On the spectra of scalar mesons from HQCD models
aa r X i v : . [ h e p - t h ] J un TAUP-287908 hep-th/yymmnnn
On the spectra of scalar mesons fromHQCD models
Oded Mintakevich and Jacob Sonnenschein
School of Physics and AstronomyThe Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University, Ramat Aviv 69978, Israel
Abstract
We determine the holographic spectra of scalar mesons from the fluctuations of the em-bedding of flavor D-brane probes in HQCD models. The models we consider includea generalization of the Sakai Sugimoto model at zero temperature and at the “high-temperature intermediate phase”, where the system is in a deconfining phase while ad-mitting chiral symmetry breaking and a non-critical 6d model at zero temperature. Allthese models are based on backgrounds associated with near extremal N c D4 branes anda set of N f << N c flavor probe branes that admit geometrical chiral symmetry breaking.We point out that the spectra of these models include a 0 −− branch which does notshow up in nature. At zero temperature we found that the masses of the mesons M n depend on the “constituent quark mass” parameter m cq and on the excitation number n as M n ∼ m cq and M n ∼ n . for the ten dimensional case and as M n ∼ m cq and M n ∼ n . in the non-critical case. At the high temperature intermediate phase we detect a decreaseof the masses of low spin mesons as a function of the temperature similar to holographicvector mesons and to lattice calculations. E-mails :[email protected] [email protected] hereas realizing confinement in dual holographic models of QCD (HQCD) is easythe incorporation of flavored chiral quarks and in particular chiral symmetry breaking ismore difficult. Sakai and Sugimoto [1] proposed a model that admits the two phenomena.It is based on placing a set of N f D8 and anti D8 probe flavor branes into the gravitymodel of near extremal D4 branes [2, 3].The mesonic spectra is one of the most important properties of hadron dynamics thatcan be “measured” in the HQCD laboratory. The low spin mesons are associate with thefluctuations of the fields that reside on the probe flavor branes, the vector mesons withthe U ( N f ) flavor gauge fields and the scalar mesons with the embedding of the probebranes . Here in this paper we focus only on scalar mesons. The motivation behindaddressing this problem are the following: (i) To verify that the meson spectrum at zerotemperature does not include tachyonic modes. Had there been such modes it wouldhave indicated that the system is unstable. Since the model of [1] is based on placingbranes and anti-branes one may be worried that the system is unstable and hence theimportance of this check. (ii) The spectrum of the scalar mesons has been determinedalready in [1]. However the attemps to derive it in generalizations of the model where theasymptotic separation of the brane anti-brane L is smaller than half of the circumferenceof the compactified direction x , namely for L ≤ πR failed for the symmetric modes [5, 6](iii) To determine the dependence of the spectrum on the excitation number n and theparameter m cq defined in (42) that is related to the constituent quark mass. In additionone naturally would like to compare the explicit ratios of meson masses that one deducesfrom any given HQCD model and the experimental data to get an indication of howwell the model describes real hadron physics. (iv) To further examine the differencesof physical properties extracted from critical models to non-critical models which werepreviously discussed in [5, 7, 8]. The spectrum of scalar mesons was extracted also inother HQCD models[9, 10, 11]. For further reading see [12] and references therein.We can summarize the outcome of the paper as follows • We were able to choose coordinates that avoid the singularities that were en-countered in previous works [5, 6] and determine the spectrum of both the anti-symmetric as well as symmetric branches. • We find that in the models examined and in particular the original model of [1] thesymmetric solutions correspond to scalar mesons of the form 0 ++ whereas the anti-symmetric solutions correspond to 0 −− mesons. This property which seems to bein common to a HQCD models based on probe branes and anti-branes, contradictthe low lying spectrum in nature. There are no low lying 0 −− mesons. High spin mesons are naturally described by semi-classical spinning string configurations[4] At zero temperature we found that the masses of the mesons M m depend on the“constituent quark mass” m cq and on the excitation number n as M m ∼ m cq and M m ∼ n α with α ∼ . M m ∼ m cq and M m ∼ n β with β ∼ .
