Holographic technicolor models and their S-parameter
aa r X i v : . [ h e p - t h ] J un TAUP-290109 hep-th/yymmnnn
Holographic Technicolor models and theirS-parameter
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 study the Peskin-Takeuchi S-parameter of holographic technicolor models. We presentthe recipe for computing the parameter in a generalized holographic setup. We then ap-ply it to several holographic models that include: (a) the Sakai-Sugimoto model and(b) its non-compactified cousin, (c) a non-critical analog of (a) based on near extremal
AdS background, (d) the KMMW model which is similar to model (a) but with D D D D S s with D D SU ( N T C ) gauge theory and with a breakdown of a flavor symmetry U ( N T F ) × U ( N T F ) → U V ( N T F ). The models (a), (c),(d) and (e) are duals of a confininggauge theories whereas (b) and (f) associate with non confining models.The S-parameter was found to be S= sN T C where s is given by 0 . λ T C , 0 . λ T C ,0 . .
50 and 0 .
043 for the (a),(b),(c),(d), (f) models respectively and for model (e) s isdivergent. These results are valid in the large N T C and large λ T C limit. We further derivethe dependence of the S-parameter on the “string endpoint” mass of the techniquarksfor the various models. We compute the masses of the low lying vector technimesons. [email protected]@post.tau.ac.il
Introduction
One of the most urgent questions in particle physics is the nature of the mechanismof electro-weak symmetry breaking (EWSB) and in particular the exact structure ofthe Higgs sector. One appealing class of models that may provide an answer to thisquestion are Technicolor models. In these models a new sector of strongly interactingfermions known as techniquarks are added to the S.M instead of the scalar Higgs. Thissector will now be responsible for the spontaneous chiral symmetry breaking ( χSB ) viaa condensate in the TeV scale. The condensate, in a similar manner to ordinary QCD,is of a tecniquark anti-techiquark operator. The techiquarks transform under certainrepresentation of the gauge group SU ( N T C ). In these models the Higgs boson is acomposite state of a techniquark and an anti-etchniquark so that the hierarchy problemis avoided. One of the most restricting demands of an EWSB model is that it shouldproduce a small Peskin-Takeuchi S-parameter[1]. This requirement comes from highprecision electro-weak measurement. The S-parameter defined as S = 16 π [(Π ′ (0) − Π ′ Q (0)] (1)is restricted to be in the range of S = − . ± . N T C D N T F = 2 of D D U ( N T F = 2) L × U ( N T F = 2) R chiral symmetry which is spontaneously brokenin the IR into a U ( N T F = 2) V symmetry. In [3], using the AdS/CFT dictionary, the For the derivation of the S-parameter its relation to electroweak measurement and the definition ofthe variables in this expression see section (2). χSB and to determine theirS-parameter as well as the low lying vector technimesons. Third to determine the generalproperties of holographic models and to compare between the holographic estimate ofthe S-parameter to phenomenological estimation of the S-parameter based on scaled upversion of QCD.We start with a five dimensional effective action derived from the DBI and CS actionsof the gauge fields on the probe branes. Assuming a background that depends only on theradial coordinate, the most general action of this nature was written down. In analogyto the derivation of [3] the expressions for the S-parameter in both the direct methodand the sum-rule method were derived. This recipe was then applied to the followingmodels: • (a) The Sakai Sugimoto model [4] which is Witten’s model [5] generalized to includeD8 prob branes and admits confinement and χSB . • (b) The uncompactified analog of the latter model. This model, which was analyzedin [6], is dual of the NJL gauge system which does not admit confinement but stillexhibits spontaneous χSB . • (c) A non critical analog of the Sakai Sugimoto model derived in [7], [8], [9] andexhibits both confining and spontaneous χSB . This model consists of N T F pairsof D − ¯ D AdS background. • (d) The KMMW model [10] which is also based on Witten’s model but with N T F = 2 D − ¯ D • (e) A holographic model [ ? ] based on the near extremal limit of D5 branes com-pactified on two circles with D − ¯ D χSB is spontaneous. • (f) The Klebanov-Witten conifold model with D − ¯ D χSB . 2e found that model (e) fails to serve as a candidate for Techinicolor/Higgs sectorsince it has a divergent S-parameter. In the rest of the models considered we found apositive S-parameter which is linear in N T C that is S = sN T C . In models (c), (d) and (f) s is just a numerical factor independent of λ T C where Λ
T C = g T C N T C . However, in theSakai Sugimoto model and its AHJK cousin s was found to be linear in λ T C (see table(1)). Recall that the results are valid in the large N T C and large λ T C limit.Holographic Technicolor was studied in recent years also in the following papers [14].The paper is organized as follows: We start in section 2 with a brief review of the physicsbehind the Peskin-Takeuchi S,T and U parameters. This section is brought for the benefitof the readers which are not familiar with those parameters. Other readers can movedirectly to section 3. Section 3 is devoted to the determination of the S-parameter inholographic models. We derive the formulae for computing the parameter from a generalholographic model reduced to five dimensions. This is done in both a direct methodas well as a sum-rule approach. In section 4 we show the way the general result isimplemented for the Sakai-Sugimoto model as was derived in [3] and we further derivein a qualitative way the dependence of the S-parameter on the string endpoint masses.In section 5 we repeat the steps of section 4 but for the uncompactified version of Sakai-Sugimoto model presented by AHJK in [6]. In section 6 we examine a holographic modelderived in [8] which is a non-critical analog of the Sakai-Sugimoto model based on N T F pairs of D − ¯ D AdS background. Section 7 isdevoted to analyzing the S-parameter of the KMMW model [10]. The model is based onWitten’s model with D − ¯ D ? ] based on the near extremal limit of D5 branes compactified on two circles with D − ¯ D D − ¯ D In the standard model, the EWSB and fermion masses are explained by the existenceof the Higgs scalar field which acquire a non zero VEV. The physics of the Higgs sectordepends on four free parameters, the coupling constants g ′ and g of the U (1) y and SU (2) L respectively, v the VEV of the Higgs field and its mass m h . We can express certainobservable quantities via these parameters such as m W = g v m Z = v q g ′ + g ; e = gg ′ √ g ′ + g ; G f = 1 √ v (2)3e used three parameters to define four observable quantities so there is a hidden relationamong them which is independent of the value of the parameters in the Lagrangian. Sucha relation can be constructed for example by using the different definition of the weakmixing angle in terms of observable quantities such as: s = sin θ w = g ′ g ′ + g = 1 − m W m Z (3)sin 2 θ = (cid:18) e √ G f m z (cid:19) / (4)Another useful