid:17) ⇒ A (cid:48) ( z ) ∼ − z , (17)this makes the integrand appearing in the NG action di-verge at z = 0 .In order to analyze the behavior of the solution nearthe boundary, the approach in [25] is employed. The NGaction is written in terms of r as a function of z , r ( z ) ,this leads to, S NG = 1 α (cid:48) ˆ z z e A ( z ) r ( z ) (cid:113) r (cid:48) ( z ) dz , (18)where r (0) = r and r ( z ) = a . The equation of motionis, ( − rA (cid:48) ( z ) r (cid:48) ) (cid:0) r (cid:48) (cid:1) + rr (cid:48)(cid:48) = 0 . (19)Near the boundary (19) implies, − lim z → r z r (cid:48) (cid:0) r (cid:48) (cid:1) + lim z → rr (cid:48)(cid:48) − lim z → (cid:0) r (cid:48) (cid:1) = 0 . (20)This last equation shows that if lim z → r (cid:48)(cid:48) ( z ) is as-sumed to be finite, then lim z → r (cid:48) ( z ) can not be infi-nite since in that case it will be impossible to cancel theterms containing r (cid:48) . Furthermore the cancellation of theterms involving r (cid:48) ( z ) , require that lim z → r (cid:48) ( z ) = 0 as z (cid:15) , (cid:15) > . In addition the cancellation of the constantterm in (20) requires (cid:15) = 0 .On the contrary, if lim z → r (cid:48)(cid:48) ( z ) is assumed to be in-finite, then lim z → r (cid:48) ( z ) → ∞ , which can be proved byintegrating the former, and again it is not possible to can-cel all the divergent terms due to their different degreesof divergence. Therefore, lim z → r (cid:48) ( z ) = 0 (21) r (cid:48) ( z ) = − za + ... ( z (cid:28) a ) . (22)Next, this asymptotics is plugged in the S subNG (13). Inthis respect it is convenient to rewrite it in the form, S subNG = 1 α (cid:48) ˆ z z e A ( z ) (cid:18) r ( z ) (cid:113) r (cid:48) ( z ) − r ( a, z ) (cid:19) − r ( a, z ) α (cid:48) ˆ z m z dz e A ( z ) . For the considered cases of A ( z ) , the second term is con-vergent because the integrand has no poles in the finiteintegration interval. The first term is also finite, indeed: lim z → e A ( z ) (cid:18) r ( z ) (cid:113) r (cid:48) ( z ) − a (cid:19) =lim z → z (cid:18) a (cid:18) c z + . . . (cid:19) − a (cid:19) = a lim z → z (cid:18) c z + . . . (cid:19) = 0 . Thus, the integrand is finite everywhere inside the finiteintegration region and therefore the integral is finite. Fur-thermore this last equation shows that the divergent termin the UV of S NG is proportional to the perimeter of theloop, which is consistent with the fact that also the sub-traction is proportional to the perimeter of the loop inthe UV, as shown by (16). IV. MODEL INDEPENDENT RESULTS
In this section some results that follow from the generalsetting described in the previous sections are considered.No approximation is involved in the derivation of theseproperties.