75 when a CS term is incorporated and β ∼ −− mesonswhich do not show up in nature. Section 6 is devoted to the spectrum of mesons abovethe deconfining phase transition in the “intermediate phase”. We summarize the resultsand raise certain open questions. The model of [2], describes the near horizon limit of N c D x direction with anti periodic boundary condition for the fermions. Into thisbackground a stack of N f D x = 0 and a stack of N f ¯ D x circle [1]. Assuming N f << N c one can overlook the modification of themetric and dilaton due to the backreaction of the background by the N f D
8- ¯ D N c D ds = (cid:18) uR D (cid:19) / (cid:20) − dt + δ ij dx i dx j + f ( u ) dx (cid:21) + (cid:18) R D u (cid:19) / (cid:20) du f ( u ) + u d Ω (cid:21) (1) F = 2 πN c V ǫ , e φ = g s (cid:18) uR D (cid:19) / , R D = πg s N c l s , f ( u ) = 1 − (cid:18) u Λ u (cid:19) V denotes the volume of the unit sphere Ω and ǫ its corresponding volume form. l s is the string length and g s a parameter related to the string coupling. The x is thecompactified direction that is asymptotically transverse to the D
8. The manifold spannedby the coordinate u, x has the topology of a cigar where its tip is at the minimum valueof u which is u = u Λ . The periodicity of this cycle is uniquely determine to be δx = 4 π (cid:18) R D u Λ (cid:19) / = 2 πR (2)in order to avoid a conical singularity at the tip of the cigar. The classical profile of the D D S D = T Z dtd xdud Ω e − φ q − det ˆ g = ˜ T Z dtd xduu vuut f ( u )( ∂ u x ) + R D u f ( u ) (3)= ˜ T Z dtd xdx u vuut f ( u ) + (cid:18) R D u (cid:19) u ′ f ( u ) (4)where ˆ g stands for the pullback metric on the D x and so its Hamiltonian isconserved. u f ( u ) s f ( u ) + (cid:18) R D u (cid:19) u ′ f ( u ) = u q f ( u ) = const (5)where we assumed that there is a point u where the curve u ( x ), which describes theprofile of the D u, x ) plane has a minimum. At that point the D D
8- ¯ D (cid:18) ∂x ∂u (cid:19) cl = 1 f ( u )( uR D ) / r f ( u ) u f ( u ) u − D x ( u ) = Z uu duf ( u )( uR D ) / r f ( u ) u f ( u ) u − u is a constant of integration setting the lowest value of u to which the D D D u also sets the asymptotic distance L betweenthe position of the D D L = Z dx = 2 Z ∞ u duu ′ = 2( R D u (cid:19) / Z ∞ dy y − / q f ( y ) r f ( y ) f (1) y − L ∝ (cid:18) R D u (cid:19) / (9)For later use we define γ = u f ( u ) u − f ( u ) u (10)The DBI action then becomes S = T Z e − φ q | det ˆ g | ∼ Z d xduγ / u / In [13] a study of the thermodynamics of the Sakai Sugimito model was carried usingthe conjecture presented in [2].The conjecture states that the thermodynamics of a field theory with a gravitational dualis determined by taking into account the contribution to the saddle point approximationfrom all the gravitational backgrounds with the correct ’UV’ asymptotic, with compacti-fied Euclidean time direction of period β = T and with anti-periodic boundary conditionfor the fermions along this direction. The temperature of the field theory is T = 1 /β andits properties are read from the manifold responsible for the most dominant contributionto the saddle point approximation namely the one that has the lowest free energy.When ever one background looses its domination to another background as we vary thetemperature, a phase transition occurs in the dual field theory.In [13] two manifolds where found to have the same ’UV’ asymptotic as the one ofSakai and Sugimoto model, the background (1), and the same configuration only withthe time and x directions interchange. ds = (cid:18) uR D (cid:19) / [ − f ( u ) dt + δ ij dx i dx j + dx ] + (cid:18) R D u (cid:19) / (cid:20) du f ( u ) + u d Ω (cid:21) (11)with f ( u ) = 1 − (cid:18) u T u (cid:19) (12)and the temperature is given by δt = 4 π (cid:18) R D u T (cid:19) / = β (13) see also [14]. N c [(2 πT ) − /R ] . This means that when the circumference of the x cycle issmaller than that of the time direction namely when T < / πR the background (1)is the dominant one, while when the opposite occurs and T > πR the action of (11)will dominates. At the temperature T = 1 / πR the two actions are the same since thetwo backgrounds are different by the labeling of the coordinates, so at T = T c = 1 / πR the system has a first order phase transition. In [13] it was argue that in the dual fieldtheory, the physical interpretation to this phase transition is a transition from confinedphase at T < / πR to deconfined at T > / πR . This can be seen via a computationof the quark anti-quark potential [15] in the two