definition is constructed from the polarization asymmetry of Z decaysinto left and right electrons A eLR = Γ( Z → ¯ e L e L ) − Γ( Z → ¯ e R e R )Γ( Z → ¯ e L e L ) + Γ( Z → ¯ e R e R ) = ( − s ∗ ) − s ∗ ( − s ∗ ) + s ∗ (5)These three different definition of the weak mixing angle coincide at tree levelsin θ w = sin θ = s ∗ (6)but their loop corrections are different. Subtracting them from each other or taking theirratios produces what is known as zeroth order natural relation which means a relationwhich doesn’t depends on the parameters of the Lagrangian. Hence, these relationsare free of any UV divergencies coming from counter terms (since these only alters theparameters of the Lagrangian), and so the only quantum corrections they receive arefinite and can be considered as predictions of the quantum structure of the theory. Inlight of (6) we can easily construct the following zeroth order natural relations: c − c = s − s ; s ∗ ( q ) − s ; s − s ∗ ( q ) (7)where we used the definitions c = 1 − s = cos θ w = m W m Z ; s = sin θ ; c = cos θ (8)Another useful zeroth order natural relation we shell use is the ratio of charged to neutral-current amplitudes denoted by ρ ∗ (0), and is equal to one at tree level. Now, we would liketo estimate the radiative corrections to these relations, and hopefully to divide them intostandard model ones and to those coming from the technicolor sector which supposedlygive the true descriptions of the Higgs sector. There are many kinds of loop correctionsto these zeroth order natural relations, in addition to the corrections to the vector boson4ropagator, there are vertex corrections, box diagrams, and diagrams with real photonemission. In strongly interacting technicolor models the techniquarks do not coupledirectly to the leptons and at low energies do not appear in the final states, the onlyplace where the new physics going to enters is through corrections to the vector bosonpropagator via its vacuum polarization where they appear in loops of techniquark andanti-techniquark pairs. Otherwise the techniquarks are not observed at low energy, hencethese corrections are called ’oblique’. We note that in general, loop contributions arenot gauge invariant one by one, but rather their sum is, but since these are the onlycontributions involving the techniquarks their gauge invariance is self evident. As wenoted earlier our goal is to sperate the radiative corrections coming from the new physicsfrom that of the standard model, if we assume that m f ≪ m z ( m f is the mass scale of thefermions at the outer legs), then we can ignore the vertex corrections and box diagramssince these are suppressed by additional factor of m f m Z relatively to the oblique correction.So we are left with the problem of separating the new physics contributions to the vacuumpolarization from the SM’s. The experimental data that we are trying to fit comes fromphysics at energy scales between Λ QCD to the TeV scale. In this range of energies theQCD is weakly coupled while the techniquarks are still in the strong coupling regime.Hence we can use perturbation theory to estimate the quarks contributions to the vacuumpolarization amplitude of the gauge fields but we cannot do so for the techniquarks. Theone loop SM oblique corrections to (7) are given by s ∗ − s = − α π ( c − s ) m t m Z + . . . (9) s − s ∗ = − α πs m t m Z + . . . Now, denoting Π IJ as the correlators of the I and J currents of SU (2) L × U (1) Y ,where only the contributions coming from the new physics are taking into account, thenafter some algebra we obtain the following form for the radiative corrections to (7) dueto the Technicolor sector: c − c = s − s = − (cid:20) e c s ( c − s ) m Z (cid:20) Π ( m Z ) − s Π Q ( m Z ) (10) − s c Π (0) − c − s c Π ( m W ) (cid:21) + e s c c − s [Π ′ QQ ( m Z ) − Π ′ QQ (0)] (cid:21) We note that the one loop vacuum polarization amplitude is proportional to m q m Z where m q is themass of the fermion in the loop, so one only consider the top quark contribution. Of course one should also consider the contributions coming from the physical Higgs boson, butsince we are replacing this sector by the technicolor sector, it is omitted [15]. ∗ ( q ) − s = (cid:20) e c − s (cid:20) Π ( m Z ) − s Π Q ( m Z ) − Π (0) m Z − ( c − s ) Π Q ( q ) q (cid:21) (11)+ e s c − s [ s Π ′ QQ ( m Z ) − c Π ′ (0) QQ + ( c − s )Π ′ QQ ( q )] (cid:21) ρ ∗ (0) − e s c m Z [Π (0) − Π (0)] (12)Thus, we see that it is both possible and natural to isolate the radiative correctionsdue to new physics from those coming from the SM fields. If the new physics included inthe vacuum polarization amplitude is associated with new heavy particles of mass scale m T C ≫ m Z , then we will see a rapid convergence of a Taylor expansion in q of theseamplitude. Thus it is natural to expand the Π IJ in powers of q , neglecting the order q and beyond: Π QQ ( q ) ≈ q Π ′ QQ (0) (13)Π Q ( q ) ≈ q Π ′ Q (0)Π ( q ) ≈ Π ( o ) + q Π (0)Π ( q ) ≈ Π ( o ) + q Π ′ (0)There are six independent coefficient in (13) but three linear combinations of them mustcancel out since there are no UV divergences in (10), (11) and (12) despite there are inthe Π IJ . The remaining three are the following: S ≡ π [Π ′ (0) − Π ′ Q (0)] (14) T ≡ πs c m Z π [Π (0) − Π (0)] U ≡ π [Π ′ (0) − Π ′ (0)]Substituting (14) into (10), (11) and (12) yields m W m Z − c = αc c − s (cid:20) − S + c T + c − s s U (cid:21) (15) s ∗ ( q ) − s = αc − s [ 14 S − s c T ] ρ ∗ (0) − αT To summarize, we concluded that under the above assumptions the dominant radia-tive corrections to (7) comes from the vacuum polarization amplitudes, and these receivecontributions from two sources, the standard model part given by (9) which is fixed and6ell known, and a part coming from a sector of new physics and are given by (15).According to (15), we have a three parameter description of the radiative corrections dueto the technicolor sector, and since the quantities in the left hand side are all observablethere are experimental bounds on the magnitude of these corrections! In this paper wefocus on pure technicolor models without an extension that could produce isospin viola-tion, in this case the T and U parameters are zero and we use the experimental boundon S alone. These experimental bounds restricts S to be in the range of S = − . ± . S = 16 π [(Π ′ (0) − Π ′ Q (0)] = − π [Π ′ V (0) − Π ′ A (0)] | (16)where Π V and Π A are define by i Z d xe − iqx hJ aVµ ( x ) J bVν (0) i = − (cid:18) g µν − q µ q ν q (cid:19) δ ab Π V ( q ) i Z d xe − iqx hJ aAµ ( x ) J bAν (0) i = − (cid:18) g µν − q µ q ν q (cid:19) δ ab Π A ( q )where J aVµ and J aAµ are the vector current and axial-vector current respectively. Usingdispersive representation with delta function resonances these could be expressed byΠ V ( − q ) = X n g V n q m V n ( − q + m V n )Π A ( − q ) = X n g A n q m A n ( − q + m A n )It follows then that S could be written as S = 4 π X n (cid:18) g V n m V n − g A n m A n (cid:19) (17)where m V n /A n and g V n /A n are the masses and decay constants of the vector/axial-vectormesons of the confined phase of the Technicolor sector. The general holographic technicolor setup