Nonlinear Realization of Chiral Symmetry Breaking in Holographic Soft Wall Models

PHYSICAL REVIEW D 102, 026013 (2020)

Nonlinear realization of chiral symmetry breaking in holographic soft wall models

† Alfonso Ballon-Bayona1,* and Luis A. H. Mamani2, 1Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro 21941-972, Brazil 2Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Rua Santa Ad´elia 166, 09210-170 Santo Andr´e, São Paulo, Brazil

(Received 16 February 2020; accepted 22 June 2020; published 13 July 2020)

We investigate nonlinear extensions of the holographic soft wall model proposed by Karch et al. [Phys. Rev. D 74, 015005 (2006)] with a positive quadratic dilaton. We consider a Higgs potential for the tachyonic field that brings a more natural realization of chiral symmetry breaking in the infrared regime. Utilizing the AdS=CFT dictionary and holographic renormalization, we find the chiral condensate as a function of the quark mass. The nonlinearity of the Higgs potential leads to a nonlinear relation between the chiral condensate and the quark mass. Solving the effective Schrödinger equations for the field perturbations, we estimate meson masses and decay constants and evaluate their dependence on the quark mass. In the axial and pseudoscalar sector we find an interesting behavior for the decay constants as the quark mass increases. We also investigate the effect of a five-dimensional running mass for the tachyonic field. We conclude that nonlinear soft wall models with a Higgs potential for the tachyon and a positive quadratic dilaton do not provide spontaneous symmetry breaking in the chiral limit.

DOI: 10.1103/PhysRevD.102.026013

I. INTRODUCTION symmetry breaking as well as quantitative predictions for hadronic phenomenology. More recently, a comple- The dynamical breaking of chiral symmetry in quantum mentary approach to nonperturbative QCD has been chromodynamics (QCD) is a fascinating phenomenon that developed. The so-called holographic QCD approach occurs in the strongly coupled (nonperturbative) regime. builds five-dimensional (5D) gravitational models dual to Chiral symmetry breaking is the mechanism responsible for four-dimensional (4D) nonperturbative quantum field dynamical generation of quark masses at low energies, and theories similar to QCD. This approach is based on the it is crucial for the description of mesons and baryons. The AdS=CFT correspondence and the gauge/gravity duality. well established order parameter associated with chiral One of the main achievements of the holographic QCD symmetry breaking is the so-called chiral condensate qq¯ , h i approach is the discovery of a universal criterion for which is the vacuum expectation value (VEV) of the quark confinement [2] based on the holographic map between mass operator. A nonzero chiral condensate signalizes 5D classical strings and 4D Wilson loops [3]. The problem chiral symmetry breaking, and it can be generated by a of chiral symmetry breaking is also fascinating in holo- nontrivial QCD vacuum. The phenomenon of chiral sym- graphic QCD. Since the chiral symmetry group UðN Þ × metry breaking is closely related to the phenomenon of f L UðN Þ is a 4D global symmetry group, it maps to 5D quark confinement [1]. f R (local) gauge symmetry group. This implies that 4D chiral The traditional approaches used to investigate the non- symmetry breaking corresponds to 5D (non-Abelian) perturbative regime of QCD are lattice QCD and the gauge symmetry breaking. A minimal 5D holographic Schwinger-Dyson equations. They have provided plenty QCD model for chiral symmetry breaking was proposed of insights on the mechanism for dynamical chiral in [4,5] and based on the so-called hard wall model [6].In that approach the breaking of the 5D non-Abelian gauge *[email protected] symmetry is induced by a 5D scalar field X that is dual to † [email protected] the quark mass operator qq¯ . The AdS=CFT dictionary maps 2 ¯ the 5D squared mass mX to the conformal dimension of qq Published by the American Physical Society under the terms of 2 −3 the Creative Commons Attribution 4.0 International license. implying a tachyonic mass mX ¼ . Further distribution of this work must maintain attribution to Although the model in [4,5] successfully introduces the the author(s) and the published article’s title, journal citation, quark mass and chiral condensate, it does not provide a and DOI. Funded by SCOAP3. dynamical mechanism for chiral symmetry in the sense that

