CColor screening masses from string models
Oleg Andreev
1, 2 L.D. Landau Institute for Theoretical Physics, Kosygina 2, 119334 Moscow, Russia Arnold Sommerfeld Center for Theoretical Physics,LMU-M¨unchen, Theresienstrasse 37, 80333 M¨unchen, Germany
We use gauge/string duality to estimate the Debye screening mass at non-zero temperature andbaryon chemical potential. We interpret this mass as the smallest one in the open string channel.Comparisons are made with the results from holography and lattice QCD.
PACS numbers: 11.25.Tq, 11.25.Wx, 12.38.Lg
LMU-ASC 24/16
I. INTRODUCTION
Understanding the properties of quarks and gluons at non-zero temperature and density is of primary importanceto both theory and experiment. One of those is the phenomenon of Debye screening which is a common feature ofone of the fundamental states of matter - plasma. Its numerical characteristics are screening masses which imply theexponential fall-off of correlation functions at asymptotically large separation.There is a long history of computing the screening masses either perturbatively or non-perturbatively. At temper-atures of a few times of the deconfinement temperature, which are the temperatures probed by heavy ion collisions,reliable calculations require numerical simulations on the lattice. However, at non-zero baryon density or, equiva-lently, non-zero baryon chemical potential those are rather limited. Gauge/string duality provides new theoreticaltools for studying strongly coupled gauge theories [2] that is a powerful motivation for using it as a possible alterna-tive. Although there still is no string theory dual to QCD, it would seem very good to understand to what extenteffective string models already at our disposal are accurate in their ability to mimic or even predict the behavior ofthe screening masses at non-zero temperature and baryon chemical potential.A suitable class of gauge invariant correlators includes the correlators of the Polyakov loops C n, ¯ n ( x , . . . , x n ; ¯ x , . . . , ¯ x ¯ n ) = (cid:104) L ( x ) . . . L ( x n ) L † (¯ x ) . . . L † (¯ x ¯ n ) (cid:105) , (1.1)with L = N trPexp (cid:104) ig YM (cid:82) /T dt A t (cid:105) . Here the trace is over the fundamental representation of SU ( N ), t is a periodicvariable of period 1 /T , with T the temperature, A t is a time component of a gauge field, and g YM is a gauge couplingconstant. These correlators are of special interest since they determine the free energy of a configuration of n heavyquarks and ¯ n heavy antiquarks placed at positions x , . . . x n and ¯ x , . . . ¯ x ¯ n , respectively [3].The present paper continues a series of studies on effective five-dimensional string theories started in [4, 5]. Itbegan with the following question. Given the prescription for computing the correlator C , ¯1 [6], what is that for C , ?Or, in other words, how to compute a diquark free energy via gauge/string duality? In Sec. III, we will attemptto give such a prescription. In doing so, the notion of a baryon vertex [7] plays a pivotal role. To cross check theprescription we make two estimates of the Debye screening mass: the first estimate from C , ¯1 and the second from C , . We go on in Sec. IV to compare our estimates with those based on holography (gravity modes) and numericalsimulations. Finally, we conclude in Sec.V with a discussion of some open problems. In particular, we address theissue of a three-loop correlator C , . Some technical details are included in the Appendices. See, for example, [1] and references therein. a r X i v : . [ h e p - ph ] N ov II. PRELIMINARIES
For orientation, we begin by setting the framework and recalling some preliminary results. We will compute theloop correlators in a 5-dimensional effective string model which is an extension of the model [4, 5] to finite temperatureand baryon chemical potential. Now, a key point is that the background geometry is described by a charged blackhole. This is a standard approach to describe the deconfined phase in AdS/CFT which is a correspondence betweenconformal field theory and string theory on Anti de Sitter (AdS) space [2]. The phenomenon of string breaking ismodeled by the black hole horizon where the effective string tension vanishes.The strings in question are elementary Nambu-Goto strings living in a five-dimensional curved space. The endpointsof each string are electrically charged with respect to a background U (1) field, with charges that are +1 / − / Our ansatz for the background fields is that ds = w ( r ) R (cid:16) f ( r ) dt + d(cid:126)x + f − ( r ) dr (cid:17) , A = (cid:0) A t ( r ) , , . . . , (cid:1) , with w ( r ) = e s r r . (2.1)It is a one-parameter deformation of the Reissner-Nordstr¨om solution in Euclidean AdS [8], with s a deformationparameter. For f = 1 it reduces to that of [4, 5] which reproduces the area and Y -laws for Wilson loops in the infraredregime. The boundary of this space is at r = 0. f and A t are some functions of r subject to the conditions f (0) = 1 , f ( r h ) = 0 , (2.2)and accordingly A t (0) = µ , A t ( r h ) = 0 . (2.3)We assume that f ( r ) decreases with r and has a positive root at r = r h such that a regular black hole horizon occurs.As usual, µ is identified with the baryon chemical potential of a dual gauge theory and an inverse period of t with itstemperature T [2].The simplest correlators, with n = ¯ n = 1 and n = 1, were computed in [9, 10] and [11]. A question to be asked iswhether the correlators for other n can be computed. At this point, it is good to remember the notion of a baryonvertex which is a higher-dimensional analog of a string junction. In ten dimensions, the baryon vertex is a wrappedfive-brane whose world-volume is a product of some internal space X and a time curve in AdS [7]. Viewed from afive-dimensional perspective, it looks like a point-like object whose action is [12] S vert = m (cid:90) dτ (cid:112) f e − s r r . (2.4)The form of the integrand follows from the world-volume term in the five-brane action but m being a result ofresummation of infinitely many terms ( α (cid:48) -corrections) is a parameter of the model. This ansatz is quite successfulbecause it allows us to describe the lattice results for the expectation value of a baryonic Wilson loop by using onefree parameter which is m [12].In the presence of an external gauge field, we extend the action by adding a coupling to A t : S vert → S vert + (cid:90) dτ A t dtdτ . (2.5)The motivation for such a form of the coupling is drawn from the AdS/CFT construction [7], where the five-braneaction has a coupling (cid:82) A ∧ G . Here G is a five-form field with 3 units of flux on X . The generalization to an The normalization has been chosen so that a baryon has a charge of +1. We discuss explicit formulas for f and A t in section IV and the Appendix C. antibaryon vertex is straightforward. It is done by changing the sign of the flux: 3 → − t = τ .In this case, the action becomes a function of rS vert = 1 T (cid:16) m (cid:112) f e − s r r + A t ( r ) (cid:17) , (2.6)which says how strongly the free-energy depends on the vertex position r .Now we would like to explain how to compute the correlator C n, ¯ n in this formalism. First, we place externalsources, n quarks and ¯ n antiquarks, at the boundary points x i and ¯ x i of the five-dimensional space. Next, we considerconfigurations in which each quark (antiquark) is the endpoint of the Nambu-Goto string. The strings in questionare oriented. For each string there is a charge +1 / − / C n, ¯ n is given by the world-sheet path integral. In practice, it can be evaluated by thesaddle-point approximation. The expectation value is then C n, ¯ n ( x , . . . , x n ; ¯ x , . . . ¯ x ¯ n ) = (cid:88) m w m e − S m . (2.7)Here the sum goes over all possible string configurations that obey the boundary conditions. S m represents theminimal action of the m -configuration whose weight is w m . This formula is a generalization of that in [13]. S m nowincludes not only the minimal area, but also the contributions from boundaries and vertices. Note that one can, infact, think of the configurations in (2.7) as those of [14] with all but the external sources dipping into the bulk. III. SCREENING MASSES FROM CORRELATORS OF POLYAKOV LOOPS
As an important illustration of these ideas, we will give a few examples with increasing complexity. We keep theform of f and A t completely general and consider the case when the black hole horizon is closer to the boundarythan the soft wall [15]. This implies that r h < √ s and the corresponding gauge theory is deconfined. In addition, wewill assume that at short distances string configurations with a minimum number of baryon/antibaryon vertices aredominant. A. Two Oppositely Oriented Loops