is similar to that of holographic QCD. Itis based on a gravity background that admits confinement in the sense of an area-law Below we discuss also models without confinement or without chiral symmetry N T C which corresponds to the rank of the dual technicolorgauge group SU ( N T C ). A set of N T F flavor probe D p branes is incorporated in thisbackground. The worldvolume of the D p flavor branes includes the four dimensionalspace-time, the radial direction and a p − S T F = S DBI + S CS = − T p Z d p +1 σe − φ q − det ( g ind + F ) + T p Z X k C k ∧ e F (18)where T p is the tension of the D p probe branes, g ind is the induced metric on thoseprobe brane, F = 2 πl s F + B ind where F is the techniflavor field strength associated with U ( N T F ) gauge symmetry and B ind is an induced B field ( if there is one) and C k is a k RR form.Since an important ingredient in the technicolor scenario is the spontaneous breakingof the technichiral symmetry, the flavor probe branes have to admit geometrically in theregion dual to the UV flavor chiral symmetry of the form U L ( N T F ) × U R ( N T F ) and in theIR a spontaneous breakdown of this symmetry to the diagonal subgroup U D ( N T F ). Thisrequires an embedding profile of the form of a U shape. Examples of such a holographicsetup are the well known Sakai Sugimoto model [4], its non-critical analog and the recentlyproposed model based on incorporating D7 flavor branes in the Klebanov Strassler model.Integrating the DBI action over the p − U ( N T F ) gauge fields. S DBI = − κ p Z d xdu [ a ( u ) F µν F µν + 2 b ( u ) F ρu F ρu ] (19)where u indicates the radial direction, Greek indices are space-time indices, the contrac-tion of indices is done with η µν , κ p , a ( u ) and b ( u ) are given by κ p ≡ − T p (2 πα ′ ) V p − g s a ( u ) ≡ g s e − φ q det ( g ind )( g ρρind ) b ( u ) ≡ g s e − φ q det ( g ind ) g ρρind g uuind (20)and where we assumed that the induced metric is diagonal and V p − is the volume ofthe compact cycle the probe brane wrap. The equations of motion associated with the8ariations of A ρ and A u are give by a ( u ) ∂ µ F µρ + ∂ u ( b ( u ) F uρ = 0 ∂ ρ F ρu = 0 (21)As was discussed above the geometrical realization of chiral symmetry implies that theprobe is of a form of a U shape with two branches. Thus the profile is a double valuedfunction of the radial coordinate, which generically is in the range ∞ ≥ u ≥ u . It isuseful to define a different coordinate z, ∞ ≥ z ≥ −∞ so that the boundary of onebranch of the probe brane say the left one, is at z = −∞ and the boundary of the rightone is at z = + ∞ . Expressed in terms of this coordinate the DBI action (19) takes theform S DBI = − κ p Z d xdz h ˆ a ( z ) F µν F µν + ˆ b ( z ) F ρz F ρz i (22)where ˆ a ( z ) = u ′ ( z ) a ( u ( z )) ˆ b ( z ) = b ( u ( z )) u ′ ( z ) (23)with u ′ ( z ) = dudz ( z ). It is clear that the corresponding equations of motion take thesame form as (21) with z replacing u and ˆ a and ˆ b replacing a and b . We continue ourdiscussion here using the u coordinates but obviously we can invert the analysis using a z coordinate.It is convenient at this point to choose the A u ( x, u ) = 0 gauge. The rest of thegauge fields A µ ( x, u ) are expanded in terms of normalizable non zero modes and non-normalizable zero modes. In addition we divide the gauge fields into vector fields V µ which are symmetric around u (or under z ↔ − z ) and axial vector fields A µ which areantisymmetric. Upon further Fourier transforming the space-time coordinate x µ → q µ the expanded fields take the form A µ ( q, u ) = V µ ( q ) ψ V ( u )+ A µ ( q ) ψ A ( u )+ X n =1 ( V nµ ( q ) ψ V n ( u )+ A nµ ( q ) ψ A n ( u )) (24)The normalizable modes are the bulk gauge fields while the non-normalizable are by thegauge/gravity dictionary sources for boundary currents. In fact as was shown in [4] thegauge transformation that sets A u = 0 requires that the zero modes include masslessmodes which are the Goldstone bosons associated with the spontaneous breakdown ofthe techniflavor chiral symmetry. These modes play obviously an important role in the9echnicolor mechanism since they will provide the mass of the electroweak gauge bosonsonce part of the techniflavor symmetry is gauged.In terms of this expansion the equations of motion (21) are1 a ( u ) ∂ u ( b ( u ) ∂ u ψ n ( u )) = − m n ψ n ( u )1 a ( u ) ∂ u ( b ( u ) ∂ u ψ ( q , u )) = − q ψ ( q , u ) (25)where we have used for the normalizable modes ∂ ρ ∂ ρ V nµ = q V nµ = m n V nµ and where wehave used ∂ ρ V nρ = 0 that follows from the equation of motion. These equations hold forboth the vector modes ψ V n as well as the axial vector modes ψ A n . Note that the eigenvalueproblem (25) becomes first order o.d.e for m n = 0 and so have only one solution whichis the odd one in accordance with the fact that the pions are pseudoscalars.Plugging the decomposition (24) into (19) we find S F = − κ p Z d qduT r (cid:18) a ( u ) (cid:20) X n =1 [ | F V nµν ( q ) | ψ V n ( u ) + | F Anµν ( q ) | ψ An ( u )]+ | F V µν ( q ) | ψ V ( u ) + | F A µν ( q ) | ψ A ( u )) + 2 F V µν ( q ) F µνV n ( − q ) ψ V ( u ) ψ V n ( u )+ 2 F A µν ( q ) F µνAn ( − q ) ψ A ( u ) ψ An ( u ) (cid:21) − b ( u ) (cid:20) | V µ ( q ) | ( ∂ u ψ V ) + | A µ ( q ) | ( ∂ u ψ A ) + X n =1 [ | V nµ ( q ) | ( ∂ u ψ V n ) + | A nµ ( q ) | ( ∂ u ψ An ) ] (cid:21)(cid:19) (26)To further reduce the action to four dimensions we have to normalize the ψ V n and ψ V n modes. This is done as follows Normalizing the gauge field as κ p Z dua ( u ) ψ V n ψ V m = δ nm (27)and the same for ψ An . Had we chosen to use the z coordinates the normalization conditionwould have same structure with ˆ a ( z ) replacing a ( u ). The resulting 4d YM action reads S F = − T r Z d q X n =1 (cid:18) | F V nµν ( q ) | + 14 | F Anµν ( q ) | − m V n | V nµ ( q ) | − m An | A nµ ( q ) | + 12 a V n F V µν ( q ) F µνV n ( q ) + 12 a An F A µν ( q ) F µνAn ( q ) (cid:19) + S source (28)where a V n = − κ p b ( u ) m V n ∂ u ψ V n | u = ∞ (29)10nd the same for a An expressed in terms of ψ An . We define S source to be the terms in(26) which involve only the source κ p Z d qdub ( u ) T r (cid:26) | V µ ( q ) | ( ∂ u ψ V ) + | A µ ( q ) | ( ∂ u ψ A ) + | F V µν ( q ) | ψ V ( u )+ | F A µν ( q ) | ψ A ( u ) (cid:27) (30)Performing an integration by parts in the first term we find κ p Z d qT r (cid:26) | V µ ( q ) | Z du∂ u [ ψ V b ( u ) ∂ u ψ V ] − ψ V ∂ u [ b ( u ) ∂ u ψ V ] (cid:27) = κ p Z d qT r | V µ ( q ) | (cid:26) [ ψ V b ( u ) ∂ u ψ V | u = ∞ u = u + q Z dua ( u )( ψ V ) (cid:27) (31)The last term cancels exactly the | F V µν ( q ) | term in (30) and there is a similar cancelationfor the axial gauge fields. Thus the leftover source term takes the form S source = − T r Z d q (cid:26) a V | V µ ( q ) | + a A | A µ ( q ) | (cid:27) (32)where a V = − κ p b ( u ) ∂ u ψ V ( u, q ) | u = ∞ (33) a A = − κ p b ( u ) ∂ u ψ A ( u, q ) | u = ∞ (34)where we have used the equations of motion and we have taken that ψ | u = ∞ = 1 for boththe vector and axial vector zero modes.The coupling between the source V ( A ) and the vector (axial) mesons fields can beread from (28) after the kinetic