2470-0010=2020=102(2)=026013(29) 026013-1 Published by the American Physical Society ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020) the chiral condensate is fixed by boundary conditions in the coupling λ < 0 never provides a minimum, whereas the infrared (IR) regime. An alternative approach was proposed positive case λ > 0 corresponds to the original Mexican hat in [7] and is known as the soft wall model. That approach potential leading to a minimum. Our results for the meson was physically motivated by experimental data on the meson spectrum and decay constants support our main conclu- spectrum indicating a set of linear Regge trajectories. In sions and also bring some pleasant surprises. In particular, holographic QCD meson masses arise as eigenvalues of we find an interesting behavior for the meson decay effective Schrödinger potentials for 5D field perturbations. constants in the axial sector as functions of the quark The soft wall model introduces a smooth cutoff in the form of mass. Our approach was partially inspired by the holo- a background scalar field ΦðzÞ, known as the dilaton. As graphic QCD model of [10], where chiral symmetry shown in [7], a positive quadratic dilaton ΦðzÞ ∼ z2 in the IR breaking maps to tachyon condensation in string theory. leads to approximate linear Regge trajectories for the Our results agree qualitatively with [10] in the regime of mesons. In the soft wall model a nontrivial dynamics for large mq but differ in the chiral limit. The holographic the tachyonic field X is driven by the presence of the dilaton model of [11] was another useful guide in our approach. field ΦðzÞ and, instead of imposing a boundary condition at The paper is organized as follows. In Sec. II we briefly an arbitrary energy scale, the chiral condensate is obtained review chiral symmetry breaking in the hard wall and soft from a regularity condition. wall models. In Sec. III we present the nonlinear extensions Despite providing a more realistic meson spectrum, the of the soft wall model based on a Higgs potential. We original soft wall model leads to a linear realization of perform a systematic analysis of the background including chiral symmetry breaking driven by an IR regularity holographic renormalization for the chiral condensate. condition. Indeed, the chiral condensate hqq¯ i is propor- Solving the effective Schrödinger equations for the 5D tional to the quark mass mq just as a consequence of the field perturbations, we obtain the meson masses and decay linearity of the tachyonic field equation. This implies, in constants. In Sec. IV we introduce a running mass for the particular, that taking the chiral limit mq → 0 in the soft 5D tachyonic field and investigate its effect on the chiral wall model does not lead to spontaneous chiral symmetry condensate, the meson spectrum, and decay constants. We breaking (the chiral condensate vanishes). This differs present our conclusions in Sec. V. Appendix A describes significantly from QCD, where spontaneous symmetry the equations for the 5D field perturbations and the dicti- breaking always occurs in the chiral limit. onary for the meson decay constants. Appendix B briefly In this work we investigate nonlinear extensions of the reviews the holographic model of [11], while Appendix C holographic soft wall model. We consider a 5D Higgs describes some complementary numerical results. potential for the tachyonic field X leading to a nonlinear differential equation for the tachyonic field. The non- II. HARD AND SOFT WALL MODELS FOR linearity allows us to find a family of solutions in the IR CHIRAL SYMMETRY BREAKING depending on only one parameter, which we call C0.For Holographic models for QCD at zero temperature satisfy fixed C0 we will be able to solve numerically the tachyon Poincar´e invariance. The 5D metric for those models is differential equation. Extracting the source and VEV usually written as coefficients for the tachyon solution near the boundary, we utilize the AdS=CFT correspondence and map those 2 2 2 2 2 ds ¼ e AsðzÞ½−dt þ dx⃗ þ dz ; ð1Þ parameters to the quark mass and chiral condensate, respectively. We will show that the nonlinearity of the where A ðzÞ is the (string-frame) warp factor given as a tachyon differential equation implies a nonlinear relation s function of the conformal coordinate z. In this work we between the chiral condensate hqq¯ i and the quark mass m . q are interested in the simplest backgrounds in which the We will show, in particular, that for the original Higgs 5D metric is just Lorentzian anti–de Sitter (AdS), potential the chiral condensate always grows with the quark 1 i.e., A ðzÞ¼− ln z. Simplifying the background as much mass, as expected in QCD; see, i.e., [8]. We will find, s → 0 as possible allows us to focus on the dynamics of chiral however, that near the chiral limit mq the nonlinear symmetry breaking. soft wall models based on a Higgs potential behave in the same way as the original soft wall model, and therefore the A. The hard wall model chiral condensate vanishes. A similar result was recently found in [9]. In the hard wall model [6] the 5D background is given by In this work we perform a systematic analysis of the 1 background and field perturbations for nonlinear soft wall 2 − 2 ⃗2 2 0 ≤ ds ¼ 2 ½ dt þ dx þ dz ;