backgrounds. Another indication to thisinterpretation is that the renormalized free energy of the low temperature phase shows a N c behavior while that of the high temperature phase shows a N c one. Hence from nowon we will denote T c = T d .At the high temperature phase there is another possible classical solution to the profileof the D x namely, x ( u ) = 0 , L . Now since the bulk free energy is the same for the two configurations of the D D y T = u u T , its value at the phase transitionturns out to be y cT ∼ . L c = 0 . (cid:18) R D u (cid:19) / , hence at the critical point y T = y cT the criticaltemperature is set by the asymptotic distance between the branes (setting R D = 1) T c = 34 π u / T = 34 π ( y cT u ) / = 0 . /L (14)The field theory sees this transition as chiral symmetry restoration at high temperature,this interpretation is natural since the D U ( N f ) × U ( N f ) global symmetry.Hence we will denote this critical temperature as T χSB . Note that this only happens atthe high temperature phase so there is still the condition T χSB = 0 . /L > / πR .So if L > . R , we find that T d is always higher than T χSB , and so deconfinement andchiral symmetry restoration phase transition happen together. We see that in this model χSB and confinement appear independently of one another as a result of the existence Of course in our model there are also D N c and the contribution of the D N c · N f which is negligible in the probeapproximation This configuration was not possible in the low temperature, but in the high temperature phase thetime circle shrink to zero at u = u Λ and so the D
5f the free parameter L coming from the 5 d nature of the field theory. A non critical model with a very similar properties to Sakai-Sugimoto model was pre-sented in [5, 7] , this model consists of non-extremal configuration of N c D D ds = (cid:18) uR AdS (cid:19) dx , + (cid:18) R AdS u (cid:19) du f ( u ) + (cid:18) uR AdS (cid:19) f ( u ) dx (15) F (6) = Q c (cid:18) uR AdS (cid:19) dx ∧ dx ∧ dx ∧ dx ∧ du ∧ dx e φ = 2 √ √ Q c ; R AdS = 152with f ( u ) = 1 − (cid:18) u Λ u (cid:19) (16)and where Q c is proportional to N c , the number of color D x is set to x ∼ x + δx ; δx = 4 πR AdS u Λ (17)Of course the curvature of order one of this background makes the leading order super-gravity an un justified approximation to string theory on this background. Neverthelessits believed that at least the extremal model due to its symmetries, is indeed a goodbackground for the study of non-critical string theory [16]. Now we place N f D S cycle and extend up to infinity in the u direction. Theproperties of the four dimensional low energy effective field theory living on the intersec-tion of these color and flavor D D u, x ) cigar and in orderto find its profile one must solve the e.o.m of the x coordinate. This e.o.m is derived For other non-critical SUGRA models with flavor see [16, 17, 18, 19, 20]. D S D = − T Z d xe − φ q − det(ˆ g ) + T ˜ a Z P ( C (5) ) (18)Following similar steps to does taken in the previous section we find x ,cl ( u ) = Z uu ( u f / ( u ) − au + au ′ ) du ′ ( u ′ R AdS ) f ( u ′ ) q u ′ f ( u ′ ) − ( u f / ( u ) − au + au ′ ) (19)where a = √ . We now turn our attention to the study of the fluctuation of the D x coordinate around its classical value and define thefluctuation ξ ( u, x µ ) as follows: x ( u, x µ ) = x ( u ) cl + ξ ( u, x µ ) (20)Substituting this into the action (3) and expanding to quadratic order in ξ we find thefollowing action for the fluctuations S ∝ Z d xdu (cid:26) u / R D γ − / η µν ∂ µ ξ∂ ν ξ + u / γ − / ( ∂ u ξ ) (cid:27) (21)where γ is defined in (10). We now introduce the following mode expansion ξ ( u, x µ ) = ∞ X n =0 f n ( x µ ) ξ n ( u ) (22)Using the symmetries along the x µ directions we have η µν ∂ µ ∂ ν f n = − m n f n (23)The e.o.m for the ξ n modes reads ∂ u [( u / γ − / ) ∂ u ξ n ] = − m n R D u / γ − / ξ n (24)7r in its canonical form (cid:26) ∂ u + (cid:20) ( 12 u − u ) γ − u (cid:21) ∂ u (cid:27) ξ n = − m n R D γu ξ n (25)For u >> u Λ , f ( u ) →