terms of the vector will be diagonalize, this is done bythe transformation ˜ V nµ = V nµ + a V n V µ ; ˜ A nµ = A nµ + a An A µ (35)Now the action in terms of the new fields is S ˜ F = − T r Z d x X n =1 (cid:18) | ˜ F V nµν ( q ) | − m V n ( ˜ V nµ − a V n V µ )+ 14 | ˜ F Anµν ( q ) | − m An ( ˜ A nµ − a An A µ ) (cid:19) + ˜ S source (36)where ˜ S source = − T r Z d q (cid:26) a V | V µ ( q ) | + a A | A µ ( q ) | (37)+ X n (cid:18) a V n | F Vµν ( q ) | + a An | F Aµν ( q ) | (cid:19)(cid:27) g V n = m V n a V n = − κ p b ( u ) ∂ u ψ V n | u = ∞ (38) g An = m An a An = − κ p b ( u ) ∂ u ψ An | u = ∞ (39)Now we have assembled all the ingredients to determine the value of the holographicS parameter. As discussed in the previous section, this can be done in two different ways.In the first method we need to compute holographically the two point functions of thevector and axial vector currents. Using the AdS/CFT dictionary this reads − Π V ( q ) ≡ hJ µV ( q ) J νV (0) i F.T = δδV ν δδV µ S DBI | V =0 = a V ( q ) (40)where V µ is the boundary value of the vector gauge field at u = ∞ . The same applies alsofor the axial vector correlator. Substituting (40) into (16) the holographic S-parameterreads S = − π (Π ′ V ( q ) − Π ′ A ( q )) | q =0 = − π ∂∂q ( a V ( q ) − a A ( q )) | q =0 = − πκ p " b ( u ) ∂∂q ( ∂ u ( ψ V ( u, q ) − ∂ u ψ A ( u, q )) q =0; u = ∞ (41)The second method is based on inserting the decay constants (38) into the expressionfor the S parameter as sum over resonance given in (42), yielding S = 4 π X n h ( a V n ) − ( a A n ) i = 4 π ( κ p ) b ( u ) X n h ( ∂ u ψ V n ) − ( ∂ u ψ A n ) i u = ∞ (42)Here we have used the gauge/gravity duality rules and derived the holographic formof the two expressions (refdefinition) and (42) that were shown in the boundary fieldtheory to be equivalent. In fact one can show directly in the gravity setup that the twoexpressions are equivalent. This was done in [3] for the sakai Sugimoto model but can bedone in a similar way for the general setup discussed in this section. The issue of whena partial sum of a small number of low lying states is a good approximation to the fullsum is discussed in [16].The determination of the S -parameter follows from the solutions of the equations ofmotion (25). The latter, as will be seen in the following sections, depend on the profile ofthe probe brane and in particular on the point with minimal value of the radial direction12 . This parameter relates to the “string endpoint mass” of the meson ( technimesonsin our case) which are defined as follows [17] m sep = 12 πα ′ Z u u Λ √− g tt g uu du (43)This mass is clearly not the current algebra or QCD mass, and in fact it is also not theconstituent mass of the meson. This mass can be thought of as m sep = ( M meson − T st L st )where T st is the string tension and L st is the length of the string. The fact that it isnot the QCD mass is easily determined from the fact that the pions associated witha probe brane profile with non trivial u are massless. Thus this mass parameter isnot related at all to the masses of particles running in the loops that determine the S parameter. Hence we should not expect the dependence of the S parameter to resemblethat of the dependence of the QCD masses. Indeed as will be seen in the sections belowthe dependence on m sep or on u will be different in the various models studied and norrelated the dependence on the QCD masses. The starting point of the hologrphic Technicolor Sakai Sugimoto model is Witten’s model[5]. The model describes the near extremal limit of N T C D x direction with anti periodic boundary condition for the fermions. Having inmind the use of the model as a hologrphic technicolor model, we use from the onset N T C and below N T F instead of N c and N f of the original model. In order to incorporatefundamental quarks in this model it was suggested in [4] to add to this background astack of N T F D N T F anti D N T F << N
T C thebackreaction of the flavor probe branes can be neglected as was shown to leading orderin N TF N TC in [ ? ]. The background which includes the metric the RR form and the dilaton isgiven by 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) (44) 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) where V denotes the volume of the unit sphere Ω and ǫ its corresponding volumeform. l s is the string length and g s is the corresponding string coupling. The techniflavorbranes are placed in such a way that the compactified x direction is transverse to themasymptotically. The manifold spanned by the coordinate u, x has the topology of a cigarwhere its tip is at the minimum value of u which is u = u Λ . The periodicity of this cycle13s uniquely determine to be δx = 2 πR = 4 π (cid:18) R D u Λ (cid:19) / = 2 πR (45)in order to avoid a conical singularity at the tip of the cigar. We also see that the typicalscale of the glueball masses computed from excitation around (44), is M gb = 1 R (46)The confining string tension in the model is given by[17] T st = 12 πℓ s √ g xx g tt | u = u Λ = 12 πℓ s (cid:18) u Λ R D (cid:19) / (47)Corresponding to u Λ one defines the following mass scale M Λ = 1 R = 32 u / R / D (48)Naively one could assume that at energies below M Λ , the dual gauge theory is effectivelyfour dimensional; however since the theory confines and develops a mass gap of order M gb ∼ M Λ there is no real separation in mass between the confined four dimensionalhadronic modes, like the glueballs and the Kaluza-Kleine excitations on the x circle.As discussers in [5] in the opposite limit where λ = g N c ≪ R one can see from loopcalculations that the scale of the mass gap is exponentially small compared to 1 /R hencethe theory does approach the 3 + 1 pure Yang-Mills theory at low energies. It is believedthat there is no phase transition when varying λ /R interpolating between the gravityregime to pure Yang-Mills. For convenience we will use from here on the freedom tore-scale the u coordinate and set u Λ = 1.The flavor probe brane are space filling in all the direction except on the cigar wherewe need to find their classical curve. In this case the problem is reduce to an o.d.e for x = x ( u ) that follows from the equation of motion associated with the DBI action ofthe D u → ∞ again transverse to the x direction. The solution of the equation of motion is found to be x ( u ) = Z uu duf ( u )( uR D ) / r f ( u ) u f ( u ) u − u is a constant of integration which determines the lowest value of u to which the D u = u generalizes the model of [4]. The interpretation of u , as the string endpointmass was discussed in section (3). Since the orientation of the D u it is actually a ¯ D D − ¯ D N T F D u → ∞ at x = x ( L )4 and N T F ¯ D x = x ( R )4 and finds thatdue to the classical equations of motion they join together at u = u . In the model of[4] x ( L )4 = 0 and x ( R )4 = π . We see that the global U L ( N T F ) × U R ( N T F ) chiral symmetryof the theory is spontaneously broken by the ground state down to U V ( N T F ). So, wegot a gravity model whose dual gauge theory admits at low energies confinement andchiral symmetry is spontaneously broken. These two qualities are of great importance inQCD phenomenology and also in building technicolor models. An HTC model means weidentify the gauge group as the technicolor SU ( N T C ) and the quarks are techniquarksand the vector field fluctuation of the D S F = − T (2 πα ′ ) V S R / D u / g s u / Z d xdu (cid:26)(cid:18) γu (cid:19) / F µν + 2 u / R D γ / F µu (cid:27) (50)Using the general discussion of section (3), the model is characterized by κ = T (2 πα ′ ) V S R / D u / g s = g N π a ( u ) = ( γu ) b ( u ) = u / R D γ / (51)where γ = u f ( u ) u − f ( u ) u (52)Solving numerically equations (25) for the present case and plugging the results into (16)we reproduced the results of [3]. For the anti-podal configuration ( u = u Λ = 1) S = 19 . κ D = 19 . g N π = 0 . λ T C N T C (53)where λ = g N T C . The authors of [3], used the values N T C = 4 and λ T C = 4 π to comparethe holographic computation with the results of [1]. This kind of comparison has to betaken with a grain of salt since the holographic result is valid in the limit of large λ T C and large N T C . 