026013-2 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020) which corresponds to a 5D Lorentzian AdS space ending in In QCD at zero temperature (and zero density) we do not an IR hard wall at z ¼ z0. A nice property of the hard wall expect any vectorial condensate, so we take the following model is that it satisfies the confinement criterion found ansatz for the 5D gauge fields: in [2]. Soon it was realized that the hard wall model allows us to estimate the spectrum of glueballs [12]. L=R A ¼ 0: ð6Þ The flavor sector of the hard wall model was introduced m in [4,5]. The key ingredients were the 4D quark mass ¯ The 5D action in Eq. (3) reduces to a one-dimensional (1D) operator qRqL as well as the left and right 4D currents μ;a ¯ γμ a action for the field vðzÞ, and the Euler-Lagrange equation J ¼ qL=R T qL=R associated with the chiral sym- ðL=RÞ takes the form metry group SUðNfÞL × SUðNfÞR. According to the AdS=CFT dictionary, the 4D quark mass operator maps ∂ 2 − 4 ∂ − 2 0 to a 5D scalar X field with mass given by the relation ½ðz zÞ z z mXv ¼ ; ð7Þ 2 Δ Δ − 4 Δ mX ¼ ð Þ, with the conformal dimension of the quark mass operator. In the extreme ultraviolet (UV) we with the exact solution have Δ 3 and this corresponds to m2 −3. This 5D ¼ X ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mass would indicate an instability in Minkowski space, Δ Δ 2 − þ Δ 2 4 and therefore the 5D field X is usually called the tachyon. vðzÞ¼c1z þ c3z ; ¼ þ mX: ð8Þ The left and right 4D currents, on the other hand, are conserved in the extreme UV, so they are mapped to 5D This has the expected scaling behavior for a 5D scalar field μ;a Δ gauge fields AL=R. dual to an operator of dimension þ and a coupling of Δ 4 − Δ In QCD chiral symmetry is broken dynamically in the IR conformal dimension − ¼ þ. In order to match to due to a nontrivial vacuum and the dynamics of the quark the quark mass operator in the extreme UV, we choose Δ 3 mass operator. An important quantity is the so-called chiral þ ¼ . The source coefficient c1 is related to the quark σ ζ ζ condensate , which is the VEVof the quark mass operator, mass mq by c1 ¼ mq , where is a normalizationffiffiffiffiffiffi constant. σ ¯ p i.e., ¼hqqi. The authors of [4,5] realized that chiral This constant is usually fixed as ζ ¼ Nc=2π to be symmetry breaking in 4D corresponds to 5D gauge consistent with counting rules of large-Nc QCD [13]. μ;a symmetry breaking of the fields AL=R driven by a nontrivial The VEV coefficient c3 will be related to the quark σ scalar field X. In this work we use the conventions of [4] condensate u. The precise relation is scheme dependent and write the 5D action as and will be obtained in the next section. In the hard wall model the VEV coefficient c3 is fixed by Z ffiffiffiffiffiffi boundary conditions at the IR wall z ¼ z0. This differs from − 5 p− 2 2 2 S ¼ d x gTr jDmXj þ mXjXj what we expect in QCD, where the chiral condensate is generated dynamically due to a nontrivial vacuum. The 1 1 ðLÞ2 ðRÞ2 hard wall model, however, has the nice feature of being the þ 2 Fmn þ 2 Fmn ; ð3Þ g5 4g5 simplest model that describes at the same time confinement and chiral symmetry breaking. where h i B. The (linear) soft wall model ðL=RÞ ∂ ðL=RÞ − ∂ ðL=RÞ − ðL=RÞ ðL=RÞ Fmn ¼ mAn nAm i Am ;An ; A shortcoming of the hard wall model is that it leads to hadronic masses that grow too fast as we increase the ∂ − ðLÞ ðRÞ DmX ¼ mX iAm X þ iXAm : ð4Þ radial number. For example, the resonances of the ρ meson 2 ∼ 2 have squared masses that grow as mρðnÞ n for large n. Note that under the non-Abelian gauge group SUðNfÞL × Experimental data, on the other hand, indicates an approxi- ðL=RÞ SUðNfÞR the left and right vector fields Am transform as mate linear dependence for the squared masses, i.e., 2 ∼ adjoint fields, whereas the scalar field X behaves as a mρðnÞ n. 2 bifundamental. From now on we take Nf ¼ and make the The quadratic dependence in the hard wall model can assumption of isospin (flavor) symmetry, i.e., mu ¼ md and be thought of as a consequence of having a metric that σ σ u ¼ d, which is a good approximation for the light quark abruptly ends at a cutoff z ¼ z0. Motivated by this problem, sector in QCD. We therefore take the following ansatz for Karch et al. [7] proposed the idea of a smooth cutoff driven the 5D tachyonic field: by a background scalar field ΦðzÞ so that the geometry does not abruptly end. This was inspired by string theory, where 1 the field Φ is known as the dilaton. Later works have XðzÞ¼ vðzÞI2 2: ð5Þ 2 × explored the backreaction of this background field on the

026013-3 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020) AdS metric and how it maps to the Yang-Mills operator Δ− Δ− 2 2 2 vðzÞ¼c˜1z U ; Δ− − 1; ϕ∞z TrF [14–16]. 2 In the soft wall (SW) model the background is given by Δ Δ þ 2 ˜ þ Δ − 1; ϕ þ c3z M 2 ; þ ∞z ; ð14Þ 2 1 2 2 2 2 ds ¼ ½−dt þ dx⃗ þ dz ; ΦðzÞ¼ϕ∞z ; ð9Þ 2 ˜ ˜ 2 −3 z where c1 and c3 are constant coefficients. For mX ¼ we Δ 1 Δ 3 have − ¼ and þ ¼ . This exact solution was first where ϕ∞ is a constant providing an IR mass scale. The obtained in [18]. In the limit x → ∞ we have Mða; b; xÞ ∼ flavor sector of the soft wall model is described by the exxa−b and Uða; b; xÞ ∼ x−a; therefore, we need to set action c˜3 ¼ 0 in order to avoid a divergent solution in the IR. The tachyon solution in Eq. (14) becomes Z ffiffiffiffiffiffi pffiffiffi − 5 p− −ΦðzÞ 2 2 2 π 1 S ¼ d x ge Tr jDmXj þ mXjXj 2 vðzÞ¼ c1zU ; 0; ϕ∞z ; ð15Þ 2 2 1 1 ðLÞ2 ðRÞ2 ffiffiffi þ 2 Fmn þ 2 Fmn : ð10Þ p g5 4g5 where c˜1 was rewritten as ð π=2Þc1. In the UV (small z), the solution in Eq. (15) behaves as The positive quadratic z dependence of Φ leads, at large z, UV 3 3 … to a harmonic oscillator form for the effective Schrödinger v ðzÞ¼c1z þ d3ðc1Þz ln z þ c3ðc1Þz þ : ð16Þ potentials for the field perturbations. This guarantees a The logarithmic term could have been anticipated from linear dependence on the radial number for the meson an asymptotic analysis using the Frobenius method. squared masses, i.e., m2 ∼ n. ðnÞ Coefficients d3 and c3 are both actually proportional to In the flavor sector we take again the ansatz (5) and (6) c1 and then vanish in the chiral limit c1 → 0. In the IR the for the background fields, and this time we obtain the tachyon solution (15) behaves as equation ffiffiffi pπ IR −2 c1 v ðzÞ¼C0½1 þ Oðz Þ;C0 ¼ pffiffiffiffiffiffiffi : ð17Þ 2∂2 − 3 2ϕ 2 ∂ − 2 0 2 ϕ ½z z ð þ ∞z Þz z mXv ¼ : ð11Þ ∞