We begin by recalling how the correlator C , ¯1 can be evaluated within the approximation (2.7). The analysis thatfollows is applicable for any N . For more explanations, see [9, 10].Consider the configurations sketched in Figure 1. For convenience, we place the sources on the x -axis at positionsseparated by l . It is straightforward to write a relation between l and the maximum value of r which holds in thecase of the connected configurations. By using the general formula (A.8) in the current context, r − = 0 and α + = 0,we get l ( ν ) = 2 r h ν (cid:90) dv (cid:112) f ( r h ν v ) (cid:0) η − − (cid:1) − , η = f ( r h ν ) f ( r h ν v ) v exp (cid:16) hν (cid:0) − v (cid:1)(cid:17) , (3.1)with l = | x − x | , ν = r + r h , and h = s r h . The parameter ν takes values in the interval [0 , The extra factor of 2 isdue to the reflectional symmetry (considered modulo string orientation) of the configurations. Note that the function l ( ν ) has a local maximum at ν = ν ∗ and vanishes at the endpoints of the interval. So l is real for h < r x r h (1) (2) (3) FIG. 1: String configurations contributing to C , ¯1 . The arrows show the string orientation chosen to go from positive tonegative charges. The horizontal line at r = r h represents the horizon. Given a solution (string profile), one can evaluate the corresponding action. For the connected configurations, itcan be expressed in terms of integrals by using equation (A.12). Explicitly S ( ν ) = 2 g νr h T (cid:90) dvv (cid:104) e hν v (cid:0) − η (cid:1) − − − v (cid:105) + 2 c . (3.2)Since the string is neutral, the dependence on the U (1) gauge field drops out. The formulas (3.1) and (3.2) areapplicable for both cases sketched in Figure 1. In the first case, where the string profile is close to the boundary, ν takes values in the interval [0 , ν ∗ ], while in the second, where the string profile looks like a spike, ν takes values in theinterval [ ν ∗ , ν = ν ∗ .For the disconnected configuration, the action reads S = − F T + 2 c , (3.3)as it follows from (A.14). S depends only on the minimal area. The reason for this is that the loops are oppositelyoriented and, as a result, the two boundary terms cancel each other.It is useful for what follows to recall some facts about the string configurations of Figure 1. The first is thatconfigurations (1) and (2) exist only if l does not exceed a certain critical value, which is equal to r h times somenumerical factor. The reason for this is that an effective string tension vanishes at r = r h so that a string breaksonce it touches the horizon. In other words, the phenomenon of string breaking is modeled by a black hole. Thesecond is that configuration (1) dominates at short distances. In that case, S (1) ∼ − /l , while the two remaining S ’sare regular at l = 0. Third, at long distances the disconnected configuration (3) dominates as in QCD.Because of scheme ambiguities, the S ’s may look not so good for a comparison to what is known in the literature.A solution can be found provided the c dependence vanishes from the correlator, and this requires a normalization of C , ¯1 . The normalized correlator is given by C , ¯1 ( l ) = (cid:104) L † ( x ) L ( x ) (cid:105)|(cid:104) L (cid:105)| . (3.4)This correlator does not suffer from linear divergences and, according to [3], determines a difference between the freeenergies: ∆ F Q ¯ Q = F Q ¯ Q − F Q − F ¯ Q which is usually called the binding free energy of a pair.Now we want to compute ∆ F Q ¯ Q at short distances. For this, we use (2.7) together with (3.2) and (3.3). Theresulting formula is At zero chemical potential and high temperature, r h ∼ /T such that lT (cid:46) ∆ F Q ¯ Q ( ν ) = 2 g r h ν (cid:90) dvv (cid:104) e hν v (cid:0) − η (cid:1) − − − v (cid:105) + 2 F − T ln w (1)1 , ¯1 , (3.5)where w (1)1 , ¯1 is a weight of configuration (1). Since the configuration needed for a description of C , or, equivalently, C , ¯1 is single, we set its weight to 1. In this form, it is easy to take the limit ν → ν with the help of(3.1). A short calculation shows that∆ F Q ¯ Q ( l ) = − α Q ¯ Q l + 2 F − T ln w (1)1 , ¯1 + o ( l ) , (3.6)with α Q ¯ Q = g (2 π ) Γ − (cid:0) (cid:1) , which is the same as in [13].How would one determine the exponential fall-off of correlators at long distances? What we must do is to revisethe formula (2.7). The most challenging task on this way is of course to find a string dual (if any) to QCD. We havenothing to say here about it and that is why we are dealing with the effective string models.One approach to seeing that the series (3.6) might be useful for gaining insight into the Debye mass involves the useof knowledge obtained from numerical simulations [1]. For our purposes, what we need to know can be summarizedas follows. The quark-antiquark pair can be in either a color singlet or a color octet state. The free energy of the pairgets the contributions from both states. At short distances, the free energy is dominated by that of the singlet state.One of the parameterizations of the singlet free energy, motivated by high temperature perturbation theory, is givenby ∆ F Q ¯ Q ( l ) = − α l e − ml , (3.7)with α and m parameters. Like in lattice gauge theory [17, 18], we call m the Debye mass.Going back to our problem, one important conclusion which emerges is that configuration (1) should correspondto the singlet state. It is perfectly possible that at long distances such a classical solution does not exist withineffective string models, but it does exist in a full string theory of QCD. We would like to go a step further, however,and assume that at short distances the singlet free energy can be well approximated by the expression (3.6). Withthe help of (3.7), we obtain an estimate for the Debye mass m = 1 α Q ¯ Q (cid:16) F − T ln w (1)1 , ¯1 (cid:17) , (3.8)which is scheme independent. Moreover, the formula is applicable to all models with the asymptotic behavior (3.6).In the case of interest, we write the Debye mass in a more detailed way m = 14 π Γ (cid:0) (cid:1) √ s (cid:18) e h √ h − √ π Erfi (cid:0) √ h (cid:1) + w T √ s (cid:19) , (3.9)where w = − ln w (1)1 , ¯1 / g . For our purposes, we can think of m as a composite function of T and µ , and treat s and w as the model parameters. In the present section, we try to keep the discussion as general as possible, deferring moredetail on h ( T, µ ) to Sec.IV.We conclude this section with a couple of remarks. First, a configuration which represents a straight string stretchedbetween the boundary and the horizon determines the expectation value of the Polyakov loop. Assuming that it isreal, one has See, e.g., [17, 18]. Note that this provides us a gauge invariant way to determine the singlet free energy. (cid:104) L (cid:105) = exp (cid:110) T (cid:16) F − µ q (cid:17) − c (cid:111) . (3.10)Second, if we replace F by (cid:104) L (cid:105) in (3.8), then it takes the form mT = 2 α Q ¯ Q (cid:16) ln (cid:104) L (cid:105) + µ q T + c −
12 ln w (1)1 , ¯1 (cid:17) . (3.11)This provides a relation between the Debye mass and the expectation value of the Polyakov loop. We will return tothis relation in Sect. IV. B. Two Similarly Oriented Loops
By considering a string stretched between two sources on the boundary, one gets the correlator C , ¯1 . What doesone do with C , ? To obtain it, one can instead consider N strings ending on N quark sources and joining at abaryon vertex in the bulk. That, in other words, is a baryonic configuration. The desired configuration is obtainedby sending N − For N = 3, one is led to suspect thatthe correlator C , can be described in terms of configurations sketched in Figure 2. r x r h (1) (2) (3) V V
FIG. 2: String configurations contributing to C , at N = 3. V is a baryon vertex. After specializing to N = 3, we proceed as before. First we evaluate the correlator C , by means of equation (2.7).The main additional step required is an analysis of gluing conditions at a baryon vertex which is presented in theAppendix B. Having done this, we then use the result to estimate the Debye mass. The goal is to cross check theestimate obtained in the previous section.Since the connected configurations are symmetric under reflection through the middle point, the side strings havean identical profile, and the middle string is stretched in the radial direction. Given this, we can write a relationbetween l and the position of the vertex. Just as in the previous section, we need (A.8) at r − = 0 but now with α + determined by the gluing condition (B.8). Putting all that together gives us l ( ν ) = 2 r h ν (cid:90) dv (cid:112) f ( r h ν v ) (cid:0) η − − (cid:1) − , η = f ( r h ν ) f ( r h ν v ) (cid:16) − θ ( r h ν ) (cid:17) v exp (cid:16) hν (cid:0) − v (cid:1)(cid:17) , (3.12) This is perfectly possible in the deconfined phase. with l = | x − x | and θ defined in (B.8). The parameter ν takes values in the interval [0 , ν max ]. Here ν max is asolution of equation f ( r h ν ) (cid:0) − θ ( r h ν ) (cid:1) = 0 such that 0 < ν max ≤ l ( ν )’s given by (3.1) and (3.12)? Both functions arecontinuous and have zeros exactly at the interval endpoints. The zeros at ν (cid:54) = 0 appear as a consequence of thecondition η = 0, which is possible for the geometry in question. Thus, the l ( ν )’s (being regular in the intervals) mustbe bounded from above. This means that the connected configurations exist only if l does not exceed the criticaldistance. In addition to these general arguments, let us be more specific and consider what is perhaps the bestunderstood example considered in [9, 10], namely the case f ( r ) = 1 − ( r/r h ) . Figure 3 shows a typical shape of l ( ν ). For configuration (1) with the side strings close to the boundary, ν takes values in the interval [0 , ν ∗ ], while for ��� ��� ��� ��� ��� ��� ν �������������� � ( ) ( ) FIG. 3: l as a function of ν at κ = − .