A. In QCD f ( z ) is even It is recalled that f ( z ) is the function appearing in thewarp factor (2). The title of this subsection means the fol-lowing. The basic hypothesis underlying this work is thatthe vacuum expectation value of the Wilson loop in QCD is given by S subNG . It will be shown below that under thisassumption, the fact that there are no odd-dimensionalcondensates [33] in QCD implies that f ( z ) = f ( − z ) . Theproof of this assertion is based on the following interme-diate result. If the expansion of α (cid:48) L S NG [ z ]( a ) as a power series in a only involves even powers of a then, f ( z ) − f ( − z ) = const. (23)Proof. Denoting by S NG [ z ]( a ) the NG action with pa-rameter a , the hypothesis is, S NG [ z ]( a ) = S NG [ z ]( − a ) . Noting that the change a → − a is, at the level of the NGaction, the same as changing z → − z implies, S NG [ z ]( − a ) = S NG [ − z ]( a ) ⇓ (23) S NG [ z ]( a ) = S NG [ − z ]( a ) . Due to this last equality if z ( r ) extremizes the NG actionso does − z ( r ) . Therefore − z ( r ) must also be a solutionof the equation of motion. The equation of motion for z ( r ) is, r z (cid:48)(cid:48) ( r )1 + z (cid:48) ( r ) + z (cid:48) ( r ) − r ( − z + dfdz ( z )) = 0 the equation of motion for − z ( r ) is, − (cid:32) r z (cid:48)(cid:48) ( r )1 + z (cid:48) ( r ) + z (cid:48) ( r ) − r ( − z − dfdz ( − z ) (cid:33) = 0 summing up these two equations leads to, dfdz ( − z ) − dfdz ( z ) = 0 ⇒ ddz [ f ( − z ) − f ( z )] = 0 ⇓ f ( − z ) − f ( z ) = const . as claimed.Now, the only solution to (23) for arbitrary z is, f odd ( z ) = f ( z ) − f ( − z ) = 0 , which shows that only even functions f ( z ) are relevantfor QCD. In particular for the warp factors considered inthe present work, the above general result implies that ifthe only non-vanishing condensates are even dimensional,the coefficients α n must vanish if n = odd . B. Condensate of dimension n > is independentof α m for m > n and z → First it is noted that α (cid:48) L S NG [ z ]( a, α ) is dimensionlessand that α n has dimension of length to the − n . There-fore if α m would contribute to the condensate of dimen-sion n < m then inverse powers of α k should appearfor some k > n . Therefore in that case α (cid:48) L S NG [ z ]( a, α ) would diverge when α k → . However the integrand in α (cid:48) L S NG [ z ]( a, α ) is well defined when any or all of the α ’svanish. Indeed the only divergence in α (cid:48) L S NG [ z ]( a, α ) isproportional to a , and appears when all the α n vanish,but something proportional to a does not contribute tothe condensates with n > . Therefore only positivepowers of the α ’s can appear and the result follows fromdimensional reasons. It should be noted that this re-sult holds for z → , otherwise since z has dimensionsof length all the dimensional arguments made above arenot valid. In conclusion, the general expression for theexpansion in powers of a of the NG action is, α (cid:48) L S NG [ z ]( a, α ) = s (0) + s (2) α a + ( s (4)2 α + s (4)4 α ) a +( s (6)2 α + s (6)2 , α α + s (6)6 α ) a + · · · , where the coefficients s ( n ) are dimensionless. V. ON-SHELL NG ACTION EXPANDED INPOWERS OF THE RADIUS a A. Condensates of dimension and The approach employed in this section is basically thesame as in [22]. That is, expand the solution of the equa-tions of motion as a power series in a , replace in theLagrangian, expand it in powers of a and then inte-grate. However they differ in some aspects. An impor-tant difference is that in this work more general curvedbackgrounds are considered. More precisely, the warpfactors given in (2)-(3) are considered for n = 1 andfor both α and α non-vanishing. The consideration of α (cid:54) = 0 is particularly relevant from the phenomenolog-ical point of view. This is so because α (cid:54) = 0 allows fora non-vanishing gluon condensate of dimension with-out having at the same time one of dimension 2 which isnot allowed in QCD [34]. The other difference concernsthe subtraction procedure which in this work is done asdescribed in section III. According to this procedure theNG action should be calculated for a loop lying at a value z of the coordinate orthogonal to the boundary. In thisrespect it is convenient to define the variable, t = (cid:113) w − ρ , w = z a , ρ = ra . In this variable