The parameter C0 characterizes the tachyon solution at In order to find an exact solution, we rewrite Eq. (11) in 2 large z. As expected, the solution is regular in the IR. This terms of a new variable x ≡ ϕ∞z and redefine the field regular solution was, however, chosen by setting c˜3 ¼ 0.In β ˜ vðxÞ as x vðxÞ. We arrive at the second order differential the next section we will introduce a nonlinear potential for equation the tachyonic field. The nonlinearity of the new differential equation will naturally lead to regular solutions in the IR 2 without the need of fixing any integration constant. 2∂2 2β − 1 − ∂ β2 − 2β − mX − β ˜ 0 x x þ½ xx x þ 4 x v ¼ : The soft wall model represents an important step toward the construction of a minimal model for describing the ð12Þ flavor sector in holographic QCD. The positive quadratic behavior of the dilaton ΦðzÞ leads naturally to a linear This equation becomes Kummer’s equation if spectrum for the mesons. Moreover, when taking backreaction into account, the same positive quadratic dilaton rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi leads to a linear spectrum for glueballs [15,16]. The soft m2 m2 Δ β2 − 2β − X ¼ 0 → β ¼ 1 1 þ X ¼ : wall model has also served as a useful guide to other 4 4 2 approaches, such as light-front holography [19]. ð13Þ III. NONLINEAR SOFT WALL MODELS

The solutions for v˜ðzÞ are of the form Mðβ; 2β − 1; xÞ Karch et al. [7] realized that the linear dependence of c3 (Kummer) and Uðβ; 2β − 1; xÞ (Tricomi). From this ana- on c1 differed significantly from what one would expect in lysis we conclude that the exact solution for vðzÞ can be QCD for the chiral condensate σq as a function of the quark written as mass mq. They suggested that the model could be improved by adding nonlinear terms to the tachyonic potential. The 2The AdS deformation by a scalar field may also be interpreted nonlinearity would lead to solutions in the IR characterized in terms of supersymmetry breaking; see, e.g., [17]. by only one parameter, C0, and the relation between the IR

026013-4 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020) parameter C0 and the UV parameter c1 would also be The VEV parameter c3 appears to be independent of c1. nonlinear. We will see, however, that in the IR the tachyon solution is In this work we investigate a nonlinear potential of the characterized by a single parameter C0. The UV parameters Higgs form c1 and c3 then can be thought of as functions of the IR parameter C0. As a consequence, c3 will depend on c1 in a 2 2 λ 4 UðjXjÞ ¼ mXjXj þ jXj : ð18Þ nonlinear fashion, as predicted in [7]. We remind the reader that the UV parameters c1 and c3 are related to the quark This time the 5D action for the flavor sector becomes mass mq and chiral condensate σq, respectively. As far as Z we are concerned, the first nonlinear realization of chiral ffiffiffiffiffiffi − 5 p− −ΦðzÞ 2 2 2 λ 4 symmetry breaking via tachyon dynamics in terms of c3 S ¼ d x ge Tr jDmXj þ mXjXj þ jXj as a function of c1 was developed in the holographic 3 QCD model of [10]. More sophisticated models that 1 ðLÞ2 1 ðRÞ2 þ Fmn þ Fmn : ð19Þ take into account the backreaction of the tachyon were g2 4g2 5 5 developed in [22–25]. In the IR it is convenient to work with the variable We will consider both cases λ < 0 and λ > 0 and also y ¼ 1=z. The differential equation (20) may be written as recover the original soft wall model in the limit λ → 0. 2 As in the previous cases, we take m ¼ −3 for the λ X ∂ 2 2 2 ϕ −2 ∂ 3 − 3 0 tachyonic field. ½ðy yÞ þ ð þ ∞y Þðy yÞþ v 2 v ¼ : ð23Þ For the background fields we take again the ansatz (5) and (6), and this time we obtain a nonlinear equation for the We first consider the power ansatz tachyon: IR α v ðyÞ¼C0y : ð24Þ 2 2 2 λ 3 ½z ∂z − ð3 þ 2ϕ∞z Þz∂z þ 3v − v ¼ 0: ð20Þ 2 Plugging this ansatz into Eq. III A, we obtain the poly- nomial equation Because of the nonlinearity of the differential equation, we cannot obtain an analytic solution. In the next subsections λ α α2 4α 3 α−2 2αϕ − 3 3α 0 we describe the asymptotic solutions for the tachyon in the C0y ð þ þ ÞþC0y ð ∞Þ 2C0y ¼ : ð25Þ UVand IR regimes and present our numerical results for the tachyon profiles. Then we investigate the meson spectrum The first term in Eq. (25) is subleading compared to the λ 0 λ 0 for the cases < and > . We finish the section by second. The third term competes with the second term only describing the meson decay constants. when α < 0. We distinguish among three cases: α > −1, α ¼ −1, and α < −1. The last case is trivial because it leads A. Asymptotic analysis to C0 ¼ 0. In the case α > −1 the second term always dominates In the UV we consider the Frobenius ansatz and we find α ¼ 0. This is the regular solution we are 3 3 5 5 looking for and it admits the following expansion, vðzÞ¼c1z þ d3z ln z þ c3z þ d5z ln z þ c5z þ …: IR 2 4 ð21Þ v ðyÞ¼C0 þ C2y þ C4y þ …; ð26Þ