083 and h = . It has a maximum at ν ∗ ≈ . ν max ≈ . configuration (2) with the side strings deeply in the interior, ν takes values in the interval [ ν ∗ , ν m ax ].Going further, the action of the middle string can be read from (A.10) at r + = r h , and that of the side strings from(A.12). Combining these with the action for the vertex, we get S ( ν ) = g νr h T (cid:18) (cid:90) dvv (cid:104) e hν v (cid:0) − η (cid:1) − − − v (cid:105) − ν √ πh Erfi (cid:0) ν √ h (cid:1) +e hν +3 κ (cid:112) f ( r h ν ) e − hν (cid:19) + 1 T (cid:16) µ q − F (cid:17) +2 c , (3.13)with κ = m g and µ q = µ . In the last step we used (A.14). Now the dependence on the gauge field does not drop out,because the configuration is charged under U (1). It comes entirely from the string endpoints on the boundary. Theconditions (2.3) and (B.2) guarantee that there is no dependence on A t coming from the bulk.Now let us consider the disconnected configuration. In this case, the total action is twice the action of the straightstring (A.14). Hence we write it as S = 2 T (cid:16) µ q − F (cid:17) + 2 c . (3.14)Having derived the corresponding actions, we are now in a position to draw the same two conclusions as before.The first is that the connected configurations exist only if l does not exceed a certain critical distance, which is equalto r h times some numerical factor. The second is that configuration (1) dominates at short distances. This requiresmore explanation. In general, the short distance behavior of S (1) is characterized by the parameter κ . The conclusionwe made is true for some values of κ . Importantly, the value of κ fitted to the lattice is one of those values [12]. If so,then S (1) ∼ − /l , while the other S ’s are regular at l = 0.The correlator C , is scheme dependent. This is why any comparison to what is known in the literature would bepointless unless it is done within the same renormalization scheme. This difficulty can be circumvented by introducinga normalized form of C , defined by C , ( l ) = (cid:104) L ( x ) L ( x ) (cid:105)(cid:104) L (cid:105) . (3.15)It follows from this definition that C , determines a difference between the free energies: ∆ F QQ = F QQ − F Q , with F QQ a diquark free energy. This is nothing else but the binding free energy of a diquark. Note that in addition to thedependence on c , the explicit dependence on µ q also cancels out.According to the above formulas, ∆ F QQ is given by∆ F QQ ( ν ) = g r h ν (cid:18) (cid:90) dvv (cid:104) e hν v (cid:0) − η (cid:1) − − − v (cid:105) − ν √ πh Erfi (cid:0) ν √ h (cid:1) + e hν + 3 κ (cid:112) f ( r h ν ) e − hν (cid:19) + F − T ln w (1)2 , , (3.16)where w (1)2 , is a weight of configuration (1). Keeping in mind that configuration (1) dominates, it is straightforwardto find the two leading terms in the expansion of ∆ F QQ in powers of l . A simple but somewhat lengthy calculationreveals that ∆ F QQ ( l ) = − α QQ l + F − T ln w (1)2 , + o ( l ) , (3.17)with α QQ = − g ξ − B (cid:0) ξ ; , (cid:1)(cid:16) κ + 12 ξ B (cid:0) ξ ; − , (cid:1)(cid:17) , ξ = √ (cid:0) − κ − κ (cid:1) . (3.18)Here B ( z ; a, b ) is the incomplete beta function. The coefficient α QQ is the same as that in the static quark-quarkpotential [12]. Importantly, it is positive for all values of κ in a narrow range which is needed to fit the lattice dataon the three quark potential.We are now in the same situation as we were with the correlator C , ¯1 . A string dual to QCD is still missing thatobstructs our ability to compute the exponential fall-off of correlators at long distances from the first principles. Hereagain, we follow the approach relying on a knowledge acquired from numerical simulations. We will not go into greatdetail about it, only what is needed for our purposes. The diquark can be in either a color antitriplet or a colorsextet state. The diquark free energy gets the contributions from both states. At short distances, the free energy isdominated by that of the antitriplet state. One of the parameterizations of the antitriplet free energy, motivated byhigh temperature perturbation theory, is given by ∆ F ¯3 QQ ( l ) = − α l e − ml , (3.19)with the same parameters as in (3.7). If so, then m is the Debye mass.Now we can draw an interesting conclusion about configuration (1). It should correspond to the antitriplet state.Thus, the string construction provides a gauge invariant way to determine the antitriplet free energy. Of course, itshould be perfectly possible, not within effective string models, but in a full string theory of QCD. In undertaking tomake an estimate of the Debye mass, a stronger assumption has to be made. Here we assume that at short distancesthe antitriplet free energy can be well approximated by the expression (3.17). Then, with the help of (3.19), we havethe following estimate for the Debye mass m = 1 α QQ (cid:16) F − T ln w (1)2 , (cid:17) . (3.20)It is scheme independent and applicable to all effective string models with the asymptotic behavior (3.17).We close this section with a few comments:(i) As a check of consistency, we can compare the two estimates. At zero temperature, nothing else but the ratiobetween α QQ and α Q ¯ Q matters. Of course, its value may be set to 1 / κ . This gives κ ≈ − . − .
083 obtained from fitting the calculated three-quark potential to the lattice data See, e.g., [19, 20]. [12]. Thus the estimates do look plausible, especially for the effective model we are pursuing. At finite temperature,in addition, a relation between the weights is required. It is (cid:0) w (1)1 , ¯1 (cid:1) = w (1)2 , . At the moment we do not know whetherthis can be demonstrated within our model. We can only appeal to lattice gauge theory, where the relative weight ofthe singlet state is , while that of the antitriplet .(ii) In the model of interest, the expression (3.20) becomes m = − ξ √ s e h √ h − √ π Erfi (cid:0) √ h (cid:1) + w T √ s B (cid:0) ξ ; , (cid:1)(cid:0) κ + ξ B (cid:0) ξ ; − , (cid:1)(cid:1) . (3.21)Here w = − ln w (1)2 , / g . Note that m is positive for the values of κ we are considering.(iii) For the correlator C , ¯2 , L is replaced by L † , that is equivalent to replacing a quark with an antiquark, and theanalysis then proceeds in the same way. IV. HOLOGRAPHIC MODELS AND LATTICE
What we have learned is the short distance behavior of the correlators which, when combined with the knowledgeacquired from numerical simulations, allows us to estimate the Debye mass m and, as a result, the exponential fall-off of the singlet and antitriplet free energies at asymptotically large distances. But what we want to know is theexponential fall-off of the correlators. For example, the connected part of C , ¯1 decays for l → ∞ as C con , ¯1 ( l ) ∼ e − Ml , (4.1)where C con , ¯1 = C , ¯1 − M is not equal to m . The reason for this is that the correlator gets a contribution from theoctet state. Thus, m is not exactly what we want, but it is the first step in the right direction. The second is todetermine M in terms of m . Here we make the second step and then compare with other results in the literature. Incontrast to the first step, it can be done in two, at first sight quite different, ways. We start again with lattice gauge theory. What is a relation (if any) of m with M ? The answer to this questionis surprising. After taking account of the remaining contribution, the octet/sextet one, the correlator shows theexponential fall-off with the screening mass M well approximated by [1] M = 2 m . (4.2)This is exactly what we need. It is noteworthy that the above relation also holds at small chemical potential [21].Another way to reach this conclusion is to consider a worldsheet path integral for the correlator C , ¯1 . The worldsheetin question is a cylinder such that each boundary is mapped into a Polyakov loop in a target space. While it is notclear how to properly define and then evaluate this integral for string theory on AdS-like geometries, in flat spaceit is well understood [22]. This is of course not what we need but can be instructive. In the limit lT → lT → ∞ the cylinder becomes very long, and it looks like that closed A bridge that would establish the relation between these two is nothing else but a string dual to QCD. This is the reason why we use the saddle-point approximation (2.7). T l lT
FIG. 4: Two limiting cases [22]. Left: The long-strip limit. Right: Closed string states propagating between two boundarystates in the long-cylinder limit. string states appear from the vacuum, propagate a distance, and then disappear again into the vacuum, as sketchedin Figure 4 on the right. Now the leading asymptotics are given by the lightest closed string states. In this examplethe relation (4.2) has a clear meaning: it is a relation between the masses of lightest closed and open string states.With this interpretation, we can understand the Debye mass m as a mass of the lightest open string state, whilethe screening mass M as that of the lightest closed string state. In the present paper, we start with the open stringchannel and make an estimate for m . Next, using the relation (4.2), we make it for M . This way is a bit longer thanone based directly on the lightest (super)gravity modes and proposed in [23], but more instructive. Both are two facesof the same coin: strings on AdS-like geometries.For future reference, we write M in explicit form M = γπ √ s (cid:18) e h √ h − √ π Erfi (cid:0) √ h (cid:1) + w √ s T (cid:19) , with γ = 12 π Γ (cid:0) (cid:1) . (4.3)We think of M = M ( h ( T, µ )) as a composite function.