the AdS solution (11) is written as, w ( t ) = t , w = za . In terms of the variable ψ ( t ) = w ( t ) the NG action isgiven by: S NG = L α (cid:48) ˆ e a α ψ + a α ψ ) t (cid:114) w − t ) ψ (cid:48) ( t ) t ψ ( t ) ψ ( t ) dt . (24) The equation of motion for this action reads: a t α ψ ( t ) − (cid:0) − t + w (cid:1) (2 t − ψ (cid:48) ( t )) ψ (cid:48) ( t ) ++16 ψ ( t ) (cid:0) a t α + a t (cid:0) − t + w (cid:1) α ψ (cid:48) ( t ) (cid:1) −− ψ ( t ) (cid:0) t − (cid:0) t + w (cid:1) ψ (cid:48) ( t ) (cid:1) −− ψ ( t ) (cid:2) − a t (cid:0) − t + w (cid:1) α ψ (cid:48) ( t ) (25) + t (cid:0) − t + w (cid:1) ψ (cid:48)(cid:48) ( t ) (cid:3) = 0 . As explained earlier, the boundary conditions to be re-quired are the following ones ψ ( w ) = w , ψ (cid:48) ( (cid:113) w ) = finite , which correspond to the loop located in the plane z = z and the surface with no cusp at r = 0 .Next a power series expansion ansatz for the solutionis considered , ψ ( t ) = (cid:88) i =0 ψ i ( t ) a i (26)Replacing in (25) and requiring the vanishing of the co-efficient in front of a i , for i = 0 this leads to, (cid:0) − t − w (cid:1) (2 t − ψ (cid:48) ( t )) ψ (cid:48) ( t ) +4 ψ ( t ) (cid:2)(cid:0)(cid:0) t + w (cid:1) ψ (cid:48) ( t ) (cid:1) + t (cid:0) − t + (cid:0) − t − w (cid:1) ψ (cid:48)(cid:48) ( t ) (cid:1)(cid:3) = 0 whose solution is the AdS one ψ ( t ) = t . For i = 1 , (cid:0) w (cid:1) (cid:0) t α + ψ (cid:48) ( t ) (cid:1) =+ t (cid:0) − t − w (cid:1) ψ (cid:48)(cid:48) ( t ) = 0 whose solution up to order O ( w ) is, ψ ( t ) = −
11 + t (cid:8) t (cid:0) − − t + t + ( − − t ) t ) w (cid:1) +2(1 + t ) (cid:0) w (cid:1) arctanh( t )+(1 + t ) (cid:0) w (cid:1) log(1 − t ) (cid:9) α . In a similar fashion the equation and its solution for ψ ( t ) are obtained.Next the NG action expansion in powers of a is com-puted. Replacing the solution (26) in the integrand of(24), expanding in powers of a and w and integratingleads to, S NG = L α (cid:48) (cid:32) (cid:112) w w − a (cid:20) − w ( 83 − (cid:21) α + a (cid:26)(cid:20)
149 (17 −
24 log 2) − w
83 ++ w
445 (599 −
744 log 2) (cid:21) α +(14 + w α (cid:27) + ... (cid:19) (27)the first term in the parenthesis is divergent in the UVlimit w → . This divergence, as will be seen in the nextsection, is canceled by the subtraction S CT . The otherterms are finite in this limit. Also, in this limit the resultfor the coefficients of a and a coincide with the ones in[22]. B. Condensates of dimension , and It is recalled that as in subsection IV A a condensateof dimension n is by definition the coefficient of a n in theexpansion of α (cid:48) L S NG [ z ]( a ) in powers of a . The calcula-tion of these condensates is done in the UV limit, z → .This procedure is valid since, according to the analysis insubsection III A, the only coefficient that diverge in thislimit, is the one corresponding to the perimeter of theloop, i.e. the coefficient of a . Taking into account thisremark, the calculation of these condensates follows thesame technique as in the previous subsection except that z = 0 is taken from the start. Their computation is pos-sible under the assumption α = 0 , i.e. no dimension twocondensate. As an example the condensate of dimension is considered. That condensate must be proportional to α . This follows from the dimensional arguments whichare considered in appendix 2. There it is shown thatfor the warp factor of the form (2), α (cid:48) S NG /L should bedimensionless, thus the coefficient of a in this quantityshould have dimension of length to the − . Next recall-ing that the dimension of α n is length to the − n , then theonly way of getting such a dimension in terms of positive[35] powers of the α ’s is by means of α α or α , thusthe assumption α = 0 leaves only α . In terms of thevariables t and ψ the action to be considered is therefore, S (6) NG = L α (cid:48) ˆ e a α ψ t (cid:114) w − t ) ψ (cid:48) ( t ) t ψ ( t ) ψ ( t ) dt (28)next an expansion in powers of a of the solution is con-sidered as in (26), replacing this in the equation of motiondetermines the coefficients ψ i ( t ) , giving, ψ (6) ( t ) = t + a α (cid:0) t − t − t − t −