Plugging this ansatz into the tachyon equation (20), we find with the subleading IR coefficients given by the UV coefficients C0 2 1 3 C2 ¼ ðC0λ − 6Þ; 2λ 4ϕ 2λ 4ϕ 2 8ϕ∞ d3 ¼ 4 c1ðc1 þ ∞Þ;d5 ¼ 64 c1ðc1 þ ∞Þ þ; 3C0 2 2 1 C4 ¼ ðC λ − 6ÞðC λ − 10Þ: ð27Þ 2λ 4ϕ −9 3λ − 20 ϕ 48 … 128ϕ2 0 0 c5 ¼ 256 ðc1 þ ∞Þð c1 c1 ∞ þ c3Þ : ∞ ð22Þ To guarantee the convergence of the series (26), we need a 2 condition of the form jC0λj=ϕ∞ < 1. In the special case 2 Note that d3 and d5 depend only on c1, whereas c5 depends C0λ − 6 ¼ 0 all subleading coefficients vanish and the 2 on c1 and c3. In the special case c1λ þ 4ϕ∞ ¼ 0 all of the subleading terms vanish and the linear solution vðzÞ¼c1z 3For earlier works on the dictionary between tachyon dynam- becomes exact. This is possible only for λ < 0 because in ics and the chiral condensate from a top-down approach, the soft wall model we have ϕ∞ > 0. see [20,21].

026013-5 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

1.2 Our numerical results for the UV parameters c1 and c3 as functions of the IR parameter C0 are displayed in Fig. 2 1.0 for different values of λ. The linear soft wall model (λ ¼ 0) is represented by the black dot-dashed line. All solutions 0.8 enjoy the symmetry ðc1;c3Þ ↔ ð−c1; −c3Þ corresponding to the symmetry v ↔ −v present in the differential 0.6

v(z) equation (20). The physical regime, of course, corresponds to c1 > 0. We see that the nonlinearity of the 0.4 tachyon differential equation leads to nonlinear relations c1ðC0Þ and c3ðC0Þ, as expected. However, in the chiral 0.2 limit (corresponding to c1 → 0) all parameters go to zero. In particular, the VEV parameter c3 (associated with the 0.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 4D chiral condensate) vanishes. This differs significantly z from QCD, where chiral symmetry is spontaneously broken in the chiral limit and therefore the chiral con- 1 λ −5 FIG. 1. Tachyon profiles for C0 ¼ and varying from (red densate is nonvanishing. Note that nonlinearity of the dashed line) to 5 (red solid line). The black dot-dashed profile in tachyon differential equation also brings saturation the middle corresponds to λ ¼ 0 (linear soft wall model). effects. In the case λ > 0 there is an upper bound in C0 (described in the previous paragraph), whereas in the case λ 0 constant solution is exact. This can be obtained only in the < we have an upper bound in c1. An upper bound in c1 case λ > 0 and, interestingly, it corresponds to the mini- implies a cutoff for the quark mass, which is not expected mum of the Higgs potential ∂U=∂v ¼ 0. in QCD. In the special case α ¼ −1 we see from Eq. (25) that the In Fig. 3 we show the VEV parameter c3 as a function first term vanishes, whereas the second and third terms are of the source parameter c1 in the physical regime. This −3 2 result will be interpreted in terms of the 4D chiral of order y , leading to the condition C0λ þ 4ϕ∞ ¼ 0. This condensate hqq¯ i as a function of the quark mass m ,as divergent solution is linear, i.e., vðzÞ¼C0z, and it is valid q only for λ < 0. This linear solution appears to be exact, and described in the next subsection. The linear relation ∼ it may be related to previous approaches to nonlinear soft c3 c1, characteristic of the linear soft wall model, is λ wall (NLSW) models such as [11]. The model of [11] is actually a good approximation at small c1. For positive briefly discussed in Appendix B. the VEV parameter c3 grows monotonically with c1, which is the expected behavior for hqq¯ i. A similar behavior was obtained in the holographic QCD model B. Numerical solution developed in [10]. The case of negative λ leads to a Once the asymptotic analysis has been done, we proceed nonmonotonic function c3ðc1Þ.InthatcasetheVEV to numerically solve the nonlinear differential equation for parameter c3 vanishes when c1 reaches its upper bound. the tachyon field (20). We may integrate Eq. (20) from the We conclude that the case λ > 0 leads to the more realistic UV to the IR using the UV asymptotic solution (21) to scenario. This could have been anticipated due to the fact extract initial conditions. By matching the numerical that λ > 0 in Eq. (18) corresponds to the Mexican hat solution to the IR analytic solution (26), we find a relation potential for the Higgs field. between the UV parameters c1 and c3 as well as the corresponding IR parameter C0. Alternatively, we can C. Holographic renormalization integrate numerically Eq. (20) from the IR to the UV using and the chiral condensate the IR analytic solution (26), and this time the matching procedure allows us to extract the UV parameters c1 and c3 In this subsection we describe the procedure of holo- in terms of the IR parameter C0. graphic renormalization for the nonlinear soft wall models We show in Fig. 1 typical profiles for the tachyon field proposed in this work. The case at hand is similar to the vðzÞ for fixed C0 and different values of λ. The linear soft case of probe branes in a fixed background; see, e.g., wall model corresponding to λ ¼ 0 is depicted by the black [10,26]. The starting point is the on-shell action. For the dot-dashed line, and we see that the effect of a nonlinear background tachyonic field the action in Eq. (19), when negative (positive) quartic coupling λ is to deform the taken on shell, can be written as tachyon profile to the right (left). In the case of λ > 0 we 2λ 6 have an upper bound C0 ¼ for the tachyon solution. S ¼ S þ S ; ð28Þ As we approach that bound, the tachyon profile becomes os Bdy Int constant in z, which is consistent with the asymptotic analysis done in the previous subsection. where