A. Sample Models
So far our discussion was general. We will now give two examples of the function f ( r ) used in the literature toillustrate the accuracy of the estimate for M .It is natural to begin with the simplest choice of f meaning that the background geometry is the slightly deformedSchwarzschild black hole in AdS space [15]. Thus, the chemical potential is zero and the function f ( r ) is f ( r ) = 1 − (cid:16) rr h (cid:17) . (4.4)The advantage of this choice is that the Hawking temperature is just proportional to the inverse of the horizon position,such that T = πr h . This allows a great deal of simplification of the resulting equations, and assists in understandingmore complicated forms of f ( r ) later on. For example, in this case, the formula (4.3) becomes MT = γ (cid:18) exp (cid:16)(cid:16) T T (cid:17) (cid:17) − √ π T T Erfi (cid:16) T T (cid:17) + w π (cid:19) , (4.5)where T = √ s π . In [15] T was interpreted as a critical temperature of the model.One might think of criticizing the above choice on the grounds that the conditions of Weyl invariance, that isessential to the consistency of string theory [22], are not satisfied. In a mathematical language, these are equationswhich determine all possible backgrounds. In our case the metric is not a solution to equation β Gµν = 0 even to leadingorder in α (cid:48) . Suppose that the opposite is true. Could one expect that at given w ( r ) there would exist a correspondingfunction f ( r )? In [24], it was shown that to leading order in α (cid:48) this is indeed the case. The two functions are relatedby a simple differential equation. Such an equation arises as a consequence of the Einstein equations and holds if thematter energy-momentum tensor obeys a special constraint. Alternatively, as explained in the Appendix C, one canderive it from the Weyl coefficient for f .1For the case w ( r ) = e s r r , the analytical solution to equation (C.3) with the boundary conditions (2.2) was foundin [25]. It is f ( r ) = 1 − − (cid:0) s r (cid:1) e − s r − (cid:0) h (cid:1) e − h . (4.6)Given this, the corresponding temperature is T ( h ) = 98 π √ s h e h − − h . (4.7)We can think of T as a function of h . Thus the screening mass is given in parametric form by M = M ( h ) and T = T ( h ). It is worth noting that the function T ( h ) behaves for h → T = π (cid:112) s h . Therefore, this limit isequivalent to the high temperature limit, which turns out to be the same for both choices of f .The above can be extended to include a background gauge U (1) field. The main additional step required is ananalysis of equation β Aµ = 0, which is a generalization of Maxwell equation by higher order α (cid:48) -corrections. To leadingorder in α (cid:48) , it is not technically difficult and presented in the Appendix C. The solutions for A t and f are given by(C.7) and (C.10), respectively. Using these solutions, the temperature and baryon chemical potential are T ( h, q ) = 98 π √ s (1 − q ) h e h − − h , µ ( h, q ) = 2 √ sr q (cid:0) − e − h (cid:1)(cid:0) h )e − h − − h (cid:1) . (4.8)Now we can think of T and µ as functions of h and q . The variable q is related to a black hole charge and takes valuesin the interval [0 , q = 0 the black hole is not charged.Finally, the screening mass is given in parametric form by M = M ( h ) together with T = T ( h, q ) and µ = µ ( h, q ). B. Numerics
Since we have estimated that the screening mass M is about given by (4.3), we can make a comparison with theknown results to decide whether it is plausible.We begin with a class of holographic models which was studied in [25]. This seems reasonable because a particularform of the warp factor considered there is similar to ours. In the holographic description the screening mass M comesfrom the lightest (super)gravity mode which in the present case is an axion. Importantly, the action of the axion fieldincludes an additional warp factor Z ( r ) parameterized by two parameters c and c . The axion mass is computednumerically for c = 1 and several values of c . In Figure 5 on the left, we plot the screening mass versus temperature.Units are set by T c = 0 . √ s and M = 3 . πT , as in [25]. Since M = M | s =0 , which is an asymptotic expression forthe limiting behavior of M at large T , we set w = 0 .
70. We see that both approaches result in a similar qualitativebehavior: the function M is monotonically increasing with T . It has a steep rise near T = T c , but is slowly varyingfor larger T . One might expect that a more quantitative agreement requires smaller values of c that will lead to asteeper rise near T = T c .We go on with a SU (3) pure gauge theory. In doing so, we write the critical temperature T c = 0 . √ σ [26] as T c = 0 . √ s , where we used σ = e gs and in the last step set g = 0 .
176 [12]. We then plot the results in Figure5 on the right. It is clear that our estimate is in qualitative agreement with the lattice. There are two importantobservations. The first observation is that the estimate based on (4.7), stemming from a more consistent background,agrees better with the lattice. The second is that the lattice results show a much steeper rise in the vicinity of thecritical point. A similar observation was made in relation to the expectation value of the Polyakov loop (correlator C , ) [11]. This might point to the need for quasi-classical corrections in (2.7). In other words, string fluctuationsbecome essential near the critical point.For practical purposes, the parametric expression for the screening mass M looks somewhat awkward. In [11], welearned that a reasonable approximation for C , ( F Q ) can be obtained by studying its high-temperature behavior.2 � � � � � � � �� � ����������������� � � � � � � �� � ����������������������� FIG. 5: Screening mass vs temperature. The solid curve is defined by (4.3) and (4.7). Left: The results of numerical holography[25]. Here w = 0 .
70 and c = 0 . w = − . It is hopefully clear from (3.11) that a similar derivation for M would proceed in essentially the same way and givea series in powers of T c T . So we expand M ( h ) and T ( h ) near h = 0 and then reduce the two equations to a singleequivalent equation MT (cid:117) γ (cid:18) w π − a T c T (cid:19) , (4.9)where we drop the higher order terms. In this formula a is given by a = s π T c such that, for the above example, itis a = (0 . π ) − ≈ . . T c ��� ��� ��� ��� ��� ��� ��� �� � ����������������������� � � � � � �� � ����������������������� FIG. 6: A comparison of different M ( T ) curves for a pure SU (3) gauge theory. Left: The solid curve is that of Figure 5, whilethe dotted curve corresponds to the power law (4.9). Right: The power law at w = − .
13 and a = 0 . the discrepancy between the two expressions is negligible. It becomes more and more visible as T approaches T c .It is tempting to fit the lattice data [17] to the power law (4.9). In Figure 6 on the right, we have plotted the result.We see that the power law is a good approximation above 1 . T c , but it fails in the vicinity of the critical temperature.In (4.9), the T term can be determined without any reference at all to concrete models (e.g., those of Section A).Let us explain it in more detail. Going back to the formulas (3.11) and (4.2), we see that the screening mass depends3linearly on ln (cid:104) L (cid:105) . On the other hand, it was suggested in [27] that the temperature dependence of L is governed bythe Gaussian ansatz (cid:104) L (cid:105) ∼ e − CT . Combining these statements, we find the T term. In fact, we might expect thatsuch a term would be another example of resummation of QCD long perturbative series. This time it is within hightemperature perturbation theory.Now we want to discuss a SU (3) gauge theory with fermions at non-zero baryon chemical potential. We restrict tothe case of two flavors. It is motivated by the fact that the singlet free energy is parameterized in the same way inboth cases, N f = 0 [17] and N f = 2 [18]. Our hope is that in the absence of first-principles methods, like lattice QCD,we can gain some important understanding of the theory near the critical line in the µT -plane. Given the formalformulas that we have just described, it seems straightforward to apply those and determine the screening mass as afunction of T and µ . To do it this way however requires a caveat. The models we are considering have no explicitdependence on quark masses. On the other hand, the dependence is visible on the lattice. This can be seen in Figure7 which shows the results of [29]. Here the screening mass was calculated at zero chemical potential and two different ��� ��� ��� ��� ��� ��� ��� ��� �� �� �������� ��� ��� ��� ��� ��� ��� ��� ��� �� �� �������� FIG. 7: Screening mass vs temperature at zero chemical potential and different quark masses [29]. The solid curve is definedparametrically by (4.3) and (4.7). We set w = − .
04. Left: Results at m PS m V = 0 .
80 and T pc = 357 MeV. Right: Results at m PS m V = 0 .
65 and T pc = 262 MeV. values of the ratio between pseudoscalar and vector meson masses at zero temperature. In contrast to that, theshape of M ( T ) defined by (4.3) and (4.7) slowly depends on T pc . Here we set s = 0 .