15or the general non anti-podal configurations we find that S is growing linearly with u . Another useful results could be obtained from Weinberg sum ruleΠ A (0) = F π = (246GeV) (54)where we assigned the thechni-pion decay constant the value of the electroweak scale inorder to reproduce the spectrum of the electroweak gauge bosons. Using (34) this gives F π = a A (0) = − κ D R − D u / γ − / ∂ u ψ A ( u ) | u = ∞ (55)Numerical integration of (25) gives F π = a A (0) = 0 . κ D M KK = 0 . M (56)where like in [3] we choused to use the values λ T C = 4 π and N T C = 4. This set theKaluza-Klein mass scale to M Λ = 1 . m ρ ≈ . m a ≈ . u which hasno 4 d interpretation.As we pointed out in the previous section, there is another way to to estimate the S-parameter by using the sum over hadronic resonance given in (42). Summing up to n = 8we find S = − κ D ≈ − . λ T C N T C (57)Thus the contribution from the eight lowest states is negative and very far from the resultfound above. This is in accordance with the statement made in section based on [16],that the higher KK modes do not decouple from the spectrum on this background andthat S n for some finite n does not produce a good approximation for S .Next we want to study the dependence of the S-parameter on u . Using (41) fordifferent values of u we found numerically that S tends to grow linearly with u . Wewill see this behavior in a more qualitative manner for large u by using scaling argumenton either one of (41) or (42). We will show how to apply this argument on the schemegiven in (42) but we note that it could be applied easily the same to (41). We start bychanging variable in (25) to y = uu and then we take the limit u >>
1, in this limit wefind that γ → ˜ γ ( y ) = y y − S F = − κ (cid:18) u u Λ (cid:19) / Z d xdy (cid:26)(cid:18) ˜ γy (cid:19) / F µν + 2 u R D y / ˜ γ / F µy (cid:27) (59)and (25) becomes y / ˜ γ ( y ) − / ∂ y ( y / ˜ γ ( y ) − / ∂ y ˜ ψ n ( y )) = − m n R D u ˜ ψ n ( y ) (60)Since the left hand side of (60) is independent of u , then so is the right one, and we find m V n/An ∼ u R D (61)Now doing the same manipulation on (38) we get g V n = − u / κ D R − y / ˜ γ − / ∂ y ψ V n ( y ) | y = ∞ (62)we might conclude that g V n/An ∼ u / R D (63)but since the normalization of the modes (27), is now taken to be: κ (cid:18) u u Λ (cid:19) / Z dy (cid:18) ˜ γy (cid:19) / ˜ ψ n ( y ) ˜ ψ m ( y ) = δ mn (64)we need to take into account that ψ ( u ) = u − / ˜ ψ ( y ) (65)Combining (62) and (65) we find g V n/An ∼ u / R D (66)Now plugging (61) and (66) into (42) we find S n ∼ u / (67)17 The AHJK model- the uncompactified Sakai Sug-imoto model
In the Sakai Sugimoto model the spontaneous breaking of the techniflavor symmetry isattributed to the U-shape configuration of the D − ¯ D u, x ) directions. It turnsout that this is a sufficient condition for having a U-shape form but it is not a necessarycondition. That is to say that there is a U shape solution even if x is not compactifiedat all. Decompactifying the x direction is achieved technically by simply substitutingone instead of f ( u ) in (118). This model was studied in [6], and the profile of the probebranes was found to be given by (49) only with f ( u ) = 1 and that the integral could bebrought to the closed form x ( u ) = 18 (cid:18) R u (cid:19) / [ B ( 916 ,
12 ) − B ( u u ; 916 ,
12 )] (68)where B ( p, q ) and B ( x, p, q ) are the complete and in complete beta functions. Theasymptotic separation between the D D L is given by L = 14 (cid:18) R u (cid:19) / B ( 916 ,
12 ) (69)In terms of the dual field theory the model is in fact physically very different fromthe Sakai Sugimoto model. It is a gravity model dual to a non-confining gauge theory.Recall that the holographic expression of the string tension is given by (47) evaluated atthe minimum value of u [17] which for the present case is u = 0 and hence the stringtension vanishes.The effective five dimensional flavored gauge theory for the present case is identicalto that of the Sakai Sugimoto model, namely, the characterization given in (51) appliesalso for the uncompactified model with the difference that now the function γ is given by γ = u u − u (70)The action of the gauge fields could be rescaled into the form S F = − T (2 πα ′ ) V S R / D u / g s Z d xdy (cid:26)(cid:18) γy (cid:19) / F µν + 2 u R D y / γ / F µy (cid:27) (71)Unlike the situation in the compactified case of the previous section, in the AHJKmodel there is no compactification scale below which the theory becomes effectively4d, so the definition of the ’t Hooft coupling of the 4d gauge theory is not so clear.18evertheless, there are two scales in the problem that could be used to construct the 4d’t Hooft coupling; L and ℓ s . In the first case we set λ = λ T C = λ L = g s N T C ℓ s L (72)where λ is the ’t Hooft coupling of the five dimensional gauge theory. The asymptoticdistance between the D D L = ≈ . R / D u / (73)and we find: κ = T (2 πα ′ ) V S R / D u / g s = 0 . T (2 πα ′ ) V S R D g s L = 0 . ℓ s g s N T C (2 π ) L = 0 . λ T C N c (2 π ) (74)Solving numerically the e.q.m of the non-normalizabe mode given in (25), and substitut-ing it into (41) we find that the S-parameter is given by S AJHK ≈ . κ = . λ T C N T C (75)and we note that the dependence on u /L is now hidden inside the definition of λ T C .Using ℓ s instead, the dimensionless ratio is λ /ℓ s and we find˜ λ T C = λ /ℓ s = g s N T C (76)so κ = g / s N / c √ π π ) (cid:18) u ℓ s (cid:19) / = √ π ˜ λ / T C N T C π ) (cid:18) u ℓ s (cid:19) / = 0 . (cid:18) u ℓ s (cid:19) / ˜ λ / T C N T C (77)And we find that the S-parameter is given by (for u ℓ s = 1) S AJHK = . λ / T C N T C (78)
AdS model In the model discussed in the previous section and in all the models we will encounter inthe following sections the flavor branes were wrapping certain non-trivial cycles on top ofspanning the Minkovski space and the radial direction. In fact the wrapped dimensionshave not played any role in the techincolor scenario and in particular in the determination Note that the gravity description is only valid for λ >> L .