1, the e.o.m simplifies and the qualitative behavior of m n canbe determined by using dimensional arguments [13]. Define the dimensionless parameter v = uu then for the limit u >> u Λ where f → γ → − v (26)The e.o.m in terms of v reads ∂ v ( v / γ − / ) ∂ v ξ n = − m n R D u v / γ − / ξ n (27)Since the L.H.S is dimensionless so must be the R.H.S and hence m n ∝ u R D (28)Using the relation (9) between u and L we find m n ∝ L (29)while the mass of the glueball is related to m gb ∼ R . For the case u Λ = u , L = πR sothe glueball and mesons masses have the same scale. However in the general case where u > u Λ there are two different scales m n ∼ L > R ∼ m gb .In order to find the exact spectrum of the eigenvalues of (25) one can use the shoot-ing technic which is implemented by demanding symmetric or anti-symmetric boundarycondition to ξ n at u = u and integrating the equation up to the u >> u region wherethe solution could be matched to its normalizable asymptotic expansion. Of course thismatching is only possible when the correct eigenvalues are being used and so one shootswith different eigenvalues until a matching is obtained . However there is a problem withthese coordinates at u = u since, dx ,cl du | u = u → ∞ ( see eq. (6)). An odd perturbationto the classical configuration will cause no change in the shape of this singularity but aneven one will, and so will also have a singular derivative.This problem is reflected in the singularity of the e.o.m (25) at u → u . To see thisbehavior explicitly we change coordinate to a dimensionless parameter z as follows u = u + u z (30)the eigenvalue problem (25) then becomes (cid:26) ∂ z + (cid:20) u zu + u z − z − γ ′ u z ( u + u z ) / γ (cid:21) ∂ z (cid:27) ξ n = − m n R D u γz ( u + u z ) / ξ n (31)8here γ ′ stands for the derivative of γ with respect to u . Since γ z → = u u (8 u − u ) z ; γ ′ z → = − u u (8 u − u ) z (32)we find that this equation has a regular singularity at z = 0!Indeed it was already noticed in [6] that by employing the ’shooting’ technic only halfof the spectrum could be found, namely only the odd modes where seen while the evenones could not be obtained, these modes that should had been obtained by setting theboundary conditions to ξ n ( z = 0) = 1 ; ∂ z ξ n ( z = 0) = 0 . (33)turned to be singular and could not be integrated. In [1] only the special case of u = u Λ was analyzed, in this case since lim u → u Λ ∂ u x cl = 0, a smooth and nonsingular transfor-mation into cartesian coordinates is allowed via u = u + u ( z + y ) ; x = R arctan( yz ) (34)The corresponding action for y is (after setting u Λ = 1) S ∼ Z d xdz (cid:20) ( ∂ µ y ) u ( z ) + u ( z ) ( ∂ z y ) + 2 y (cid:21) (35)inserting the expantion y = P n =1 ϕ n ( x µ ) y n ( z ) the e.o.m for y n is ∂ z y n + 2 z z ∂ z y n − y n z = m n (1 + z ) / y n (36)which is non-singular. For the more general case of u > u Λ we were not able to find asimilar coordinate transformation and hence we follow a different approach desribed inthe next section. As we have seen above, we could not obtain the even modes of the fluctuation aroundthe classical curve because dx ,cl du diverges at u = u . The issue of choosing a directionalong which one should anlayze the fluctuations, has been discussed in the context of thestringy description of the Wilson like [21]. It was found that the safest approach is to use If the classical curve x ,cl was odd, then the odd mode would become singular. u = u is along the u direction.Thus from here on we study the fluctuation in the u direction, that is u ( x , x µ ) = u cl ( x ) + ξ ( x , x µ ) (37)our classical curve would be u cl ( x ) and as can be seen from (6) we have du cl dx | x =0 = 0so the point u ( x = 0) = u pause no problem now! The quadratic action for thesefluctuation is (after setting u Λ = 1) S = 12 Z dx (cid:26) a u f ( ∂ x ξ ) + 1 u f ( ∂ µ ξ ) (38) − (11 u + 18 a + 3 u − u − a ( u + u ) − u )2 u f ξ (cid:27) where a = u f ( u ) and it should be understood that u = u cl ( x ) and its formal expres-sion is u ( x ) = Z x dx f ( u )( uR D ) / vuut f ( u ) u f ( u ) u − ∂ x ξ n − ( 11 u + 9 uf ) u x ∂ x ξ n − f u m n a ξ n (40)+ (11 u + 18 a + 6 u − u − a ( u + u ) − u )2 a u ξ n = 0where u x = ∂ x u cl . Since there is no analytic expression for the integral in (39), weobtained u ( x ) numerically during the integration of eq. (40) when ’shooting’ to find theeigenvalues of (40).The resulted spectra are summarized in figures (1), (2) and (3). The following propertiescharcterize these spectra • The first observation one can make is that for u = u Λ our results for the symmetricand anti-symmetric lowest lying states match those of [1]. m s = 3 . m as = 5 . • The figures (1) and (2) describe the dependence of the squared mass of the firstexcited symmetric and anti-symmetric states as a function of the “constituent quarkmass” defined in [5] and [4], as follows m cq = 12 πα ′ Z u u Λ √− g tt g uu du = 12 πα ′ Z u u Λ f − / ( u ) du (42)10his parameter relates to the constitutent quark mass and not to the current