24 log(1 + t ) (cid:1) (29)replacing in (28) gives the following contribution propor-tional to a , α (cid:48) L S NG | a = 35 α a For the case of the dimension condensate, for dimen-sional reasons, only α and α are relevant. The actionto be considered thus involves only these two coefficientsin the warp factor. The solution can be obtained as apower series in a up to order a , having an expressionconsiderably more lengthy than (29), which is given inappendix C. The contribution proportional to a to the NG action is given by, α (cid:48) L S NG | a = − − α − α ] a For the case of the dimension condensate, for dimen-sional reasons, only α , α and α are relevant. The ac-tion to be considered thus involves only these coefficientsin the warp factor. The solution can be obtained as apower series in a up to order a , having an expressionconsiderably more lengthy than (29), which is given inappendix C. The contribution proportional to a to theNG action is given by, α (cid:48) L S NG | a = 134725 [(2999 − α α + 175 α ] a VI. THE GLUON CONDENSATE, UVUNIVERSALITYA. The computation of the subtraction
The subtracted NG action is, S subNG = S NG − S CT where, S CT = (cid:112) a + z α (cid:48) ˆ z m z dz e A ( z ) = L (cid:112) a + z α (cid:48) ˆ z m z dz e (cid:80) n =1 α n z n z (30)and z m denotes the minimum of e A ( z ) . Here the com-putation is done for the case where only α and α aredifferent from . In this case, z m = 12 (cid:115) (cid:112) α + 4 α α − α α (31)Since the integrand in (30) is well behaved in the inte-gration region then the exponential in the integrand canbe expanded before doing the integral [36] Proceeding inthis way leads to, α (cid:48) L S CT = (cid:113) a + z ˆ z m z dz e α z +2 α z z = (cid:113) a + z ˆ z m z dz (cid:20) z + 2 α + (cid:0) α + 2 α (cid:1) z + (cid:18) α α α (cid:19) z + · · · (cid:21) = (cid:113) a + z (cid:20) − z + 2 α z + 23 (cid:0) α + α (cid:1) z + 415 α (cid:0) α + 3 α (cid:1) z + · · · (cid:21) z m z (32)it is to be noted that the − /z appearing in the lastequality, when evaluated at z = z and multiplied bythe factor (cid:112) a + z , cancels the divergent term, when z → , appearing in the on-shell NG action in (27). B. The subtracted on-shell NG action S subNG is defined in (13). Using (27) , (32) and keepingup to terms of order z , leads to, α (cid:48) L S subNG = − a (cid:18) z m − z m α (cid:19) + a (cid:18) α (cid:19) + a ( α
149 (17 −
24 log 2) + α
149 ) − a z α + z (cid:20) α (cid:18) − log 16 (cid:19) + α
245 (1198 − − α (cid:21) (33)the first two lines corresponds to the terms that survive inthe UV limit z → . The IR scale z m should be replacedby its expression (31). In this respect it is worth noticingthat the contribution of that scale is proportional to a ,therefore a change in that scale only changes the coef-ficient of the perimeter. For α = 0 the results for thecoefficients of a and a coincide in the UV limit withthe ones in [22].Next the expression of S subNG as a power series in a inthe UV limit z → , is given for the case α = 0 , α (cid:48) L S subNG | z = − z m a + α a + 35 α a − (cid:2) ( − α − α ] a + 134725 [(2999 − α α + 175 α ] a (34)Due to the proof in IV B these results are unchangedby considering additional α ’s in the expression (3). Itmeans that the coefficients of a n in this expression areexact, the inclusion of additional terms in the expansionof the warp exponent do not change their value. This isa strictly UV result, it is only valid in the limit z → .It shows that if the expectation value of Wilson loops arerelated to minimal areas in the dual theory, as assumed,then the expectation values of gauge invariant operatorsin QCD can be used to systematically build the QCDdual background. In particular, since there is no dimen-sion two gauge invariant operator in pure QCD then thecoefficient of a should be zero [37]. Thus, under theseconditions, this absence implies α = 0 . The case of thecoefficient of a is different since in QCD there is a gaugeinvariant quantity of dimension , which is the expec-tation value of (cid:104) F µν F νµ (cid:105) . This coefficient is related tothe gluon condensate, and its value fixes the value of α .This procedure can be continued for higher order termsin the expansion. Higher dimensional condensates fix thevalues of higher index α i coefficients, once the ones withlower indices are known. This is clearly exemplified byexpression (34). C. Computation of the gluon condensates
