An AdS/QCD model from Sen's tachyon action
aa r X i v : . [ h e p - ph ] J u l Preprint typeset in JHEP style - HYPER VERSION
CCTP-2010-2
An AdS/QCD model from Sen’s tachyon action
Ioannis Iatrakis a , Elias Kiritsis a ∗ , ´Angel Paredes b a Crete Center for Theoretical Physics, Department of Physics, University of Crete,71003 Heraklion, Greece b Departament de F´ısica Fonamental and ICCUB Institut de Ci`encies del Cosmos,Universitat de Barcelona, Mart´ı i Franqu`es, 1, E-08028, Barcelona, Spain.
Abstract:
We construct a new, simple phenomenological model along the lines ofAdS/QCD. The essential new ingredient is the brane-antibrane effective action includ-ing the open string tachyon proposed by Sen. Chiral symmetry breaking happens becauseof tachyon dynamics. We fit a large number of low-spin meson masses at the 10%-15%level. The only free parameters involved in the fits correspond to the overall QCD-scaleand the quark masses. Several aspects of previous models are qualitatively improved.
Keywords:
Gauge-gravity correspondence, Tachyon Condensation, QCD, SpontaneousSymmetry Breaking. ∗ On leave of absence from APC, Universit´e Paris 7, (UMR du CNRS 7164). ontents
1. Introduction 12. The model 13. Fitting the meson spectrum 44. Conclusions 7
1. Introduction
Understanding the strong dynamics underlying many observations related to the stronginteraction remains an unsolved problem. Much progress has been made to date withdifferent methods but new insights are always welcome. A recent development that hasled to reconsidering the strong interaction has been the AdS/CFT duality. This has beenapplied to obtain new insights on QCD phenomenology. In this paper, we focus on themeson spectrum - (see [1] for a review of the gauge-gravity literature on the issue).There are two main ways to address the problem at hand. Top-down approaches usestring theory from first principles in order to build dual theories as close as possible toQCD. Notable examples are [2], [3]. On the other hand, bottom-up approaches, startingfrom the works [4, 5, 6], use known QCD features to develop holographic models that areonly inspired by string theory. The model we present here is more of the second kind, butincludes the main stringy ingredients we expect from first principles, namely the effectiveaction controlling the chiral dynamics. Our main observation is that merging this stringyinput of top-down holographic models for flavor allows us to improve the existing bottom-up models both qualitatively and quantitatively.
2. The model
Background and action
In [7], it was shown, quite generally, that effective actions for brane-antibrane systemsderived from string theory [8] encode a set of qualitative features related to chiral symmetrybreaking and QCD at strong coupling. The goal of this paper is to build a concrete modelwithin the framework of [7]. We will consider the simplest smooth gravitational backgroundthat is asymptotically AdS, while having a confining IR in the same spirit as [9]. Thisturns out to be the
AdS soliton, which was shown to be a solution to the two-derivative– 1 –pproximation of subcritical string theory and used as a toy model for certain aspects of4d Yang-Mills in [10]. The metric reads: ds = R z (cid:2) dx , + f − dz + f Λ dη (cid:3) (2.1)with f Λ = 1 − z z . The coordinate η is periodically identified and z ∈ [0 , z Λ ]. The dilaton isconstant and we do not write the RR-forms since they do not play any role in the following.We now consider a D4- ¯D4 pair, located at fixed η in this background . We write the actionproposed by Sen [8] as S = − Z d xdzV ( | T | ) (cid:16)p − det A L + p − det A R (cid:17) (2.2)The objects inside the square roots are defined as: A ( i ) MN = g MN + πα ′ h F ( i ) MN + (( D M T ) ∗ ( D N T ) + ( M ↔ N )) i (2.3)where M, N = 1 , . . . ,