45 GeV as it follows from the ρ meson Regge trajectories [30]. However one common feature does emerge: a rise in the vicinity of the pseudocriticaltemperature.The next case where lattice results are still available is that of small baryon chemical potential. Like in latticegauge theory [21], we expand M ( T, µ ) in powers of µT M ( T, µ ) = ∞ (cid:88) n =0 M n ( T ) (cid:16) µT (cid:17) n , (4.10)for µT (cid:28)
1. The Taylor coefficients M n can directly be derived from (4.3) and (4.8). The first two are given in theAppendix D. Note that when we make this expansion, odd powers of µT do not appear because h is an even functionof µ . This is in agreement with the lattice.The alternative to what we have described is to expand the Debye mass. In this case, the formula (4.2) saysthat m n = M n . Such a relation was checked numerically [21] for a few values of n . It holds in a wide range oftemperatures with a lower bound close to T c . This is a good piece of evidence that the relation (4.2) is true. For a discussion of this issue and its connection to quadratic corrections, see [28]. Of course, the case of physical quark masses is of primary interest but what happens there is not exactly known. m and m . In [21], it wasobserved that the perturbative result m m = π remains reliable even for T as low as 2 T c . Let us confront it withour findings. To do so, we plot the results in Figure 8, on the left. In contrast to Figure 7, agreement with the lattice ��� ��� ��� ��� ��� ��� ��� �� � ������������������������� � � � � ��� ��� ��� ��� ��� ��� ��� ��� μμ �� ������������� μ FIG. 8: Left: Ratio m m versus temperature. The lattice data are taken from [21]. The curve is defined parametrically by (4.7)and (D.4). We set w = − . r = 6 .
00, and T c = 0 . √ s . Right: The screening mass versus baryon chemical potential. Thesolid curve is defined by (4.3) and (4.8), while the dashed curve by (4.12). We set r = 2 .
00 and µ pc = 2 √ s . is very good. Therefore we might expect that the the explicit dependence on quark masses, or the number of flavors,would be multiplicative.In view of the good agreement with the lattice, it appears natural to use the perturbative result for purposes ofdetermining r in terms of w . Then, from (D.5), we learn that it is written as r = 2 √ (cid:16) w π (cid:17) − . (4.11)This reduces the number of parameters to just one, which is w .One can analyze in a similar fashion the opposite case Tµ (cid:28)
1. It follows from (4.8) that temperature is zero at q = 1. In that case, M is a function of µ . It is given in parametric form by equations (4.3) and (4.8). In Figure 8on the right, we plot the resulting prediction. What emerges is more or less similar to that of Figure 5: the function M is monotonically increasing with µ . It has a steep rise near µ = µ pc , but is slowly varying for larger µ . One mightexpect that such a behavior would indicate that a phase transition occurs near µ = µ pc .Actually, as discussed above, one can try to approximate M ( µ ) by a power law. To see this, we expand M ( h ) and µ ( h ) near h = 0 and then reduce these equations to Mµ (cid:117) γπ r (cid:18) − b µ pc µ (cid:19) , (4.12)with b = r s µ pc . We drop the higher order terms in the expansion. As seen from Figure 8, this becomes a goodapproximation for µ larger than 2 . µ pc . Interestingly, the leading correction is again quadratic.Next, we expand the screening mass in powers of Tµ In [21], the screening masses are expanded in powers of the quark chemical potential and therefore m q n = 3 n m n . M ( T, µ ) = ∞ (cid:88) n =0 M n ( µ ) (cid:18) Tµ (cid:19) n . (4.13)In contrast to the Taylor series (4.10), it contains all integer powers. The coefficients M n can be derived from (4.3)and (4.8). The first two are given in the Appendix D. Note that the coefficients of the Taylor series of m ( T, µ ) arerelated to M n as m n ( µ ) = M n ( µ ). In that case, the plot m m ( µ ) has a shape similar to that in Figure 8 on the leftso that this ratio also tends to a constant value as µ goes to infinity.Finally, there is no difficulty to understand the general case. This means that the screening mass is definedparametrically by equations (4.3) and (4.8), with h and q the parameters. In Figure 9, we plot M ( T, µ ). The
FIG. 9: M as a function of T and µ . We set r = 3 . T pc = 0 . √ s , and µ pc = 3 √ s . normalization is chosen to be M = γπ (cid:16) ( µ r ) (cid:104)(cid:113) ( µ r ) + ( πT ) − πT (cid:105) − + w T (cid:17) . It is an asymptotic expression for thelimiting behavior of M at large T and µ . As before, M shows a steep rise in the region close to the origin and becomesslowly varying for larger values of T and µ . One might expect that such a steep rise would indicate the deconfinementcritical line in the T µ -plane. If so, its shape, which looks like that of the bottom edge in Figure 9, seems to be similarto that of [31]. In the framework of effective (field theory) models, the phase structure of two flavor QCD was studiedat different quark masses. Interestingly, the obtained results show that the Cabibbo-Parisi pattern of the phasediagram holds for a wide range of masses.A natural question to ask is, what about N f = 3? Obviously, Figure 9 shows no sign of crossover. The reason iseither it is far from the critical line in the T µ -plane, or the model has nothing to do with it. There are also manyother questions to ask. For instance, what can it say about quarkyonic matter? Unfortunately, no real answers willbe given here. Our goal is to give an example how the screening mass can be estimated in the effective string theoryapproach and gain some insight that will help us with the further development of the model, as well as with findinga string description of QCD. See, e.g., [32] and references therein. V. CONCLUDING COMMENTS (i) It is common practice in phenomenology to use Lipkin’s rule which postulates that at zero temperature thequark-quark and quark-antiquark potentials are related by V QQ = V Q ¯ Q . Such a simple relation is motivated by theresults of perturbative QCD. But one fact makes it difficult to take this seriously: the linear combination V QQ − V Q ¯ Q is scheme-dependent.There is one important situation in which the problem of scheme-dependence can be avoided. This is the caseof binding free energies at finite temperature and non-zero baryon chemical potential. As we saw in Sec.III, in thedeconfined phase the binding free energies of two-quark states are scheme-independent. From this point of view thecomparison between ∆ F Q ¯ Q and ∆ F ¯3 QQ may prove to be instructive. So far, we have exploited the two leading terms inthe short distance expansion of the binding free energies. What those suggest is a Lipkin-like relation ∆ F ¯3 QQ = ∆ F Q ¯ Q .Now a question arises: What happens at larger separations? In Figure 10 on the left, we display our results for the ��� ��� ��� ��� ��� � � - ��� - ��� - ��� - ������ Δ �� V x x x x yrr h FIG. 10: Left: Binding free energies of two-quark states at T = 0 . √ s and µ = 0. The solid curve corresponds to ∆ F ¯3 QQ and the dashed one to ∆ F Q ¯ Q . We set g = 0 . w = − .
05, and κ = − . V is a baryonvertex. The quarks are placed at x i . For negative values of κ , gravity pulls the vertex toward the boundary that blunts the tipof the configuration. binding free energies. Here we set the values of the parameters so as to fit the model to the data for a pure SU (3)gauge theory as close as possible. Clearly, for larger l the binding free energy ∆ F ¯3 QQ grows a bit faster than ∆ F Q ¯ Q .Thus our model predicts ∆ F ¯3 QQ ≥
12 ∆ F Q ¯ Q . (5.1)The reason for this is the higher order terms in the expansion. For instance, the coefficient in front of the linear termin the expansion of ∆ F ¯3 QQ , which is an effective string tension inside a heavy diquark [12], turns out to be larger thanwhat is expected from Lipkin’s rule. One can think of (5.1) as a lower bound on ∆ F ¯3 QQ .(ii) One might expect that things would be not so different for multi-loop correlators, except some technical points.We will not try to compute a generic correlator here. Rather, our goal is to briefly discuss new features related tothree-quark interactions. So we want to consider a correlator of three similarly oriented loops.The analysis proceeds in the following way. Let x i be at the vertices of an equilateral triangle of length l . Thestring configuration which dominates at small l is sketched in Figure 10, on the right. If one normalizes the correlatoras C , ( l ) = (cid:104) L ( x ) L ( x ) L ( x ) (cid:105)(cid:104) L (cid:105) , (5.2)then it determines a difference between the free energies: ∆ F Q = F Q − F Q . The analysis of the correlator can beperformed along the lines of [12]. It shows that the short distance behavior of ∆ F Q is7∆ F Q ( l ) = − α Q l + 3 F − T ln w (1)3 , + o ( l ) , (5.3)with α Q = − √ g B (cid:0) κ ; , (cid:1)(cid:16) κρ − + 14 B (cid:0) κ ; , − (cid:1)(cid:17) , ρ = 1 − κ . (5.4)Here B ( z ; a, b ) = B ( a, b ) + B ( z ; a, b ) and B ( a, b ) is the beta function. w (1)3 , is a weight of configuration. The coefficient α Q is the same as that in the static three-quark potential at zero temperature [12].The above formula has to be understood in a formal sense if l is close to a critical value above which the baryonconfiguration of Figure 10 does not exist. We are in that situation here as in Sec.III. The bottom line is that we areunable to compute the exponential fall-off of correlators at long distances from the first principles. Once again, wefollow the approach relying on a knowledge acquired from numerical simulations. Let us summarize what is neededfor the purposes of the present paper. The three-quark system can be in a color singlet, two symmetrically mixedoctet as well as in a color decuplet state. The three-quark free energy gets the contributions from all these states. Atshort distances, it is dominated by that of the singlet state. One of the parameterizations of the singlet free energy,is motivated by the ∆-law which relates it to the binding free energy of the diquark system in the antitriplet channel[19]. Explicitly ∆ F Q = 3∆ F ¯3 QQ . (5.5)Like in the case of the there-quark potential at zero temperature, it should be a good starting point, especially atsmall and intermediate quark separations.Going back to our problem, one immediate conclusion is that the baryon configuration of Figure 10 should corre-spond to the singlet state. If so, then combining (5.3) with (5.5) and (3.19) yields m = 1 α Q (cid:16) F − T ln w (1)3 , (cid:17) , (5.6)which is one more estimate of the Debye mass.Since we have made the third estimate, it is time to ask about its consistency with the first two. At zero temperature,it is required α Q ¯ Q = α QQ = α Q . As seen from Figure 11 on the left, this system of equations has no exactsolutions. However, if one sets α eff = ( α qq ( − . α Q ¯ Q ), then its discrepancy from the others does not exceed - ���� - ���� - ���� - ���� - ���� ���� ���� κ ��������������������� α α QQ _ α QQ α Q T l
FIG. 11: Left: α Q ¯ Q , α QQ , and α Q in units of g . The intersection points are at κ ≈ − . , − . , − .