For the soft wall case α i = δ i, α the results are thesame as in [22].For the z case α i = δ i α . Eq. (33) shows that in thiscase the coefficient of a in S subNG is, L α (cid:48) α . Using the expression of G in terms of this coefficientappearing in [22] leads to the following expression for α , L α (cid:48) α = π G , (35)which according to the value of G = 0 . GeV in [26]leads to, L α (cid:48) α = 0 . GeV . In the appendix it is shown how an additional relationbetween L α (cid:48) and α can be obtained by means of com-puting the string tension for the linear potential betweenstatic quarks [38] in the case α i = δ i α . This relationis, σ = α (cid:48) L √ e √ α where σ is the string tension mentioned above. Taking forit the slope in the linear term of the Cornell potential[27],i.e. σ = 0 . GeV leads to, L α (cid:48) = 1 . , α = 0 . GeV For the case α i = δ i α the coefficient of a in S subNG is, L α (cid:48) α (36)In [12] there is an estimation of the following dimension expectation value given by, (cid:104) g f abc F aαβ F bβδ F cδα (cid:105) (cid:119) . GeV in addition in [28] a relation between this expectationvalue and the coefficient of a in the expansion of a cir-cular loop on its radius a is derived, (cid:104) W ( C ) (cid:105)(cid:99) a = π N c (cid:104) g f abc F aαβ F bβδ F cδα (cid:105) this should be equal to the coefficient (36). Furthermoreas shown in appendix 4 the string tension for this caseis given by, σ = α (cid:48) L (6 α e ) / thus the following two relations involving L α (cid:48) and α areobtained, L α (cid:48) α = π N c . GeV . GeV = α (cid:48) L (6 α e ) / where as before the string tension is taken to be σ =0 . GeV . Solving these equations for L α (cid:48) and α , leadsto, L α (cid:48) = 1 . , α = 0 . GeV VII. CONCLUDING REMARKS
In this work the calculation of the minimal areabounded by a circular loop lying at a certain value z of the radial coordinate z has been considered. This sur-face is embedded in a 5-dimensional space with a globalmetric which in conformal coordinates depends only onthe warp factor e A ( z ) . The connection of this calculationwith QCD observables is shown in the following scheme, Global metric NG ←→ M in. area a − exp. ←→ QCD condensates where a − exp. goes for the expansion of the minimalarea in powers of the loop radius a . The continuation ofthis scheme to the left would require the knowledge of agravity-string theory from which the warp factor couldbe obtained. In this respect it is worth remarking thatif such a theory would include a dilaton field then thewarp factor e A ( z ) considered in this work correspondsto the string frame warp factor [24]. The arrows in theabove scheme go in both directions, trying to indicatethat these connections could be employed in both ways.That is, knowledge of QCD condensates could be em-ployed to obtain warp factors as in (35) and, in the otherdirection, details of a higher dimensional theory wouldgive information about QCD.Regarding the connection between the minimal areaand the condensates, it is emphasized that an impor-tant ingredient for this connection is the subtraction em-ployed. This subtraction involves both UV and IR di-vergences, the first already present in the AdS case aretreated as in [17] and maintaining conformal invariance,the second coming form the consideration of confiningwarp factors, require an IR scale which is argued to begiven naturally by the location of the minimum of thesewarp factors. In this respect, it is important to realizethat the approximations employed are well suited for thecalculation of the first coefficients in the expansion inpowers of the radius a for the subtracted NG action.Finally it is noted that the techniques employed in thiswork are not restricted to the particular family of warpfactors (2). Any other choice that can be made conver-gent by the subtractions appearing in section III would work [39]. If this is not the case other subtractions shouldbe considered.This work was supported by DOE Contract No. DE-AC05-06OR23177 under which JSA operates the ThomasJefferson National Accelerator Facility, and by the Na-tional Science Foundation (USA) through grants PHY-0855789 and PHY-1307413 (J.L.G.), and by CONICET(Argentina) PIP Nº 11220090101018 (R.C.T.). J.L.G.thanks the Instituto Balseiro and the Centro At\’omicoBariloche for hospitality and support during the earlystages of this project.