19f the S parameter. This naturally calls for models without the wrapped cycle and ingeneral with as less as possible extra dimensions. Models of this kind are the non-criticalgravitational models. Such a model that may serve as a non-critical dual of QCD wasproposed in [9]. This model is based on a non-critical SUGRA background presented in[8],[7] which can be viewed as the backreaction of N T C coincident D N T F 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 (79) F (6) = Q c (cid:18) uR AdS (cid:19) dx ∧ dx ∧ dx ∧ dx ∧ du ∧ dx (80) e φ = 2 √ Q c ; f = 1 − (cid:18) u Λ u (cid:19) ; R AdS = 152 ℓ s (81)where Q c = N TC π with N T C being the number of D x is taken to be periodicwhere to avoid conical singularity its periodicity is set to x ∼ x + δx ; δx = 4 πR AdS u Λ (82)We also define M Λ as a typical scale below which the theory is effectively four dimensional: M Λ = 2 πδx = 5 u Λ R AdS = u Λ R AdS is a constantindependent of g s N T C , hence the curvature is always of order one and there is no way togo to a small curvature regime.Into this background a set of N T F pairs of D − ¯ D x , ...x , x ). In the corresponding brane configuration, one cannot separatethe color and flavor branes, namely the strings that connect the two types of branes arenecessarily of zero length, hence it is dual to a field theory system with chiral symmetry.The profile of the probe branes is determined by solving the equations of motion thatfollow from their DBI action. Unlike the critical case, here there is a priori an additionalCS term on top of the DBI action (18) of the form S CS = T R P ( C (5) ). However, forreasons given in [18] including the CS term yields unphysical results, hence from here on20e shall set the CS term to zero. Thus using only the DBI action the profile of the probebranes is found to be x ,cl ( u ) = Z uu ( u f / ( u )) du ′ ( u ′ R AdS ) f ( u ′ ) q u ′ f ( u ′ ) − u f ( u ) (84)Again for convenience we define γ = u u f ( u ) − u f ( u ) (85)We also rescale u to set u Λ = 1.In terms of the general discussion of section (3), the model is characterized by κ nc = T ( nc )4 (2 πα ′ ) e − φ R AdS = s
52 3 N T C π ∼ . N T C a ( u ) = γ b ( u ) = R − AdS u γ / (86)Once we obtained the solution for the non-normalizable mode by numerical integrationof (25) we plug it into the holographic definition of S (41), and got an estimation of S,for the antipodal configuration u = u Λ = 1: S = 10 . κ nc = 0 . N T C (87)The dependence of S on u for u > u Λ = 1 is drawn in figure (1). It is obvious from thefigure that at large u S is a constant independent of u . The asymptotic value it takesis Sκ nc ≃ .
54. This behavior will be derived below also qualitatively.Next we would like to compute the S parameter using the sum rule formula of (42). Tocompute S , the sum over first eight resonance, we need on top of the low lying massesalso the corresponding decay constants. These are determined by solving numerically(38). Substituting the values of the masses and of the decay constants into (42) andsumming up to n = 8 we find S = 8 . κ nc = 0 . N T C (88)According to [16] it was anticipated that the higher KK modes will decouple from thespectrum and that S n for some finite n will produce a good approximation for S .21 masses of first four vector mesonsmasses of first four axial vector mesons u n Non-critical model: In blue, S vs. u . In green , vector mesons (even modes) m n vs. u .In red , axial vector mesons (odd modes) m n vs. u . Figure 1: S in the non critical model vs. u . The linearity of the vector (axial) mesonsmasses in u could be seen from their doted green (red) plots for the first 8 modes( u Λ = 1). 22or the general case u > u Λ = 1, the S-parameter seems to be almost independentof u as could be seen in figure (1).In order to see the S-parameter dependence on u in a more qualitative manner werepeat the scaling argument we used in section (4). By changing to the dimensionlessvariable y = uu we find that after taking the limit u ≫ γ − / ∂ y ( y ˜ γ − / ∂ y ˜ ψ n ( y )) = − m n R AdS u ˜ ψ n ( y ) (89) − κ nc ( R AdS ) − y ˜ γ − / ∂ y ˜ ψ V n | y = ∞ = g V n u (90)where we noted that in this limit γ → ˜ γ ( y ) = y /u y − u , so the right hand isindependent of it as well, and we find m V n/An ∼ u R AdS (92) g V n/An ∼ u R AdS (93)Plugging these into (42) we see that indeed the S-parameter is independent of u in thelimit u >> M Λ . As before taking the techinipiondecay constant to equal the electro-weak scale, we find using the numerical integration(246GeV) = Π A (0) = F π = − κ nc R − AdS u γ − / ∂ u ψ A ( u, | u = ∞ = 0 . κ nc M (94)and the corresponding mass scale is M = (246GeV) . N T C (95)For N T C = 4 the scale is found to be 2 . m ρ T = 1 . m a T = 2 . The KMMW model with D and anti- D flavorbranes As was emphasized in the last two sections, to have a chiral flavor symmetry of the form U L ( N T F ) × U R ( N T F ) one has to place a set of N T F probe branes and anti-branes in sucha way that the strings that stretch between them and between the original technicolorbranes that constitute the background cannot have a non-trivial length. That required D D D D U ( N T F ) × U ( N T F ) is spontaneously broken toa diagonal symmetry U D ( N T F ). The dual of such a field theory can be realized by placinga stack of D D D /D − ¯ D D − ¯ D D − ¯ D
8. The profile of the probe branes is determined by solving theequations of motion for the three coordinates transverse to the branes. This was done byKruczenski et al in [10]. The D S inside the S and curve along the cigar spanned by ( u, x ) coordinates. Obviouslyall the parameters of the background are those of [5] as was described in section (4). Onthe other hand, the induced metric is now ds = (cid:18) uR D (cid:19) / [ − dt + δ ij dx i dx j ]+ R / D u / d Ω + (cid:20)(cid:18) uR D (cid:19) / f ( u )( ∂ u x ) + (cid:18) R D u (cid:19) / f ( u ) (cid:21) du (97)where we still sets u Λ = 1. The curve of the D − ¯ D u, x ) is found via the DBI action to be x ( u ) = u / f ( u ) / Z uu dx x / f ( x ) q x f ( x ) − u f ( u ) (98)In the terminology of section (3) this model is characterized by κ nc = − T (2 πα ′ ) V R D g s = N T C (2 π ) ≈ . N T C a ( u ) = γ u b ( u ) = u R D γ / (99)where γ ( u ) = u u f ( u ) − u f ( u ) (100)24ntegrating Numerically the equations of motion for the non-normalizable mode wefind that for the antipodal configuration u = u Λ = 1 S = 4 π ddq ( a V ( q ) − a A ( q )) | q =0 ≈ . κ = 20 . × . N T C = . N T C (101)For the general case u > u Λ = 1, we find a slow decrease of S towards the asymptoticalvalue S ≃ .