algebra( QCD) mass, since even when it is turned on the fluctuations that correspond to thepions are massless. In fact the quantity dual of the constituent quark mass shouldassociate with m cq plus a constant term which is independent of u since alreadyfor u = u Λ the mesons are massive and hence there is a non-trivial constituentmass. This assignment is also in accordance with the semi-classical description ofhigh spin mesons [4] and their stringy split into two lower mass mesons [22]. ¿Fromthese figures we see that indeed for for u > u Λ the square of the mass of the scalarsgrows linearly with m cq . This is to be contrasted with the results found in [5] forvector mesons of non-critical models where the masss itself is found to be linearwith the m cq ( see also down in section (5).) • We have also determined the spectrum of the higher excited mesons both the sym-metric as well as the anti-symmetric ones. The dependence of the squared masseson the excitation number for various values of m cq is drawn in figure (3). The linearfit to this curves are given by m n = 3 . . n . m cq = 0 (43) m n = 10 . . n . m cq = 9 . m n = 15 . . n . m cq = 14 . m ∼ n behavior. We thus seethat the scalar meson spectra that follow from the model of [1] do not correspondto stringy modes. This is of course of no surprise since it follows from a low energyeffective field theory and not from a semi-classical treatment. • Last by not least we see from figure (1) that the lowest scalar excitation remainnon-tachyonic for all values of u which serves as an partial evidence for the stabilityof the Sakai Sugimoto model. We would like now to find the masses of the scalar modes associated with the fluctuationsof the probe brane around the classical profile in the non-critical gravity background of[7]. Using the background (15) in an effective action that includes only the DBI. The CSterm S CS ∼ Z D C ∼ Z D u R AdS (44)11A) (B)Figure 1: (A) The mass squared m of the lowest exited symmetric mode as a functionof m cq ( R D = u Λ = 1)Figure 2: (B) The mass squared m of the lowest exited antisymmetric mode as a functionof m cq ( R D = u Λ = 1)Figure 3: (A) The tower of the mesons squared mass m n in the low temperature phase( R D = u T = 1) 12ould contribute upon a substitution of the form (37) for u and expanding a linear andquadratic term to the action, thus affecting the spectrum. However, it was found in [8]that including this CS does not yield a sensible thermal phase diagram and hence wediscuss separately an effective action that includes only a DBI action and one with boththe DBI and CS terms. We start first with the former case: Analyzing the spectrum in asimilar manner to the analysis of section (3) we find that the fluctuations are subjectedto the following eigenvalue equation ∂ u ( u γ − / ) ∂ u ξ n = − R AdS m ′ n u γ / ξ n (45)Like in the critical case, for u >> u Λ the qualitative behavior of m ′ n can be seen bychanging the variable u into the dimensionless parameter y = uu . At the limit u >> u Λ we find that f ( u ) → γ → u ( y − y ) (46)and find that in terms of the dimensionless parameter y the e.o.m is now ∂ y ( y γ − / ) ∂ y ξ n = − R AdS m ′ n y u γ / ξ n (47)Since the L.H.S is dimensionless so is the R.H.S and we find m ′ n ∝ u R AdS (48)Note that due to the different background now L ∼ R AdS u and hence again we get that m ′ n ∼ L . However in terms of m cq the asymptotic behavior is that m ′ n ∼ m cq and not m ′ n ∼ m cq as was the case for the mesons of the critical model.Repeating the exact same steps as for the critical case we find that the quadraticaction for fluctuation in the x direction around the classical curve leads to an e.o.mwhich is singular at u = u and as a consequence the attempt carried in [5] to obtain thespectrum of the even modes had indeed failed. And so like in the critical case we turn tostudy the fluctuation in the u direction instead. The action for the fluctuation is then S = 12 Z dx (cid:26) a / u f ( ∂ x ξ ) + a / R AdS u f ( ∂ µ ξ ) (49) − a / ( u + 36 a − u + 14 u + 48 u − a u − a )2 u f (cid:27) and indeed this action leads to a regular e.o.m at u ( x = 0) = u . ∂ x ξ − ( 14 u + 15 u f ) u x ∂ x ξ + u f R AdS a η µν ∂ µ ∂ ν ξ
13 ( u + 36 a − u + 14 u + 48 u − a u − a u )2 u a ξ = 0 (50)Using the shooting technic we found the eigenvalues of different modes of the fluctuationfor various values of m cq , our finding are summarize in figures (4),(5). One can see thatthe masses m ′ and m ′ grow linearly with m cq as expected from (48). At u = u Λ = 1 wefind . m ′ s = 1 .