5, the field strengths F ( i ) MN = ∂ M A ( i ) N − ∂ N A ( i ) M and the covariantderivative of the tachyon is D M T = ( ∂ M + iA LM − iA RM ) T . The active fields in (2.2), (2.3)are two 5-d gauge fields and a complex scalar T = τ e iθ , which are dual to the low-lyingquark bilinear operators that correspond to states with J P C = 1 −− , ++ , − + , ++ -; see[7] for details. In the action of [8], the transverse scalars (namely η in the present case)are also present. We have discarded them when writing (2.3) since they do not have anyinterpretation in terms of QCD fields. Accordingly, even if the background (2.1) is six-dimensional, the holographic model for the hadrons is effectively five-dimensional and, infact, its field content coincides with those of [5], [6]. For the tachyon potential we take, asthe simplest possibility, the one computed in boundary string field theory for an unstableD p -brane in flat space [12], although one should keep in mind that this expression for V isnot top-down derived for the present situation. In the present conventions, V = K e − π τ ,where K is an overall constant that will play no role in the following since it does notenter the meson spectrum computation (it is important though, in the normalization ofcorrelators when computing for instance decay constants [5], [6], [13]). The tachyon massis m T = − α ′ and we will impose: R = 6 α ′ in order to have m T R = −
3. This shouldnot be interpreted as a modification of the background due to the branes, but just as a(bottom-up) choice of the string scale that controls the excitations of those branes, suchthat the bifundamental scalar T is dual to an operator of dimension 3, as in [5], [6]. Sincethe AdS radius is not parametrically larger than α ′ , the two-derivative action cannot be acontrolled low enegy approximation to string theory. This is the main reason why a modelof this kind cannot be considered of top-down nature. Notice the value of R we take differsfrom the one used in [10]. In five-dimensional holographic models of QCD, the flavor branes are expected to be a D4- ¯D4 system,[11]. – 2 – he tachyon vacuum and chiral symmetry breaking
As shown in [7], an essential ingredient of the present framework is that the generation ofthe correct flavor anomaly on the flavor branes requires the tachyon modulus τ to divergesomewhere. Therefore, τ must have a nontrivial vev which breaks the chiral symmetry.From the action (2.2) we obtain the equation determining τ ( z ): τ ′′ − π z f Λ τ ′ + ( − z + f ′ Λ f Λ ) τ ′ + (cid:18) z f Λ + π τ ′ (cid:19) τ = 0 (2.4)where the prime stands for derivative with respect to z . Near z = 0, the solution can beexpanded in terms of two integration constants as: τ = c z + π c z log z + c z + O ( z ) (2.5)where, on general AdS/CFT grounds, c and c are related to the quark mass and con-densate (see [13] for a careful treatment). From (2.4), we find that τ can diverge only at z = z Λ . There is a one-parameter family of diverging solutions in the IR: τ = C ( z Λ − z ) − πC ( z Λ − z ) + . . . (2.6)The interpretation is the following: for a given c (namely quark mass ) fixed in the UV(near z = 0), the value of c (namely the quark condensate) is determined dynamicallyby requiring that the numerical integration of (2.4) leads to the physical IR (near z = z Λ )behavior (2.6). Hence, for any value of c , one can obtain numerically the function for thevev h τ i . Meson spectrum: Numerical results
There is a rather standard method for computing the meson spectrum in holographicmodels, see [1] for a review. Each bulk field is dual to a boundary operator and itslinearized perturbation can be obtained after expanding (2.2). By looking at normalizablefluctuations of the bulk fields, one typically encounters discrete towers of masses for thephysical states with the corresponding quantum numbers. Thus, for a fixed value of c ,we have a Sturm-Liouville problem for each bulk mode. This can be solved numerically,using a standard shooting technique. By computing the different towers at different valuesof c , we found the following expressions to be very good approximations to the numericalresults, in the range 0 < c < z Λ m (1) V = 1 .
45 + 0 . c , z Λ m (2) V = 2 .
64 + 0 . c z Λ m (3) V = 3 .
45 + 0 . c , z Λ m (4) V = 4 .
13 + 0 . c z Λ m (5) V = 4 .
72 + 0 . c , z Λ m (6) V = 5 .
25 + 0 . c . (2.7) For the present work, we will just use that c is proportional to m q . Finding the proportionalitycoefficient requires normalizing the action and fields as in [14]. In practice, we have been able to perform numerics in a controlled manner only for 0 ≤ c < – 3 –or the axial vectors: z Λ m (1) A = 1 .
93 + 1 . c , z Λ m (2) A = 3 .
28 + 1 . c z Λ m (3) A = 4 .
29 + 0 . c , z Λ m (4) A = 5 .
13 + 0 . c z Λ m (5) A = 5 .
88 + 0 . c , z Λ m (6) A = 6 .
55 + 0 . c . (2.8)For the pseudoscalars: z Λ m (1) P = q . c + 5 . c , z Λ m (2) P = 2 .
79 + 1 . c z Λ m (3) P = 3 .
87 + 1 . c , z Λ m (4) P = 4 .