026013-6 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

3 15

2 10

1 5 1 3 c c 0 0 –1/2 –3/2

–1 –5

–2 –10

–3 –15 –4 –2 0 2 4 –4 –2 0 2 4

C0 C0 pffiffiffiffiffiffiffi FIG. 2. Numerical results for (left panel) c1 and (right panel) c3 as functions of C0 in units of ϕ∞. The blue and red solid lines (dashed lines) correspond to λ ¼ 2 and λ ¼ 5 (λ ¼ −2 and λ ¼ −5). The black dot-dashed line corresponds to λ ¼ 0 (linear soft wall model).

Z 1 − 4 Π Π − −3 −ΦðzÞ∂ 1. Counterterms and covariant subtraction SBdy ¼ d x zðz0Þvðz0Þ; z ¼ z e zv; 2 Plugging the UV asymptotic solution (21) into the ð29Þ surface term (29), we see that it splits into divergent and finite pieces

Z Div Fin λ SBdy ¼ SBdy þ SBdy; ð31Þ 4 −5 −ΦðzÞ 4 SInt ¼ d xdzz e v : ð30Þ Z 8 1 Div 4 2 −2 2 2 S d x c z c c λ 4ϕ∞ z0 ; Bdy ¼ 2 ½ 1 0 þ 1ð 1 þ Þ ln ð32Þ S is a boundary term (the AdS boundary is located at Z Bdy 1 1 z ¼ z0 with z0 → 0), and Π ¼ ∂L=∂ð∂ vÞ is the conjugate Fin 4 4 z z S d x c λ 4c1c3 : Bdy ¼ 2 4 1 þ ð33Þ momentum in z. The bulk term SInt appears because of the nonlinear term in the tachyon potential. We have omitted in Eq. (33) the terms that vanish in the limit z0 → 0. The bulk term (30) can not be split in a simple 14 way, but, from Eq. (21), we find the divergent piece Z 12 λ Div − 4 2 SInt ¼ d xc1 ln z0 ð34Þ 10 8 3 c 8 and simply define the finite piece as –3/2 6 Fin − Div SInt ¼ SInt SInt : ð35Þ 4 In order to cancel the UV divergences (32)–(34) in a 2 consistent way, we introduce the covariant counterterms Z 0 ffiffiffiffiffiffi 0.0 0.5 1.0 1.5 2.0 2.5 3.0 − 4 p−γ 2 Sct ¼ d x fa1v ðz0Þ –1/2 c1 2 4 þ ln z0½a2Φðz0Þv ðz0Þ − a3λv ðz0Þ FIG. 3. VEV parameter c3 as a function of the source parameter pffiffiffiffiffiffiffi Φ 2 − λ 4 c1, in units of ϕ∞, for different values of λ. The blue and red þ a4 ðz0Þv ðz0Þ a5 v ðz0Þg: ð36Þ solid lines (dashed lines) correspond to λ ¼ 2 and λ ¼ 5 (λ ¼ −2 and λ ¼ −5). The black dot-dashed line corresponds to λ ¼ 0 The first three terms in Eq. (36) are the minimum required (linear soft wall model). to cancel the UV divergences. The last two terms in