51 ; m ′ as = 2 . . (51)which is in agrement with [5]. Again we also studied the dependence of the mass onthe excitation number, the results are summarized in figure (6) and are: m n = 1 .
51 + 2 . n . m cq = 0 (52) m n = 13 . . n . m cq = 9 . n as well as the dependence on m cq the scalar meson spectra admits a different behavior than that of the critical modelof [1]. A similar behavior has been observed for the vector mesons in [5].Next we consider the case where the effective action includes both the DBI and CSterms. Including now the CS term (with its full strengh ˜ a = 1) the quadratic action forthe fluctuation becomes S = 12 Z dx (cid:26) B / u f ( ∂ x ξ ) + B / R AdS u f ( ∂ µ ξ ) (53) − B / ( u + 36 B − u + 14 u + 48 u − Bu − B )2 u f − √ u ξ (cid:27) where B = ( u f / ( u ) − u + u ) and the e.o.m is then ∂ x ξ − ( 14 u + 15 u f − u B / ) u x ∂ x ξ + u f R AdS
B η µν ∂ µ ∂ ν ξ + ( u + 36 B − u + 14 u + 48 u − Bu − Bu )2 u B ξ + 20 u √ B / ξ = 0 (54)With the Chers Simon taking into account the dependence of the mass squared onthe excitation number is now to be read from figure (9) to be: m n = 2 .
07 + 5 . n . m cq = 0 (55) m n = 16 .
49 + 1 . n . m cq = 9 . m cq is described in figures (7) and (8). Our results are for R AdS = 1. To keep contact with the results in [5] we had renormalized the masses by the factor coming fromthe change of variables u → z . q q (A) (B)Figure 4: (A) The mass m ′ of the lowest exited symmetric mode of the non-criticalmodel as a function of m cq ( R AdS = u Λ = 1).Figure 5: (B) The mass m ′ of the lowest exited antisymmetric mode of the non-criticalmodel as a function of m cq ( R AdS = u Λ = 1).Figure 6: The tower of mesons masses m ′ n in the non-critical model15 q q . m q Non critical model with CS term (a=1) :
The mass of the first excited anti-symmetric mode vs. m q . (A) (B)Figure 7: (A) The mass m ′ of the lowest exited symmetric mode of the non-criticalmodel with CS term included as a function of m cq .Figure 8: (B) The mass m ′ of the lowest exited antisymmetric mode of the non-criticalmodel with CS term included as a function of m cq . n2 4 6 8 10 12M n m q = 0 , 9.3 in RB order Figure 9: The tower of mesons masses m ′ n in the non-critical model with CS term included16 Parity and charge conjucation
In order to compare the resulting spectra from both the critical and non-critical models,we first have to identify the “quantum numbers” of the states that correspond to thefluctuations. More explicitly we have to determine the operations in the gravity mod-els which correspond to charge conjugation and parity transformations. In the modelof [1] they were defined as follows: The charge conjugation operation associates withexchanging the left and right handed quarks which maps into the interchange of a D D z → − z . Parity transformation in the five-dimensional space-time spanned by x i , z where i = 1 , , x i , z ) → ( − x i , − z ).For the generalized set up with u > u Λ we can still define the coordinate z as follows u = u + u Λ z (56)Note the difference with respect to (30) since here we take z to have dimension of length.With this definition of the z coordinate the discrete transformations of [1] remain in tact.The effective action on the probe brane has to be invariant under both parity and chargeconjugation. The DBI part (21) is quadratic in ξ and hence cannot determine the righttransformation of the fluctuation modes. The situation with the CS term is different.Recall that the CS term has the form S CS ∼ Z D F ∧ F ∧ C = Z S F ∧ F ∧ Z d xdzC = Z S F ∧ F Z d xdzξ ( x µ , z ) (57)The last part we have used the explicit form of the C C = ξ ( x µ , z ) dx ∧ ...dx ∧ dz (58)In order for this term in the action to be invariant under parity and charge conjugationit is clear that ξ ( x µ , z ) has to be even under both charge confugation and parity trans-formation. Now since ξ ( x, z ) = P n f n ( x µ ) ξ n ( z ) we conclude that the map between thefluctuation modes and sclar particles is the following symmetric ξ n → ++ mesonsantisymmetic ξ n → −− mesons (59)For the non-critical model again the DBI action does not determine the transforma-tions of ξ under parity and charge conjugation. We have argued above based on [8] thata CS term of the form (44) should not be incorporated. Thus there is no way to thisorder to determine the transformation of ξ .17ithout the constraint from the CS term we may have that ξ is even or odd undercharge conjugation and parity transformtaions. In the latter case the assignments of (59)have to be reversed, namely symmetric ξ corresponds to 0 −− and antisymmetric ξ to 0 ++ Next we want to compare the spectra to mesons observed in nature. It is well knowthat scalar mesons in nature are either 0 ++ or pseudo sclars of the form 0 − + and there areno observed low lying mesons of the form 0 −− . Thus there is a serious mismatch betweenthe holographic scalar mesons extracted from models with flavor branes anti-branes ofcritical models and with the observed mesons in nature. We will come back to this issuein the conclusions. The background that corresponds to the deconfined phase namely
T > / πR is given in(11). As was shown in [13] this deconfined background can admit also a phase where chiralsymmetry is broken, the so called “intemediate phase” We now analyze the spectrum ofthe scalar mesons in this phase. Since the procedure of extracting the scalar meson isidentical to that of the low temperature analysis of the previous sections we present thefinal results for the spectra of masses. The spectra are presented in figures (10), (11) and(12). The main features that these spectra admit are the following • As can be seen, at the phase transition T = T d the value of the masses are (for thevalues u T = 1 , u = 8) m s ( T = T d ) = 8 .