Appendices1. The subtraction for Dp -branes inspired warpfactors As an example of other warp factors of interest , thefollowing are considered, A ( z ) = − n log (cid:16) zL (cid:17) + f ( z ) (A.1)The case n = 1 is the one already studied in III A. Forbackgrounds generated by a stack of Dp -branes, one oftenarrives to metrics with n ≤ . These warp factors divergein the UV, the leading singularity is, A ( z ) ∼ − n log (cid:16) zL (cid:17) ⇒ A (cid:48) ( z ) ∼ − n Lz (A.2)the equation of motion near the boundary implies, − lim z → r n z r (cid:48) (cid:0) r (cid:48) (cid:1) + lim z → rr (cid:48)(cid:48) − lim z → (cid:0) r (cid:48) (cid:1) = 0 which in a similar way as in III A leads to the followingasymptotic behavior, lim z → r (cid:48) ( z ) = 0 r (cid:48) ( z ) = 1 a (1 − n ) z + ... ( z (cid:28) Inserting this in S subNG , shows that the leading behavior ofthe integrand in the NG action is governed by, lim z → e A ( z ) (cid:18) r ( z ) (cid:113) r (cid:48) ( z ) − a (cid:19) =lim z → z n (cid:18) a (cid:18) c z + . . . (cid:19) − a (cid:19) = a lim z → z n (cid:18) c z + . . . (cid:19) = 0 This implies that for n > the regularization proceduredoes not work since the expression diverges. However n ≤ corresponds to the metrics obtained from top-bottom approaches with stacks of Dp -branes. A concreteexample can be found in [29], where the area of the cir-cular loop is found to be, S NG | Dp = 12 πα (cid:48) ˆ a (cid:18) − p z (cid:19) − p − p r (cid:112) z (cid:48) dr (A.3)The regularization procedure ensures the convergence ofthe subtracted area except for the cases p = 4 and p = 5 .
2. Admissible monomials in the expansion of theNG action solution
The NG action times α (cid:48) is an area and therefore hasdimension of length squared. Making explicit the firstterm in (A.1) it is written as follows, α (cid:48) S NG = L n ˆ a e (cid:80) k =1 α k z k z n r (cid:112) z (cid:48) dr therefore the integral in this last equation should havedimensions of length to the power − n . In particular for the case n = 1 (deformation of AdS), it should bedimensionless. This integral depends on a , z ,which havedimension of length, and the α ’s. In this respect it isuseful to note that the α k has dimensions of length tothe power − k . Any monomial contributing to α (cid:48) L n S NG of the general form, a j z m α lk will have vanishing coefficient unless, j + m − k · l = 2 − n The same general conclusions are valid for α (cid:48) L n S CT .