49 so S virtually independent of u .Now, we want to estimate the S-parameter using the sum over hadronic resonancegiven in (42) and see its agreement with (101). This requires the values of the decayconstants of each of the vector and axial-vectoes mesons, and these are given as in ( ?? )by: g V n = m V n a V n = − κ R − u γ − / ∂ u ψ V n | u = ∞ (102) g An = m An a An = − κ R − u γ − / ∂ u ψ An | u = ∞ (103)We plugged this into (42) and summed up to n = 8 and found S = 8 . κ = . N T C (104)We see that as in the Sakai-Sugimoto model, the higher KK modes doesn’t decouple fromthe spectrum and S n for some finite n doesn’t produce a good approximation for S .We repeat the scaling argument to determine qualitatively the dependance of theS-parameter on u we. Changing to the dimensionless variable y = uu in eq. (25) and(102), then in the limit u >> y ˜ γ − / ∂ y ( y ˜ γ − / ∂ y ˜ ψ n ( y )) = − m n R u ˜ ψ n ( y ) (105) − κ R − u ˜ γ − / ∂ u ψ V n | u = ∞ = g V n u (106)where we denoted γ → ˜ γ ( y ) = y y − u , so the right hand isindependent of it as well, and we find m V n/An ∼ u R (108)25 V n/An ∼ u R (109)A brief look at (42) tells as that at the limit u ≫
1, the S-parameter will exhibitindependency of u .As in the previous cases we determine the compactification scale of the model by equatingthe technipion decay constant to the electro-weak scale. Using numerical integration wefind Π A (0) = F π = − κ ( R ) − u γ − / ∂ u ψ A ( u ) | u = ∞ = 0 . M κ = (246GeV) (110)For N T C this gives M KK = 1 . m ρ = 0 . m a = 1 . D branes compactified on two circles with D − ¯ D flavor branes Another interesting model in the context of HQCD is given by the near horizon limitof the non-extremal background of N c D N T F D − ¯ D D D SU ( N c ) gauge group in doublets of SU ( N T F ). The fields in thisbackground are given by ds = uR ( η µν dx µ dx ν + dx + f ( u, u Λ ) dx ) + Ru du f ( u, u Λ ) + Rud Ω (111)where f ( u, u Λ ) = (cid:18) − u u (cid:19) ; R = g s N c α ′ (112)and the dilaton and 3 form field strength are given byexp( φ ) = g s uR ; F = 2 R g s Ω (113)In this model x and x are compact x = x + 2 πR x ; x = x + 2 πR x (114)where in order to avoid conical singularity we must set R x = R . We choose the D − ¯ D M and S directions and26urves on the ( u, x , x ) space. We choose to parameterize the curve by the u coor-dinate ( x ( u ) , x ( u ) , u ) where the functions x ( u ) and x ( u ) will be determine by theminimization of the DBI action S D = T V V g s Z duu vuut ( ∂ u x ) + f ( u, u Λ )( ∂ u x ) + R u f ( u, u Λ ) (115)and we find (from here on we will set u Λ = 1) x ( u ) = P R Z uu du ′ s u ′ f ( u ′ ) (cid:18) u ′ − P − P f ( u ′ ) (cid:19) (116) x ( u ) = P R Z uu du ′ s u ′ f ( u ′ ) (cid:18) u ′ − P − P f ( u ′ ) (cid:19) (117)For detailed description of the solutions as a function of the integration constants P and P see [ ? ]. The induced metric is therefore ds = (cid:18) uR (cid:19)(cid:20) η µν dx µ dx ν + (cid:18) ( ∂ u x ( u )) + f ( u, u Λ )( ∂ u x ( u )) + R u f ( u, u Λ ) (cid:19) du + R d Ω (cid:21) = (cid:18) uR (cid:19)(cid:20) η µν dx µ dx ν + R u γ ( u ) du + R d Ω (cid:21) (118)where we defined γ ( u ) ≡ u f ( u ) (cid:18) u − P − P f ( u ) (cid:19) (119)The DBI action for the gauge fields on the D S F = − R (2 πα ′ ) T Ω g s T r Z d xdu (cid:18) γ / ( u ) F µν F µν + 2 u R γ / ( u ) F µu F µu (cid:19) (120)Using mode decomposition as in (24), the equation of motion for the ψ n are γ − / ∂ u ( u γ − / ∂ u ψ n ( u )) = − m n R ψ n ( u ) (121)and for the non-normalizable part it is γ − / ∂ u ( u γ − / ∂ u ψ ( u, q )) = − q R ψ ( u, q ) (122)27or the antipodal case P = P = 0 , u = 1, (121) and (122) will simplify to f / ∂ u ( u f / ∂ u ψ n ( u )) = − m n R ψ n ( u ) (123)and f / ∂ u ( u f / ∂ u ψ n ( u, q )) = − q R ψ n ( u, q ) (124)After the change of variable u = 1 + z , we find ∂ z ((1 + z ) ∂ z ψ n ( z )) = − m n R ψ n ( z ) (125)and ∂ z ((1 + z ) ∂ z ψ ( q , z )) = − q R ψ ( q , z ) (126)One can transform (125) into a standard Schr¨odinger form via the transformation z = sinh ( x ) ; ψ n ( x ) = 1 q cosh ( x ) Ψ n ( x ) (127)and find (cid:18) ∂ x −
14 (1 + 1 cosh ( x ) ) (cid:19) ψ n = − m n R ψ n (128)This eigenvalue problem has only two normalizable solutions and so the sum in (42) runsonly on these two modes.Substituting (24) into (120) we get 4 d YM action for the gauge fields with the kineticterms canonically normalized provided the SU ( N ) generator obey T r [ T a T b ] = δ ab andthe ψ n are normalized as R (2 πα ′ ) T Ω g s Z duψ V n ψ V m γ / ( u ) = δ nm (129)2 R ( πα ′ ) T Ω g s Z duψ An ψ Am γ / ( u ) = δ nm (130)According to our prescription in section (3) the boundary terms , (29),(33) will be givenby a V n = − κ ( m n R ) − u γ − / ∂ u ψ V n | u = ∞ (131) a An = − κ ( m n R ) − u γ − / ∂ u ψ An | u = ∞ (132)28nd a V = − κ R − u γ − / ∂ u ψ V ( u, q ) | u = ∞ (133) a A = − κ R − u γ − / ∂ u ψ A ( u, q ) | u = ∞ (134)Where we defined κ = u Λ R (2 πα ′ ) κ Ω g s (135)The correlators of the vector and axial-vector currents are given by the AdS/CFT pre-scription (40), and we findΠ V ( q ) ≡ hJ µV ( q ) J νV (0) i F.T = − a V ( q ) (136)and Π A ( q ) ≡ hJ µA ( q ) J νA (0) i F.T = − a A ( q ) (137) S = − π ddq (Π V − Π A ) | q =0 = − π ddq ( a V ( q ) − a A ( q )) | q =0 (138)For the antipodal configuration ( u = u Λ = 1) (133),(134) becomes a = − κR − u f / ∂ u ψ ( u, q ) | u = ∞ = − κ ( R ) − u z f / ∂ z ψ ( u, q ) | z = ∞ (139)using uf / = z we find a = − κR − (1 + z ) ∂ z ψ ( u, q ) | z = ∞ (140)The asymptotic behavior of ψ V ( u, q ) could be read from (126) by expanding it in powersof z − . Keeping only the leading order term (126) becomes ∂ z ( z ∂ z ψ n ) = − q ψ ( q , z ) (141)This has the form of an Euler equation and can be solved using ψ ( q , z ) = z α q whichleads to an equation for α q α q ( α q −
1) + 2 α q + q = α q + α q + q = 0 (142)with the roots α q = − ± √ − q − ± ∓ q (143)So we have two asymptotic behavior α − q = − q and α + q = − q . One correspond tothe even mode and one to the odd mode. Plugging these solution into (138) would leadto a diverging S-parameter! 29 The conifold model with D − ¯ D flavor branes So far we have discussed the S parameter of holographic technicolor models that are basedon gravity backgrounds with a cigar like structure of the sub-manifold that includes acoordinate compactified on an S circle and the radial direction. This structure of thebackground ensures the confining nature of the dual gauge theory and the U shapesolutions for the probe brane profile implies the spontaneous breaking of its flavor chiralsymmetry. Recently, another type of a holographic model that admit these two featuresand which is based on the conifold geometry was proposed in [12] and [13]. In this sectionwe show that this model also fits the general framework discussed in section (3) and wedetermine the S parameter of the holographic techincolor scenario based on this model.In fact we can discuss two such models. One based on the conifold geometry which isa conformal model that does not admit confinement [12] and a one which relates to thedeformed conifold [13] which is a confining model. To simplify the analysis we discusshere the former but a similar type of calculation can be done also to the latter. Thus weconsider here the conifold background. The flavor probe brane is taken to be a D D x µ , the radial direction u and thethree-sphere parameterized by the forms f i (or alternatively w i ). The transversal spaceis given by the two-sphere coordinates θ and φ . The classical profile depend only on theradial coordinate.The 10 d metric is: ds = u R AdS dx µ dx µ + R AdS u ds (144)with the 6 d metric given by ds = dr + r (cid:18)
14 ( f + f ) + 13 f + ( dθ − f ) + (sin( θ ) dφ − f ) (cid:19) (145)and the AdS radius is R AdS = πg s N T C ℓ s . Because the background has no fluxes exceptfor the C form the Chern-Simons terms do not contribute and the action consists onlyof the DBI part S DBI ∝ Z duu u (cid:16) θ u + sin θφ u (cid:17)! / . (146)Here the subscript u stands for the derivatives with respect to u . Setting θ = π/ √ φ ( u ) ! = (cid:18) u u (cid:19) . (147)There are two branches of solutions for φ in (147) with φ ∈ [ − π/ ,
0] or φ ∈ [0 , π/ u = 0 we have two fixed ( u -independent) solutions at φ − = − √ π and φ + = √ π .30he induced 8 d metric in this case is that of AdS × S as one can verify by plugging dφ = dθ = 0 into (145). For non-zero u the radial coordinate extends from u = u (for φ = 0) to infinity (where φ ( u ) approaches one of the asymptotic values φ ± ). The inducedmetric has no AdS × S structure anymore. Notice that unlike the case of the SakaiSugimoto model, here the D θ, φ ) two-sphere. This is due to the the conical singularity at the tip, so the S does notshrink smoothly.It is convenient to define a new dimensionless radial coordinate z = uR AdS s (1 − u u ) (148)so that the D7 probe brane stretches along positive z and the anti-brane along negative z . In terms of these the gauge fields action is given by S KW = κ T r Z d xdz (cid:20) F µν q z + u R AdS ) + 16 (cid:18) z + ( u R AdS ) (cid:19) / F µz (cid:21) (149)This form of the action translate into the following parameters in the framework forcomputing the S parameter are κ = 0 . N T C ˆ a ( z ) = 1 q z + u ˆ b ( z ) = 16 (cid:18) z + ( u R AdS ) (cid:19) / (150)Repeating the procedure of determining the S parameter using (41) we find S = 0 . N T C (151)and the result using the sum-rule (42) is S = 0 . N T C (152)For comparison we substitute N T C = 4 to yield S = 0 .