026013-7 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

Eq. (36) are finite counterterms associated with the holographic dictionary for the chiral condensate then takes renormalization scheme dependence of the 4D dual theory. the form The counterterms action in Eq. (36) can be split into divergent and finite pieces: δS δS hqq¯ i¼ζ os þ ct ; ð43Þ δc1 δc1 Div Fin Sct ¼ Sct þ Sct ; ð37Þ ffiffiffiffiffiffi ζ p 2π Z where ¼ Nc= is the normalization constant intro- Div − 4 2 −2 duced in the previous section. We identify the first term, Sct ¼ d x a1c1z0 δ ∂ Sos vðz0Þ 2 a1 4 ¼ − Πzðz0Þ þ ð2a1 þ a2Þc ϕ∞ þ − a3 c λ ln z0 ; δc1 c1 1 2 1 3 −2 4 ϕ − 3λ ð38Þ ¼ c1z0 þ c1 ∞ 2 c1 ln z0 Z 1 3 Fin − 4 2 2ϕ − 4λ þ 3c3 þ c1∂ c3 þ c λ; ð44Þ Sct ¼ d x½ a1c1c3 þ a4c1 ∞ a5c1 : ð39Þ c1 4 1

The renormalized action is then defined as as the bare contribution and the second term, δS 3 S ¼ S þ S þ S : ð40Þ ct − −2 − 4 ϕ 3λ − − ∂ Ren Bdy Int ct ¼ c1z0 c1 ∞ þ c1 ln z0 c3 c1 c1 c3 δc1 2 3 The UV divergences cancel for − 2a4c1ϕ∞ þ 4a5c1λ; ð45Þ 1 1 as the counterterm contribution. From Eqs. (43)–(45) we a1 ¼ ;a2 ¼ 1;a3 ¼ − ; ð41Þ 2 8 see that the UV divergences cancel, and we arrive at the final expression for the chiral condensate: and the renormalized action takes the form 1 3 Fin Fin Fin hqq¯ i¼ζ 2c3 − 2a4c1ϕ∞ þ c λð1 þ 16a5Þ : ð46Þ SRen ¼ SBdy þ SInt þ Sct : ð42Þ 4 1

The coefficients a4 and a5 reflect the scheme dependence 2. The chiral condensate and the renormalized of the chiral condensate. Since we already know that c3 has Hamiltonian a very similar behavior to the QCD chiral condensate, we The conformal field theory (CFT) deformationR due to suspect that the natural choice for fixing the scheme would 4 the quark mass operator has the form d xmqhqq¯ i. The be choosing a4 ¼ 0 and a5 ¼ −1=16. In Fig. 4 we plot the

40 0

30 –10

–20 20

–30 10

–40 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

FIG. 4. [Left (right) panel] The renormalized Hamiltonian (chiral condensate) as a function of the UV parameter c1 (quark mass) in pffiffiffiffiffiffiffi units of ϕ∞. The scheme was fixed by setting a4 ¼ 0 and a5 ¼ −1=16. The blue and red solid (dashed) lines correspond to λ ¼ 2 and λ ¼ 5 (λ ¼ −2 and λ ¼ −5), respectively. The black dot-dashed line corresponds to λ ¼ 0 (linear case).

026013-8 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020) renormalized Hamiltonian and the chiral condensate, in TABLE I. The mass of the vector mesons (in MeV) obtained in pffiffiffiffiffiffiffi units of ϕ∞, for that particular scheme. As promised, we the soft wall model, compared against the holographic model [11] find a nonlinear relation between the chiral condensate and and experimental results from PDG [29]. the quark mass. As described in the previous subsection, n SW [7] GKK [11] ρ experimental [29] the model with λ > 0 provides a more realistic description 1 776 475 776 1 of the chiral condensate. This conclusion will be supported 1282 37 by the analysis of the meson spectrum performed in the 2 1097 1129 3 1344 1429 1465 25 next subsection. It can be checked that the chiral con- 4 1552 1674 1720 20 densate could have been obtained directly using the 5 1735 1884 1909 30 ¯ −∂ Ren ∂ relation hqqi¼ H = mq. 6 1901 2072 2149 17 We finish this subsection with a general conclusion for 7 2053 2243 2265 40 the nonlinear soft wall models presented in this work. As described in the previous subsection, the VEV parameter c3 → 0 ⊥;a a always vanishes in the limit c1 . Regardless of the transverse Aμ and longitudinal parts ∂μϕ . The fields renormalization scheme, this result implies that the chiral ⊥;a Aμ will describe a tower of axial-vector mesons, whereas qq¯ condensate h i always vanishes in the chiral limit, in the fields ϕa will couple to the fields πa and describe a sharp contrast to QCD. This means that the nonlinear tower of pseudoscalar mesons. The equations for the vector models at hand never lead to spontaneous symmetry ⊥;a ⊥;a sector Vμ , scalar sector S, axial-vector sector Aμ , and breaking. This result will be confirmed later when we pseudoscalar sector (πa, ϕa) are obtained from a second conclude that pseudo-Nambu-Goldstone (NG) modes are order expansion of the action (19), as described in absent in the spectrum of pseudoscalar mesons. An original Appendix A. solution to this problem was found in [11,27], where the dilaton profile was modified drastically near the boundary 1. Spectrum of the vector sector to allow for a description of spontaneous symmetry breaking. The authors of [11,27] realized that the dilaton profile We start with the equation of motion (A9) (the flavor had to be negative in the UV to guarantee spontaneous index is hidden in the following analysis). After performing μ μ symmetry breaking. In this work we take a more the Fourier transform Vμðx ;zÞ → Vμðk ;zÞ on Eq. (A9), □ → 2 conservative approach of a positive quadratic dilaton to where we have set mV, the equation may be written avoid instabilities in the fluctuations associated with the in the Schrödinger form through the transfor- − ξ BV ψ − Φ 2 ξ meson spectrum. Nevertheless, the model in [11,27] mation Vμ ¼ μe vn , where BV ¼ðAs Þ= and μ deserves further systematic study. That model is briefly is a (transverse) polarization vector.4 The effective reviewed in Appendix B. Schrödinger equation reads