3. Solutions needed to obtain the condensates ofdimensions and ψ (8) ( t ) = t − a (cid:0) t (cid:0) − t + t (cid:1) + 4 log(1 + t ) (cid:1) α + 1945 a (cid:8) − (cid:2) π + t [ − t (cid:0) t (cid:0) − t − t + 135 t (cid:1)(cid:1) ] + 13440 t log 2 − log (1 − t ] log( 21 + t ) − t ) (cid:0) −
478 + 105 t (cid:0) t + t (cid:1) +210 log(1 + t )) − (1 + t )] α − (cid:0) t (cid:0) −
24 + 12 t + 6 t + 4 t + 3 t (cid:1) +24 log(1 + t )) α } ψ (10) ( t ) = t − a (cid:0) t (cid:0) − t + t (cid:1) + 4 log(1 + t ) (cid:1) α − a (cid:2) π + t ( − t (cid:0)(cid:0) t (cid:0) − t − t + 135 t (cid:1)(cid:1)(cid:1) + 13440 t log 2 − − t ) log( 21 + t ) − t ) (cid:0) −
478 + 105 t (cid:0) t + t (cid:1) + 210 log(1 + t ) (cid:1) − ( 1 + t α − a (cid:0) t (cid:0) −
12 + 6 t + 3 t + 2 t (cid:1) + 12 log(1 + t ) (cid:1) α + a (cid:18) (cid:0) − π + 15120 (log 2) − t ( − t (cid:0) t (cid:0) t (cid:0)
240 + t (cid:0) − t − t + 420 t (cid:1)(cid:1)(cid:1)(cid:1) +60480 log 2) − − t ) log( 21 + t ) + 24 log(1 + t ) (cid:0) −
922 + 21 t (cid:0)
48 + 42 t + 15 t + 2 t (cid:1) +252 log(1 + t )) + 30240 Li (1 + t )) α α − (cid:0) t (cid:0) −
120 + 60 t + 30 t + 20 t + 15 t + 12 t (cid:1) +120 log(1 + t )) α ) , where Li denotes the dilogarithm function.
4. Linear potential between static quarks
The string tension is given by the value at its minimumof the function [23], f ( z ) = α (cid:48) L e A ( z ) α n = α δ n it is given by, f ( z ) = α (cid:48) L e − log z + α z ) its minimum and the corresponding string tension are, z = 1 √ α / , σ = f ( z ) = α (cid:48) L √ e √ α which has the right units since α has units of length tothe minus , thus √ α has units of energy squared as corresponds to a string tension.For the case α n = α δ n the minimum and string ten-sion are given by, z = 1(6 α ) / , f ( z ) = α (cid:48) L (6 α e ) / [1] G. ’t Hooft. A Planar Diagram Theory for Strong Inter-actions. Nucl.Phys. , B72:461, 1974.[2] J. M. Maldacena. The Large N limit of superconformalfield theories and supergravity.
Adv.Theor.Math.Phys. ,2:231–252, 1998.[3] E. Witten. Anti-de Sitter space and holography.
Adv.Theor.Math.Phys. , 2:253–291, 1998.[4] S.S. Gubser, I. R. Klebanov, and A. M. Polyakov.Gauge theory correlators from noncritical string theory.
Phys.Lett. , B428:105–114, 1998.[5] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri,and Y. Oz. Large N field theories, string theory andgravity.
Phys.Rept. , 323:183–386, 2000.[6] S. S. Gubser. Dilaton driven confinement. 1999.[7] T. Sakai and S. Sugimoto. Low energy hadron physics inholographic QCD.
Prog.Theor.Phys. , 113:843–882, 2005.[8] L. Da Rold and A. Pomarol. Chiral symmetry breakingfrom five dimensional spaces.
Nucl.Phys. , B721:79–97,2005.[9] J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov. QCDand a holographic model of hadrons.
Phys.Rev.Lett. ,95:261602, 2005.[10] J. Polchinski and M. J. Strassler. The String dual of aconfining four-dimensional gauge theory. 2000.[11] C. Csaki and M. Reece. Toward a systematic holographicQCD: A Braneless approach.
JHEP , 0705:062, 2007.[12] M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov.{QCD} and resonance physics. theoretical foundations.
Nuclear Physics B , 147(5):385 – 447, 1979.[13] P. Colangelo and A. Khodjamirian. QCD sum rules, amodern perspective. 2000.[14] A. Di Giacomo and G.C. Rossi. Extracting the VacuumExpectation Value of the Quantity alpha / pi G G fromGauge Theories on a Lattice.
Phys.Lett. , B100:481, 1981.[15] Paul E.L. Rakow. Stochastic perturbation theory andthe gluon condensate.
PoS , LAT2005:284, 2006.[16] T. Banks, R. Horsley, H.R. Rubinstein, and U. Wolff.Estimate of the gluon condensate from Monte Carlo cal-culations.