17 and S = 0 . M Λ is givenby M = (246GeV) . N T C (153)so that for N T C = 4 we get M Λ = 1 . m ρ = 3 . TeV , m a = 5 . TeV.31
In this paper we have examined a variety of technicolor models through their holographicduals. We have focused mainly on the S-parameter of these models. For that purpose wepresented the method used in [3] to deduce the holographic S-parameter and showed howto apply the technique to general (suitable) background and then applied it on severalmodels. Generically Technicolor models admit a confinement behavior and spontaneousflavor chiral symmetry breaking. Indeed some of the models we have chosen, the SakaiSugimoto model, the non-critical model and the model based on D D U ( N T F ) × U ( N T F ) → U V ( N T F ). However, it isnot a symmetry of chiral fermions but rather a symmetry of Dirac fermions. From thepoint of view of the S-parameters there is not much difference between the models thatadmits both confinement and spontaneous flavor chiral symmetry breaking to the othermodels.The direct estimation of the Peskin-Takeuchi S-parameter for a strongly interactingsector is still a grave problem in technicolor model-building. But as was shown in [3]for the Sakai-Sugimoto model, and also in the present paper a reliable estimate for theS-parameter is with in reach if the field theory has a gravity dual. Strictly speaking thelatter applies only for large N T C and large λ T C .The results of the S-parameter and the low lying technivector mesons is summarizedin table 1.In general the S-parameter is a function of all the free parameters of the theory N T C , λ
T C , N
T F and u or instead the “string endpoint” masses defined in (43). As forthe dependence on N T C and λ T C there is a striking difference between the Sakai-Sugimotomodel both the compactified and the uncompactified and the rest of the models. Whereasin the former models S depends linearly on the product of N T C λ T C , in the latter modelsit does not depend on λ T C but rather it is linear only in N T C . The dependence of theS-parameter on u in some of the models is drawn in figure 2.We can see from this figure that while the S-parameter in Sakai-Sugimoto model (andalso its uncompactified cousin) grows linearly with u , the D − D AdS + D u parameteris related to the string endpoint mass which is given roughly by M m − T st L where M m isthe mass of the corresponding meson, T st is the string tension and L is the length of32 on-critical D4-D6 D4-D8U_010 20 30 40S/k 010203040
The S-parameter of the D4-D8, D4-D6 and the non-critical models verses U_0.
Figure 2: S-parameter of the
AdS + D D − D
8, and D − D u ( u Λ = 1). 33 ( u /u Λ = 1) S ( u /u Λ = 1) m ρ m a D − D . λ T C N T C − . λ T C N T C . √ λ TC N TC TeV . √ λ TC N TC TeV D − D . λ T C N T C - - - D − D . N T C . N T C . / N / T C . / N / T C
AdS + D . N T C . N T C . / N / T C / N / T C D − D D
7) 0 . N T C . N T C . TeV / N / T C . TeV / N / T C D − D ∞ - − − Table 1: The S-parameter of the six models, N T F = 2, S is given by the AdS/CFTdictionary (41), S is the sum over the first eight modes in (42).the stringy meson. This mass parameter has nothing to do with the current algebra or“QCD” mass and hence one cannot compare it to the dependence on the mass found in[1] at the weak coupling regime.The dependence on N T F is more tricky. If one naively embed the U (2) ∈ U ( N T F insuch a way that the generator of SU (2) for instance T is just one and -minus one in theupper terms along the diagonal, then there is no dependence of the S parameter on N T F since it relates to the electroweak currents that are affected only by the upper 2 × N T F × N T F matrices. However, if we generalize the models discussed in the paperwith only a single factor of SU ( N T F = 2), to a set of N TF of such group factors, thisshould yield an S parameter which is N TF times bigger than the one of a single groupfactor. The holographic realization of such a scenario is by taking N TF pairs of U shapeflavor probe brane and distribute them along the radial direction, namely assign to eachof them a different u . In the non-critical, KMMW and KW+ D u and hence a summation over all the pairs of U-shape flavor branes isdefinitely justified. However for the Skai Sugimoto model and his uncompactified cousin,the S-parameter depends linearly on u and thus a naive summation is incorrect. Onecan of course introduce very small differences in the values of the u associated with eachpair and in this way the summation result will be a reasonable approximation.We demonstrate these results for the AdS + D S = 10 . κ nc = 10 . s
52 3 N T F N T C π = 0 . N T F N T C (154)and in the D − D S = 20 . κ = 20 . N T F N T C π ) = . N T F N T C (155)Of course holography is not the only way to estimate the S-parameter, in [1] a fewphenomenological formula where suggested in order to estimate the S-parameter of tech-nicolor models with QCD like dynamics. The starting point of their formulas is to useeq. (42) with the masses and decay constants of the techni-hadrons given by assumingthe large- N rescaling relations between these to their QCD counterparts. They foundby summing over the first two hadronic resonance ρ T C and a T c , that for a model with SU ( N T C ) technicolor gauge group and N T F SU (2) doublets S is given by S ≈ . N T F N T C S ,AdS ≈ . N T F N T C N scaling relations m ρ T ≈ N T C N T F F π m ρ f π ; m a T ≈ N T C N T F F π m a f π (158)using the data m ρ ≈ m a ≈ F π ≈ (246GeV) ; f π ≈ (92GeV) (159)we find for N T C = 4 and N T F = 2 m ρ T ≈ . m a T ≈ . M Λ = 2 . m ρ T ≈ . M Λ = 1 . m a T ≈ . M Λ = 2 . cknowledgements It is a pleasure to thank Ofer Aharony for very useful conversations and for his commentson the manuscript. We are also grateful to Stanislav Kuperstein for fruitful discussions.This work was supported in part by a centre of excellence supported by the Israel ScienceFoundation (grant number 1468/06), by a grant (DIP H52) of the German Israel ProjectCooperation, by a BSF grant, by the European Network MRTN-CT-2004-512194 and byEuropean Union Excellence Grant MEXT-CT-2003-509661.
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