82% that looks quite good in the light of effective theory. At finite temperature, it is required, in addition, that (cid:0) w (1)1 , ¯1 (cid:1) = w (1)2 , = (cid:0) w (1)3 , (cid:1) . Like in Sec.III, we have no idea how to demonstrate, if possible, those relations. We canonly appeal to lattice gauge theory, where the relative weights are , , and , respectively.The last point to be mentioned here is the large distance behavior of the correlator. It is convenient to express C , in terms of the connected parts C , ( l ) = 1 + 3 C con , ( l ) + C con , ( l ) , (5.7)where C con , = C , − Since our technique doesn’t allow us to compute the large l behavior directly, we will assumethat, like in the case of the two-point correlator, in the large l limit C con , can be described in terms of closed stringstates propagating in flat space, as sketched in Figure 11 on the right. The solid lines here are propagators like thatof Figure 4. The leading asymptotics is given by the lightest closed string state. If a cubic interaction vertex is local,then it is C con , ( l ) ∼ e −√ Ml , (5.8)as l → ∞ . Thus, C con , is subleading to the first two terms in (5.7).At high temperatures, perturbation theory predicts that the prefactor in C con , is of order l − . This suggests that C con , ∼ (∆ F Q ) , which when combined with (4.2) and (5.8) gives∆ F Q ( l ) ∼ e − √ ml . (5.9)Of course, the above arguments are heuristic. Note, however, that they are similar to those in the case of the two-loopcorrelator, where C con , ¯1 ∼ (∆ F Q ¯ Q ) and, as a result, M = 2 m [17].What conclusions can one draw from the exponential decay law (5.9)? First, ∆ F Q decays for l → ∞ a bit fasterthan ∆ F ¯3 QQ . Indeed, one has √ ≈ . >
1. If so, then the relation (5.5) is not valid at large quark separations. Thesituation here reminds us of one story: the three-quark potential at zero temperature and its fitting by the ∆ and Y -laws, according to which the Y -law always holds at long distances. Curiously, the ratio between the exponentialdecay constants is the same as that between the coefficients in front of the linear terms in the ∆ and Y -laws. It is √ = σ ∆ σ Y , with σ ∆ = √ σ and σ Y = σ . Second, one can think of ∆ F Q as a complicated function whose asymptoticbehavior is given by equations (5.5) and (5.9). If so, then it follows that∆ F Q ≥ F ¯3 QQ . (5.10)A word of caution here is that the model we are considering is somewhat crude to account for a small deviationeven at short distances. Therefore, it would be very interesting to see the results of high-precision numerical studiesof the quark free energies. (iii) By now there are strong indications that the SU (3) theory of quarks and gluons has a dual description interms of quantum strings. Because the precise formulation of the latter is not known, we can only gain useful insightand grow with each success of the effective string model already at our disposal. Despite its efficiency [4, 11, 12],one might think of criticizing this model on several grounds including, (a) use of ad hoc background geometry whoseconsistency remains an open question within the α (cid:48) -expansion of string theory, (b) inability to account for correctionsarising from string fluctuations to the classical expression (2.7), (c) use of string theory to describe QCD with a finitenumber of colors and flavors, and (d) a lack of a satisfactory framework for string perturbation theory in the presenceof baryon (antibaryon) vertices. We have nothing to say at this point, other than that we hope to return to theseissues in future work. Note that it decays for l → ∞ as C con , ∼ e − Ml . It took some time to check this statement numerically. See, e.g., [33] and references therein. Unfortunately, the present precision [19] does not allow one to make certain conclusions about the relations between the free energies. Acknowledgments
We are grateful to C. Bachas, S. Finazzo, Y. Maezawa, J. Noronha, and P. Weisz for helpful discussions andcorrespondence. We also thank the Arnold Sommerfeld Center for Theoretical Physics for the warm hospitality. Thiswork was supported by Russian Science Foundation grant 16-12-10151.
Appendix A: A static Nambu-Goto string with fixed endpoints
The goal here is to generalize the results of [12] to the background fields of (2.1). This will enable us to build someof the string configurations in concrete examples. We set r h < / √ s because it guarantees that the correspondinggauge theory is deconfined [15].As usual, the Nambu-Goto action is given by S NG = 12 πα (cid:48) (cid:90) dσ (cid:90) dτ √ γ , (A.1)with γ an induced metric on a world-sheet with Euclidean signature. However, this is not the whole story. Since thequarks and antiquarks are regarded as string endpoints and the U (1) background field is introduced to mimic thebaryonic charge in gauge theories, there are additional coupling terms to the Nambu-Goto action S b = 13 (cid:90) dτ A t dtdτ (cid:12)(cid:12)(cid:12)(cid:12) σ =0 − (cid:90) dτ A t dtdτ (cid:12)(cid:12)(cid:12)(cid:12) σ =1 . (A.2)In other words, the string endpoints are electrically charged with respect to the background gauge field, with a chargethat is +1 / − / P and B in the xr -plane,as shown in Figure 12. This means that at the endpoints we have r xB ( x + , r + ) P ( x − , r − )0 α − α + r h FIG. 12: A string stretched between a quark placed at P and an antiquark placed at B . α ± denote the tangent angles, whichare both assumed to be positive. The horizon is at r = r h , while the boundary is at r = 0. x (0) = x − , x (1) = x + , r (0) = r − , r (1) = r + . (A.3) In our approach free strings are associated with mesons. t = τ and x = aσ + b , where a = x + − x − and b = x − . Moreover, in (A.1) and (A.2) the integrands are time-independent, so the integration over t simply givesa factor of 1 /T . We then have S = 1 T (cid:18) g (cid:90) x + x − dx w ( r ) (cid:112) f ( r ) + ( ∂ x r ) + 13 A t ( r − ) − A t ( r + ) (cid:19) . (A.4)For convenience, we use the shorthand notation g = R πα (cid:48) and ∂ x r = ∂r∂x .Obviously, the variation of the action (A.4) includes the contributions from the endpoints. In our study, we areinterested in the configurations shown in Figures 1 and 2. In all those cases, there are no such contributions if thefield r is subject to the Dirichlet boundary condition on the boundary of space and the Neumann boundary conditionat string junctions. The latter deserves a separate discussion. We will return to this issue after analyzing the equationof motion following from the above action.Since the integrand in (A.4) does not depend explicitly on x , the corresponding Euler-Lagrange equation has thefirst integral I = wf (cid:112) f + ( ∂ x r ) , (A.5)which takes the form I = wf (cid:112) f + tan α ± (cid:12)(cid:12)(cid:12)(cid:12) r = r ± (A.6)at the endpoints, where tan α ± = ∂ x r | x = x ± .In general, α + can take both positive or negative values. For our purposes, we need only positive ones such that afunction r ( x ) describing a string profile is increasing on the interval [ x − , x + ], as sketched in Figure 12. In this case,(A.5) yields a differential equation ∂ x r = (cid:0) f w I − f (cid:1) that can be easily integrated over the variables x and r . So,we get (cid:96) = x + − x − = (cid:90) r + r − dr √ f (cid:0) η − − (cid:1) − , with η = I f w . (A.7)After a rescaling of r , the integral becomes (cid:96) = r + (cid:90) r − r + dv √ f (cid:0) η − − (cid:1) − . (A.8)Having found the solution, we can now compute the corresponding action. It includes the minimal area andboundary contributions. The result is S = 1 T (cid:18) g r + (cid:90) r − r + dv w (cid:0) − η (cid:1) − + 13 A t ( r − ) − A t ( r + ) (cid:19) . (A.9)In the case of a string stretched along the r -direction, the integral can be evaluated analytically, yielding S = 1 T (cid:18) g (cid:20) √ π s (cid:16) Erfi( √ s r + ) − Erfi( √ s r − ) (cid:17) + e s r − r − − e s r r + (cid:21) + 13 A t ( r − ) − A t ( r + ) (cid:19) . (A.10)Another special case is a string with P on the boundary of space, i.e. when r − = 0. In this case, the integral (A.8)remains finite at v = 0, while (A.9) tends to infinity. We regularize it by imposing a cutoff (cid:15) such that r ≥ (cid:15) . In thelimit (cid:15) →