36 ; m as ( T = T d ) = 45 .
96 (60)while in the low temperature phase at the point of phase transition with u Λ → u T = 1, u = 8 the masses are m s ( T = T d ) = 8 .
40 ; m as ( T = T d ) = 46 .
00 (61)We see a very small jump in the masses at the transition point, the same as wasseen for the vectors in [6] • While in the low temperature confining phase the masses of the mesons are tem-perature independent since the background in this phase does not depend on thetemperature, the masses of the mesons do depend on the temperture in the inter-mediate deconfined phase. As was observed in Lattice simulations and was foundalso for holographic vector mesons [6], the masses decrease as a function of the tem-perature. The symmetric mesons decrease at the chiral symmetry phase transitiontemperature T = T χSB to ∼
60% persent of their values whereas the antisymmetric18nes to ∼ • Note that it is only consistent to increase the temperature up to where the nextphase transition occur and chiral symmetry is restored.This happens at T = T χSB (for the choice u = 8 we found that T χSB = 2 . T d ),then the merged together D
8- ¯ D D
8- ¯ D
8. We canalso see from figure (10) that if we continue to increase the temperature beyond T χSB then at some point the scalar mode becomes Tachyonic, signaling that thisbackground is no longer stable at this temperature as indeed we know. • Like in the low temperature we had also checked the squared masses dependenceon the excitation number see figure (12), this was found to be: m n = 8 . . n . T = T d (62) m n = 7 . . n . T = 2 T d I this paper we had overcome technical problems faced in [22, 5] and succeeded to obtainthe holographic mass spectra of the scalars in the low and intermediate phases of thechiral symmetry broken phase of the critical model and also of those of the non-critical.Let us summarize the results of this work and mention certain open directions. • There is a difference between the dependence of the mass of the scalar mesons onthe “constituent mass parameter” m cq . In the ten dimensional models one finds a m ∝ m cq relation (see figures (1),(2) for the first two excited modes), whereas forthe non-critical model the relation is m ∝ m cq (see figures (4),(5) and (7),(8)). • Both the critical models and the non-critical one do not admit a Regge/stringybehavior of M n ∼ n . This is not unexpected since the stringy excitations is notvisible in the low energy effective field thoery. • One can compare the ratio of the low lying mesons both vector and scalar mesonsto those observed in nature. Table (1) present such a comparison. It is interestingto note that turning on a constituent mass m cq improves the ratios with respect tothose for zero m cq . 19A) (B)Figure 10: (A) The mass squared m ( T ) of the lowest exited symmetric mode as afunction of T /T d ( u = 8 , R D = 1 and R = 2 / m ( T ) of the lowest exited antisymmetric mode as afunction of T /T d ( u = 8 , R D = 1 and R = 2 / m n in the intermediate phase20able 1: A comparison to the experimental data where the best fitted m cq is presentedvs. m cq = 0 (for the critical case we have found that there is no improvement in ratios ofthe vectors so we have left these entries empty.).experiment D4-D8 at m cq = 0 / .
38 Non-critical at m cq = 0 / . m v, /m v, m v, /m v, m s /m v, m v, /m s • The hologrphic spectra of the critical models admit a branch of scalar mesons of thetype 0 −− . These does not exist in nature. It seems to be a severe shortcoming ofthese holographic models. This difference cannot be attributed to the fact that weconsider large N c . It will be interesting to investigate the question of how genericis this situation and whether one can construct a mechanism to project it out fromthe low lying spectra. • The behavior of the scalar mesons at finite temperature in the intermediate phaseis similar to that for the vector meson in the model of [6]. However the decrease ofthe mass with increasing temperature is more dramatic for the scalar mesons. It isinteresting to check if a similar phenomenon occurs also in lattice simulations.