77 + 1 . c z Λ m (5) P = 5 .
54 + 1 . c , z Λ m (6) P = 6 .
24 + 0 . c . (2.9)For the scalars: z Λ m (1) S = 2 .
47 + 0 . c , z Λ m (2) S = 3 .
73 + 0 . c z Λ m (3) S = 4 .
41 + 0 . c , z Λ m (4) S = 4 .
99 + 0 . c z Λ m (5) S = 5 .
50 + 0 . c , z Λ m (6) S = 5 .
98 + 0 . c . (2.10)It turns out that meson masses increase linearly with c . Namely, they increase linearly withthe bare quark mass, as expected from an expansion in m q and in qualitative agreementwith lattice results, see for instance [15],[16],[17]. The exception, of course, is the firstpseudoscalar for which m π is proportional to √ m q (for small m q ), as expected from theGell-Mann-Oakes-Renner relation. Actually, the behaviour m π = q b m q + d m q was alsofound in the lattice [15].
3. Fitting the meson spectrum
We now proceed to make a phenomenological comparison of the results of (2.7)-(2.10) tothe experimental values quoted by the Particle Data Group [18]. Obviously, we can onlymodel those mesons with J P C = 1 −− , ++ , − + , ++ . From [18], we will just take thecentral value quoted for each resonance. We do not discuss decay widths here (in the strict N c → ∞ limit they are of course zero). Isospin 1 mesons
We start by looking at mesons composed of the light quarks u and d . In particular, wediscuss the isovectors. In table 1, we show all the mesons listed in the meson summarytable of [18] under light unflavored mesons which have isospin 1 and the J P C ’s present inour model. The only exception is a (980), which is considered to be a four-quark state [18].We have fitted the parameters of the model to these observables by minimizing the rmserror ε rms = 100 × √ n (cid:16)P O (cid:0) δOO (cid:1) (cid:17) , where n = 8 − z − = 522MeV , c , l = 0 . P C
Meson Measured (MeV) Model (MeV)1 −− ρ (770) 775 762 ρ (1450) 1465 1379 ρ (1700) 1720 18061 ++ a (1260) 1230 10150 − + π π (1300) 1300 1462 π (1800) 1816 20260 ++ a (1450) 1474 1295 Table 1:
A comparison of the results of the model to the experimental values for light unflavoredmeson masses. with ε rms = 12%.In table 2, we display the resonance masses with isospin 1 listed in [18] under otherlight unflavored mesons . These are namely states considered as “poorly established thatthus require confirmation”. For the results given by our model, we use (3.1) and thereforeno further parameter is fitted here. For this set of observables we get ε rms = 24%, wherewe have inserted n = 11 − ρ (2150) as the fourth memberof the ρ -meson tower . We have not included ρ (1570) in table 2 because its excitationnumber is smaller than ρ (1700), which was included in the previous fit. In case ρ (1570)gets confirmed as a member of this tower, the fit should be redone. We observe that themodel tends to consistently overestimate the masses of the excited axial vectors and pions.This is connected to the fact the model yields a Regge slope for axial mesons larger thanthe one for the vectors [7]. If the experimental results of table 2 are confirmed, one shouldthink of improving the model in order to ameliorate this discrepancy. s ¯ s states A nice feature of the present model is that it incorporates the dependence of the hadronmasses on the quark mass. This allows us to study s ¯ s states. More precisely, it allows us todiscuss “hypothetical states” with quark content s ¯ s assuming no mixing with other states.In the real world, the mixing for pseudoscalars and scalars is important (see chapter 14 of[18]), and therefore it is not possible to compare directly the outcome of the model to theexperimental results. Nevertheless, as in [19], we can estimate the masses of these “hypo-thetical” s ¯ s mesons from the light-strange and light-light mesons. Then, using quotationmarks for the hypothetical states, and using the quark model classification (table 14.2 of[18]) we can write m (“ η ”) = q m K − m π , m (“ φ (1020)”) = 2 m ( K ∗ (892)) − m ( ρ (770)), m (“ η (1475)”) = 2 m ( K (1460)) − m ( π (1300)), etc. Keeping the value of z Λ found in (3.1), Indeed, there is much more experimental evidence for ρ (2150) than for ρ (1900) or ρ (1570). We thankS. Eydelman for very useful explanations. – 5 – P C
Meson Measured (MeV) Model (MeV)1 −− ρ (1900) 1900 2159 ρ (2150) 2150 2467 ρ (2270) 2270 27461 ++ a (1640) 1647 1721 a (1930) 1930 2245 a (2096) 2096 2686 a (2270) 2270 3073 a (2340) 2340 34230 − + π (2070) 2070 2493 π (2360) 2360 28990 ++ a (2020) 2025 1952 Table 2:
A comparison of the results of the model to the experimental values for other lightunflavored meson masses. we fit the value of c associated with the strange quark to the “experimental” values oftable 3, obtaining. c ,s = 0 .