Nucl.Phys. , B190:692, 1981.[17] J. M. Maldacena. Wilson loops in large N field theories.
Phys.Rev.Lett. , 80:4859–4862, 1998.[18] S. Rey and J. Yee. Macroscopic strings as heavy quarksin large N gauge theory and anti-de Sitter supergravity.
Eur.Phys.J. , C22:379–394, 2001.[19] L. Hung, R. C. Myers, and M. Smolkin. Some calcula-ble contributions to holographic entanglement entropy.
Journal of High Energy Physics , 2011:1–45, 2011.[20] U. Gursoy and E. Kiritsis. Exploring improved holo- graphic theories for QCD: Part I.
JHEP , 0802:032, 2008.[21] J. L. Goity and R. C. Trinchero. Holographic models andthe QCD trace anomaly.
Phys.Rev. , D86:034033, 2012.[22] O. Andreev and V. I. Zakharov. Gluon Condensate,Wilson Loops and Gauge/String Duality.
Phys.Rev. ,D76:047705, 2007.[23] Y. Kinar, E. Schreiber, and J. Sonnenschein. Q anti-Q potential from strings in curved space-time: Classicalresults.
Nucl.Phys. , B566:103–125, 2000.[24] U. Gursoy, E. Kiritsis, and F. Nitti. Exploring improvedholographic theories for QCD: Part II.
JHEP , 0802:019,2008.[25] C. R. Graham and E. Witten. Conformal anomaly of sub-manifold observables in AdS/CFT correspondence.
Nu-clear Physics B , 546:52–64, April 1999.[26] E.M. Ilgenfritz. Large order behavior of Wilson loopsfrom NSPT.
PoS , ConfinementX:301, 2012.[27] E. Eichten, K. Gottfried, T. Kinoshita, J. Kogut, K. D.Lane, and T.-M. Yan. Spectrum of Charmed Quark-Antiquark Bound States.
Physical Review Letters ,34:369–372, February 1975.[28] M. A. Shifman. Wilson Loop in Vacuum Fields.
Nucl.Phys. , B173:13, 1980.[29] P.N. Kopnin and A. Krikun. Wilson loops in holographicmodels with a gluon condensate.
Phys.Rev. , D84:066002,2011.[30] C Fefferman and C R Graham. Conformal invariants in:The mathematical heritage of élie cartan. In
Astérisque1985, Numero Hors Serie , pages 95–116, 1984.[31] Recalling that near the UV boundary the relation be-tween the conformal coordinate z and Fefferman-Graham[30] coordinate ρ is ρ = z , then a polynomial in z , asconsidered in this work, corresponds to an expression in-volving integer and half integer powers of ρ . However seesection IV where it is shown that half integers powers of ρ can not appear for a theory describing QCD.[32] Indeed, suppose there were a minimal surface withboundary at z < z m that extends to values of z > z m ,then, since the warp factor necessarily grows for thesevalues(recall that z m is a minimum), a surface stoppingat z m will have less area than the one originally supposedto be minimal, which is a contradiction.[33] By definition a condensate of dimension n is the coeffi-cient of a n in the expansion of α (cid:48) L S NG [ z ]( a ) in powers ofa.[34] In this assertion the effect of renormalons is neglected.This assumption is supported by the results in [15, 26].[35] See IV B [36] For the case considered, α n = δ n α , the integral in (30)can be explicitly calculated as, ˆ z m z dz e α z z = 14 (cid:32) E ( − z α ) z − E ( − z m α ) z m (cid:33) = 1 z + (cid:32) − E ( − z m α )4 z + Γ( − ) ( − α ) / / (cid:33) ∼ = − α z − α z + O ( z ) where E ν ( z ) denote the exponential integral and the lastapproximate equality is an expansion in powers of z . From this expression it is clear that the coefficients ofpositive powers of z are the same as the ones obtainedexpanding the integrand in (30).[37] See footnote [30] .[38] This argument is based on the important fact that thecoefficient in front of the NG action, i.e. L α (cid:48)(cid:48)