Acknowledgments
We would like to thank Kasper Peeters, Tadakatsu Sakai and Marija Zamaklar for usefuldiscussions, and specially to Ofer Aharony for many insightful conversations. This workwas supported in part by a center of excellence supported by the Israel Science Foundation(grant number 1468/06), by a grant (DIP H52) of the German Israel Project Cooperation,by a BSF grant and by the European Network MRTN-CT-2004-512194
References [1] T. Sakai and S. Sugimoto. Low energy hadron physics in holographic qcd. Preparedfor 2004 International Workshop on Dynamical Symmetry Breaking, Nagoya, Japan,21-22 Dec 2004.[2] Edward Witten. Anti-de sitter space, thermal phase transition, and confinement ingauge theories.
Adv. Theor. Math. Phys. , 2:505–532, 1998.213] Nissan Itzhaki, Juan Martin Maldacena, Jacob Sonnenschein, and ShimonYankielowicz. Supergravity and the large n limit of theories with sixteen super-charges.
Phys. Rev. , D58:046004, 1998.[4] Martin Kruczenski, Leopoldo A. Pando Zayas, Jacob Sonnenschein, and Diana Va-man. Regge trajectories for mesons in the holographic dual of large-N(c) QCD.
JHEP , 06:046, 2005.[5] Roberto Casero, Angel Paredes, and Jacob Sonnenschein. Fundamental matter,meson spectroscopy and non-critical string / gauge duality.
JHEP , 01:127, 2006.[6] Kasper Peeters, Jacob Sonnenschein, and Marija Zamaklar. Holographic meltingand related properties of mesons in a quark gluon plasma.
Phys. Rev. , D74:106008,2006.[7] Stanislav Kuperstein and Jacob Sonnenschein. Non-critical, near extremal ads(6)background as a holographic laboratory of four dimensional ym theory.
JHEP ,11:026, 2004.[8] Victoria Mazo and Jacob Sonnenschein. Non critical holographic models of thethermal phases of QCD. 2007.[9] Johanna Erdmenger and Ingo Kirsch. Mesons in gauge / gravity dual with largenumber of fundamental fields.
JHEP , 12:025, 2004.[10] J. L. Hovdebo, M. Kruczenski, David Mateos, Robert C. Myers, and D. J. Winters.Holographic mesons: Adding flavor to the AdS/CFT duality.
Int. J. Mod. Phys. ,A20:3428–3433, 2005.[11] E. Antonyan, J. A. Harvey, and D. Kutasov. Chiral symmetry breaking from inter-secting D-branes.
Nucl. Phys. , B784:1–21, 2007.[12] Johannaa Erdmenger, Nick Evans, Ingo Kirsch, and Ed Threlfall. Mesons inGauge/Gravity Duals - A Review.
Eur. Phys. J. , A35:81–133, 2008.[13] Ofer Aharony, Jacob Sonnenschein, and Shimon Yankielowicz. A holographic modelof deconfinement and chiral symmetry restoration.
Annals Phys. , 322:1420–1443,2007.[14] Andrei Parnachev and David A. Sahakyan. Chiral phase transition from stringtheory.
Phys. Rev. Lett. , 97:111601, 2006.[15] Y. Kinar, E. Schreiber, and J. Sonnenschein. Q anti-q potential from strings incurved spacetime: Classical results.
Nucl. Phys. , B566:103–125, 2000.2216] Igor R. Klebanov and Juan Martin Maldacena. Superconformal gauge theories andnon-critical superstrings.
Int. J. Mod. Phys. , A19:5003–5016, 2004.[17] Dan Israel. Non-critical string duals of N = 1 quiver theories.
JHEP , 04:029, 2006.[18] U. Gursoy, E. Kiritsis, and F. Nitti. Exploring improved holographic theories forQCD: Part II.
JHEP , 02:019, 2008.[19] Sameer Murthy and Jan Troost. D-branes in non-critical superstrings and dualityin N = 1 gauge theories with flavor.
JHEP , 10:019, 2006.[20] F. Bigazzi, Roberto Casero, Angel Paredes, and A. L. Cotrone. Non-critical stringduals of four-dimensional CFTs with fundamental matter.
Fortsch. Phys. , 54:300–308, 2006.[21] Y. Kinar, E. Schreiber, J. Sonnenschein, and N. Weiss. Quantum fluctuations ofWilson loops from string models.
Nucl. Phys. , B583:76–104, 2000.[22] Kasper Peeters, Jacob Sonnenschein, and Marija Zamaklar. Holographic decays oflarge-spin mesons.