317 (3.2)The rms error for this set of observables ( n = 6 −
1) is ε rms = 10%. J P C
Meson Measured (MeV) Model (MeV)1 −− “ φ (1020)” 1009 876“ φ (1680)” 1363 14741 ++ “ f (1420)” 1440 12100 − + “ η ” 691 725“ η (1475)” 1620 16470 ++ “ f (1710)” 1386 1403 Table 3:
A comparison of the results of the model to the hypothetical states with s ¯ s quark content. Further comments
We end this section by commenting on the estimates given by the model on two otherphysical quantities. Without fixing the proportionality coefficient between c and thequark mass, from (3.1), (3.2), we can infer the ratio of the strange quark to the light quarkmass: m s m u + m d ≈ c ,s c ,l ≈
25. Moreover, the background studied here experiences a first orderdeconfinement phase transition in full analogy with [9]. The deconfinement temperatureis given by T deconf = π z Λ ≈
208 MeV. Both of these values are close to the experimentalvalues. – 6 – . Conclusions
We have built a new phenomenological model for the meson sector of QCD. In this paperwe have discussed the mass spectrum. We note the simplicity of the construction, whoseessential point is the use of Sen’s action [8] including the open string tachyon field. Wehave applied it to one of the simplest backgrounds exhibiting confinement [10]. Despitethe minimal input, we have found the following interesting qualitative properties: • The model includes towers of excitations with J P C = 1 −− , ++ , − + , ++ , namely alllow-lying operators that do not need a dual excited stringy state. • Chiral symmetry breaking is consistently realized. Moreover, the value of the quarkcondensate is computed dynamically and is not a tunable input. Hence, the num-ber of tunable parameters coincides with those present in QCD: they are just thedynamically generated scale and the quark masses. • We find Regge trajectories for the excited states m n ∝ n , as in the soft wall model[20]. This allows good predictions for the higher excitations, as opposed to the hardwall model [5, 6]. Notwithstanding, the Regge slope for axial vectors is bigger thanthe one for vectors. This fact requires further study. • Our model incorporates the increase of the vector meson masses due to the increaseof quark masses, as m ρ ≈ k + k m π for small m π .Previous AdS/QCD models present some of these properties, but as far as we know,no existing model is able to encompass all of them, see [21] for recent related discussions.We briefly comment on the three benchmark models: the Sakai-Sugimoto model [3] missesthe first and third points listed above, the hard wall model [5, 6] misses the third one andpartially the second one; and the soft wall model [20] misses the second one. All of thesemodels [3, 5, 6, 20] and variations thereof fail to get the fourth point (although it is worthmentioning that D3D7 models with abelian flavor symmetry do capture the physics of thisfourth point, see section 6.2.3 of [1]).Moreover, the quantitative matching shown in tables 1 and 3 with the central valuesof the meson resonances is excellent, at the 10%-15% level. This is a typical accuracy ofAdS-QCD-like models (a recent example is [22], which accounts for excited spin states ofthe ρ and ω families). Since the systematic error produced by quenching is of the orderof 10% [23] and the differences between quenched lattice computations with N c = 3 and N c = ∞ are again of the order of 10% [16, 17], it would be unexpected to get a betteraccuracy from any model of the kind presented here.It would be of utmost interest to generalize the set-up to the non-abelian case, allowingseveral quark flavors, but this is beyond the scope of the present work. Acknowledgments
This work was partially supported by a European Union grant FP7-REGPOT-2008-1-CreteHEP Cosmo-228644. The research of A.P is supported by grants FPA2007-66665C02-– 7 –2 and DURSI 2009 SGR 168, and by the CPAN CSD2007-00042 project of the Consolider-Ingenio 2010 program. I.I. has been supported by Manasaki Graduate Scholarship and thePhysics department of the University of Crete until September 2009 and by A.S. OnassisFoundation Graduate Scholarship from October 2009.
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