0, the regularized expression behaves like1 S R = g T (cid:15) + S + O ( (cid:15) ) . (A.11)Subtracting the (cid:15) term and letting (cid:15) = 0, we get a finite result S = 1 T (cid:18) g r + (cid:90) dv (cid:20) w (cid:0) − η (cid:1) − − r + v − r + (cid:21) + µ q − A t ( r + ) (cid:19) + c , (A.12)where µ q is a quark chemical potential and c is a normalization constant. It is dimensionless and related to the oneusually used in the literature by a simple rescaling. In the last step, the boundary condition satisfied by A t ( r ) at r = 0 and relation µ = 3 µ q were used.We similarly deal with the expression (A.10). As a result, we obtain S = 1 T (cid:18) g (cid:20) √ π s Erfi( √ sr + ) − e s r r + (cid:21) + µ q − A t ( r + ) (cid:19) + c . (A.13)For r + = r h , when a string stretched between the boundary and the horizon, this expression simplifies to S = 1 T (cid:16) µ q − F (cid:17) + c , with F = g (cid:20) e s r r h − √ π s Erfi( √ s r h ) (cid:21) . (A.14) Appendix B: Gluing conditions
The string solutions we discussed in the Appendix A provide just the basic blocks for building multi-string config-urations. By analogy with what was done in [12], we need certain gluing conditions for such blocks in the presence ofthe background fields (2.1). Here we will illustrate the idea with a simple example, as in Figure 13, that suffices forcomputing the free energy of a quark pair (diquark). x fe r h − l/ l/ e e r V (0 , , r + ) y FIG. 13: Three strings meeting at a string junction (vertex) placed at V . The black hole horizon is at r = r h . A physical way of interpreting the gluing conditions is to say that a static string configuration must obey thecondition that a net force vanishes at any vertex. We then write We denote vectors by boldface letters. (cid:88) i =1 e i + f = 0 , (B.1)where e i is a tangent vector at a string endpoint which represents a force exerted by the i string on the vertex and f is a gravitational force exerted on the vertex.In the presence of the U (1) gauge field, the zero force condition is to be completed by the neutrality condition (cid:88) i =1 q i + Q = 0 , (B.2)where q i is a charge of an endpoint of the i string that ends on the vertex. It is +1 or − Q is a charge of the vertex that is +3 or − U (1) background gauge field. The neutrality condition (B.2) implies that the termslinear in A t ( r + ) cancel each other. This simplifies the problem of finding the variation of the total action with respectto a location of the vertex and associated zero force condition. The former is nothing else but the Neumann boundarycondition at r = r + .With this simplification, we can straightforwardly extend the analysis of [12] to the black hole geometry (2.1).Accordingly, we take S = (cid:88) i =1 S i + S vert , (B.3)where S i is the Nambu-Goto action of the i -string and S vert is that of the vertex.Since we are interested in the static configuration, it is convenient to choose gauge conditions t i ( τ i ) = τ i , x i ( σ i ) = a i σ i + b i , (B.4)with ( τ i , σ i ) the world-sheet coordinates. Then the action of the i -string takes the form S i = g T (cid:90) dσ i w (cid:113) f a i + (cid:0) ∂ σ i r (cid:1) , (B.5)where a partial derivative with respect to σ i is conveniently denoted ∂ σ i .For the configuration in question, the boundary conditions on the fields are given by x (0) = x i (1) = 0 , x (0) = − x (0) = − (cid:96)/ , r (0) = r h , r (0) = r (0) = 0 , r i (1) = r + . (B.6)With these boundary conditions, a short calculation shows that a = b = 0 and a = − b = − a = b = (cid:96)/ xr -plane and, as a consequence, the equationsfor the x and y components of the net force are trivial. Thus, we need only consider the variation with respect to r + keeping in mind that δr i (1) = δr + . After some straightforward computation, we obtain2 tan α + (cid:112) f + tan α + − κr + e − s r ∂ r + (cid:16) e − s r r + (cid:112) f (cid:17) = 0 , (B.7)where κ = m g and α + = α + = − α + . The last term results from the gravitational force exerted on the vertex.Importantly, the resulting equation allows us to express α + in terms of r + . We havetan α + = f θ − θ , with θ = 1 − κr + e − s r ∂ r + (cid:16) e − s r r + (cid:112) f (cid:17) . (B.8)3 Appendix C: Consistency Issues
It is well known that Weyl invariance is essential to the consistency of string theory [22]. Apart from being anecessary requirement for a two-dimensional field theory on a worldsheet, this is very attractive for phenomenologyas it allows one to reduce a number of free parameters. As an important illustration of some of these ideas, we willdiscuss the relation between w , f , and A t in (2.1).For our purposes in this paper, what we need to know is that the Weyl coefficient for f is essentially the renor-malization group beta function β f [22]. Given the beta function for the metric β Gµν = Λ dG µν d Λ , with Λ a worldsheetscale, the beta function for f is determined by the components G tt and G ii which respectively represent the time andspatial (any one) components of the metric. The formula reads β f = 1 w R (cid:16) β Gtt − f β Gii (cid:17) . (C.1)Here i takes any value from 1 to 3.As a warmup, consider the case A t = 0. In the context of the α (cid:48) expansion of β Gµν , the leading term is given by theRicci tensor [22] and one gets β f = α (cid:48) w R (cid:16) R tt − f R ii (cid:17) . (C.2)By letting β f = 0 and then R tt − f R ii = 0, one further deduces ∂ r f + 32 ∂ r ww ∂ r f = 0 , (C.3)with ∂ r = ∂∂r . This provides the desired relation between w and f . Such a relation occurs in models of Einsteingravity coupled to matter , if the matter energy-momentum tensor obeys the constraint T tt = f T ii . (C.4)Then R tt − f R ii = 0 is a consequence of the Einstein equations.When we add the U (1) gauge field, we gain a Weyl coefficient β Aµ . To leading order in α (cid:48) , it is given by β Aµ = α (cid:48) ∇ ν F µν . (C.5)For the background fields (2.1), we have, by letting β Aµ = 0, ∂ r (cid:0) w ∂ r A t (cid:1) = 0 . (C.6)This equation with the boundary conditions (2.3) can immediately be integrated with the result A t ( r ) = µ − c R s (cid:16) − e − s r (cid:17) , (C.7) In the case of dilaton gravity, it was discussed in detail in [24]. The difference with [34] stems from the fact that we are dealing with the 5 d background metric. Also, the imaginary unit i is excludedfrom the solution. c is a positive constant. This form of the solution is particularly convenient as it allows one to make a contactwith the Reissner-Nordstr¨om solution of [8] in the limit s → β Gµν receives A -dependent corrections. To leadingorder in α (cid:48) , there are contributions from two terms: α (cid:48) G λσ F µλ F νσ and α (cid:48) G µν F . The latter is clearly irrelevant forour purposes, but the former matters because it has impact on β f . The contribution coming from the coefficient β Gtt is c α (cid:48) f R w ( ∂ r A t ) . Then equation (C.2) becomes β f = α (cid:48) w R (cid:18) R tt − f R ii + c π g f w − (cid:0) ∂ r A t (cid:1) (cid:19) . (C.8)Here we used the fact that α (cid:48) = R / π g . By letting β f = 0, we arrive at the desired equation ∂ r f + 32 ∂ r ww ∂ r f − c c π g R w − = 0 . (C.9)Like in the previous case, it can be integrated. Taking into account the boundary conditions (2.2), we obtain f ( r ) = 1 − A (cid:16) − (cid:0) s r (cid:1) e − s r (cid:17) − B (cid:16) − (cid:0) s r (cid:1) e − s r (cid:17) . (C.10)Here A = c c π g R s and B = − A (1 − (1+2 h )e − h )1 − (1+ h )e − h , with h = s r h . Note that at c = 0 the solution reduces to that of [25].With the expressions for A t and f , the temperature and baryon chemical potential are defined as usual T = 14 π (cid:12)(cid:12)(cid:12) ∂ r f (cid:12)(cid:12)(cid:12) r = r h , µ = A t (0) . (C.11)We think of c and h as two variables and of c as a parameter. However, it is convenient for practical purposes tointroduce a new variable q = A (9 + (7 + 6 h )e − h − − h ) and to express T and µ in terms of q and h : T √ s = 98 π (1 − q ) h U ( h ) , µ √ s = 2 √ r qZ ( h ) , (C.12)with r = (cid:114) π g c , U ( h ) = e h − − h , Z ( h ) = 9 + (7 + 6 h )e − h − − h − − h + e − h . (C.13)In this form, c combines with g to form a new parameter r to which we can assign arbitrary values. It is easy toguess the domain of the functions T and µ . It is defined by inequalities 0 ≤ q ≤ < h <
1. In particular, if q = 0, then µ = 0 and if q = 1, then T = 0.Finally, it is worth noting that the behavior of T and µ near h = 0 is T √ s = 1 − q √ h , µ √ s = r q √ h . (C.14) Here we are assuming a numerical factor c which can be traced into a ratio between the gauge and gravitational couplings in theeffective Lagrangian R + Λ + gF of [8] or, explicitly c = π g R g . See also [35]. Appendix D: Taylor Series of M Our goal here is to compute the first few coefficients of the Taylor series of M ( T, µ ) about the points ( T,
0) and(0 , µ ). The higher order coefficients can be computed in a similar fashion.Before getting started with the Taylor series of M , let us rewrite the relations (C.12) in a more convenient form8 xU = 9 h (cid:0) − Zy (cid:1) , (D.1)where x = π T √ s and y = √ r µ √ s . It now remains to obtain h as a function of x and y . We are unable to do soanalytically because the above equation is highly nonlinear. Therefore we use the Taylor expansion of h that sufficesfor our present needs.We start by illustrating the procedure in the case of small y . The form of (D.1) implies that h is an even functionof y . So we expand h as h ( x, y ) = h ( x ) + h ( x ) y + O ( y ) . (D.2)Inserting (D.2) into (D.1) and equating the terms with identical powers of y on the two sides of equation gives x = 98 h U − , h = 23 h ZUU + h (1 − e h ) . (D.3)The first equation says that x is a function of h , while the second equation represents a recursion relation for h once h is known. Here again, we are unable to analytically find h as a function of x .We are now ready to compute the two leading coefficients of the Taylor series (4.10). Taking the solution for h andusing (4.3), after a short computation we obtain M ( h ) = γπ √ s (cid:18) h − e h − √ π Erfi (cid:0)(cid:112) h (cid:1) + 9 w π h U − (cid:19) , M ( h ) = 9 γ √ s π r h e h ZU − h (e h − − U . (D.4)Thus, the Taylor coefficients are given by the parametric equations (D.3) and (D.4), with h a parameter.It is instructive to see what happens to these coefficients as h goes to zero. This limit is of interest because itcorresponds to the high temperature limit, as follows from (C.14). For the ratio between M and M , we get M M = π r (cid:0) w + π (cid:1) , (D.5)which is simply a constant. What is important for us is that (D.5) shows a similar behavior to that known fromperturbative QCD.We can now carry out a precisely analogous computation for the case of small x . In that case, we expand thefunction h as usual h ( x, y ) = h ( y ) + h ( y ) x + O ( x ) . (D.6)After inserting it into equation (D.1) and equating the terms with identical powers of x , we find y = Z − ( h ) , h ( y ) = 89 h − (e h − U Z (e h − Z − − h U . (D.7)The first equation gives y as a function of h , while the second gives a recursion relation for h .6Given the solution (D.7), we can compute the two leading coefficients of the Taylor series (4.13) by using the sameformula as before, with the result M ( h ) = γπ √ s (cid:18) h − e h − √ π Erfi (cid:0)(cid:112) h (cid:1)(cid:19) , M ( h ) = 2 √ γ r √ s Z (cid:18) w π + 49 h − e h (e h − U Z − h U − (e h − Z (cid:19) . (D.8)Thus, the Taylor coefficients are given by the parametric equations (D.7) and (D.8), with h is a parameter.For completeness, we note that in the limit h →
0, now corresponding to large chemical potential, the ratio between M and M tends to a constant value. Explicitly M M = r (cid:16) w + 12 π (cid:17) . (D.9) [1] P. Petreczky, J.Phys. G39 , 093002 (2012); A. Mocsy, P. Petreczky, and M. Strickland, Int.J.Mod.Phys.A , 1340012(2013); N. Brambilla et al., Eur. Phys. J.C , 2981 (2014).[2] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep. , 183 (2000).[3] L.D. McLerran and B. Svetitsky, Phys.Lett.B , 195 (1981).[4] O. Andreev and V.I. Zakharov, Phys.Rev.D , 025023 (2006).[5] O. Andreev, Phys.Rev.D , 065007 (2008).[6] S.-J. Rey, S. Theisen, and J.-T. Yee, Nucl.Phys.B , 171 (1998); A. Brandhuber, N. Itzhaki, J. Sonnenschein, and S.Yankielowicz, Phys.Lett.B , 36 (1998).[7] E. Witten, J. High Energy Phys. 07 (1998) 006.[8] A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, Phys.Rev.D , 104026 (1999).[9] O. Andreev and V.I. Zakharov, J. High Energy Phys. 0704 (2007) 100.[10] P. Colangelo, F. Giannuzzi, and S. Nicotri, Phys.Rev.D , 035015 (2011).[11] O. Andreev, Phys.Rev.Lett. , 212001 (2009).[12] O. Andreev, Phys.Lett.B , 6 (2016); Phys.Rev.D , 105014 (2016).[13] J.M. Maldacena, Phys.Rev.Lett. , 4859 (1998).[14] G.C. Rossi and G. Veneziano, Nucl.Phys.B , 507 (1977); J. High Energy Phys. 1606 (2016) 041.[15] O. Andreev and V.I. Zakharov, Phys.Lett.B , 437 (2007).[16] D.J. Gross and H. Ooguri, Phys.Rev.D , 106002 (1998); K. Zarembo, Phys.Lett.B , 527 (1999).[17] O. Kaczmarek, F.Karsch, E. Laermann, and M. L¨utgemeier, Phys.Rev.D , 034021 (2000).[18] O. Kaczmarek and F. Zantow, Phys.Rev.D , 114510 (2005).[19] K. H¨ubner, O. Kaczmarek, and O. Vogt, PoS LAT2005 (2006) 194.[20] M. D¨oring, K. H¨ubner, O. Kaczmarek, and F. Karsch, Phys.Rev.D , 054504 (2007).[21] M. D¨oring, S. Ejiri, O. Kaczmarek, F. Karsch, and E. Laermann, Eur.Phys.J. C , 179 (2006).[22] J. Polchinski, String theory, Vol.1, Cambridge Univ. Press, (1998).[23] D. Bak, A. Karch, and L.G. Yaffe, J. High Energy Phys. 0708 (2007) 049.[24] U. Gursoy, E. Kiritsis, L. Mazzanti, and F. Nitti, J. High Energy Phys. 0905, 033 (2009).[25] S.I. Finazzo and J. Noronha, Phys.Rev.D , 115028 (2014).[26] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, Nucl.Phys.B , 419 (1996).[27] E. Megias, E. Ruiz Arriola, and L.L. Salcedo, J. High Energy Phys. 01 (2006) 073.[28] S. Narison and V.I. Zakharov, Phys.Lett.B , 355 (2009).[29] Y. Maezawa et al, Phys.Rev.D , 091501 (2010).[30] O. Andreev, Phys.Rev.D , 107901 (2006).[31] N. Cabibbo and G. Parisi, Phys.Lett.B , 67 (1975).[32] T. Kahara and K. Tuominen, Phys.Rev.D , 114509 (2002); C. Alexandrou, Ph. deForcrand, and O. Jahn, Nucl. Phys.Proc.Suppl. , 667 (2003); N. Sakumichi and H. Suganuma, Phys.Rev.D , 034511(2015).[34] O. Andreev, Phys.Rev.D , 087901 (2010).[35] P. Colangelo, F. Giannuzzi, S. Nicotri, and F. Zuo, Phys.Rev.D88