½−∂2 þ V ψ ¼ m2 ψ ; ð47Þ D. Meson spectrum (λ < 0) z V vn V vn The non-Abelian Higgs action in Eq. (19) describes the where the potential is given by dynamics of the scalar tachyonic field X as well as the left ðL=RÞ 2 2 and right gauge fields Am . At the beginning of this VV ¼ð∂zBVÞ þ ∂z BV: ð48Þ section we analyzed the background tachyonic field X ¼ 1 In this case the problem has an exact solution [7]: 2 vðzÞ (the background gauge fields were set to zero). Now ðL=RÞ we consider the perturbations of the fields X and A .As 2 m m ¼ 4ϕ∞ð1 þ nÞ;n¼ 0; 1; 2; …: ð49Þ described in Appendix A, the perturbations associated with V the tachyonic field are the scalar field S and the pseudo- At this point the free parameter is ϕ∞, and we may fix πa scalar fields . The former will describe a tower of scalar the value of this parameter by comparing the first vector mesons, whereas the latter will be related to the pseudo- state with the corresponding experimental value of the ρ 2 scalar mesons (pions in the case of Nf ¼ ). On the other meson, as was done in, e.g., [28]. We obtain the value 2 hand, the gauge field fluctuations will be written as ϕ∞ ¼ð388 MeVÞ . ðL=RÞ a a a a Am ¼ Vm Am, with Vm ¼ VmT and Am ¼ AmT The spectrum obtained is shown in Table I, labeled as identified as the 5D fields dual to the vectorial and axial SW, and compared against the holographic model of [11] currents in the chiral symmetry group. and experimental data [29]. The details of [11] are given in a As described in Appendix A, the vector field Vm Appendix B. a a a decomposes as ðVz ;VμÞ, and Vμ will describe a tower a 4 of 4D vector mesons. The axial-vector field Am decom- We are introducing again the string-frame warp factor a a a − poses as ðAz ;AμÞ and, in turn, Aμ decomposes into As ¼ ln z.

026013-9 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

2. Spectrum of the scalar sector Now we proceed to calculate the spectrum of the scalar mesons. This sector is obtained from the fluctuations of the tachyon field, see Eq. (A1), where Sðx; zÞ represents the scalar field related to the scalar mesons. After performing the Fourier transform Sðxμ;zÞ → Sðkμ;zÞ on Eq. (A11), 2 where we set □ → ms, we arrive at the following equation:

−3 Φ 3 −Φ 2 e Asþ ∂ ðe As ∂ Sðk; zÞÞ þ m Sðk; zÞ z z s 2 2 3 2 − As λ 0 e mXðzÞþ2 v ðzÞ Sðk; zÞ¼ ; ð50Þ FIG. 5. Potential of the Schrödinger equation for ϕ∞ ¼ ð388 MeVÞ2 and three different values of the parameter λ. 2 −3 where mXðzÞ¼ . We rewrite the last equation in a Schrödinger form, redefining the scalar modes as Sn ¼ − e BS ψ ðzÞ, where B ¼ 3A =2 − Φ=2. Thus, we get sn S s 3. Spectrum of the axial-vector sector μ μ After performing the Fourier transform A⊥ðx ;zÞ → 2 2 μ μ 2 −∂z ψ s þ VSψ s ¼ msψ s ; ð51Þ □ → n n n A⊥ðk ;zÞ on Eq. (A13), with mA, redefining the − ξ BA ψ axial-vector mode as Aμ ¼ μe an , where BA ¼ with the Schrödinger potential given by ðAs − ΦÞ=2, we arrive at the Schrödinger equation −∂2ψ ψ 2 ψ z an þ VA an ¼ mA an ; ð53Þ 2 2 2 2 3λ 2 ∂ ∂ As VS ¼ð zBSÞ þ zBS þ e mX þ 2 v ðzÞ : ð52Þ with the Schrödinger potential given by

2 2 2 2 2 ∂ ∂ As For λ ¼ 0 the potential (52) reduces to the one obtained in VA ¼ð zBAÞ þ zBA þ g5e v ðzÞ: ð54Þ [18]. Notice how the parameter λ controls the minimum value of the potential (52), as shown in Fig. 5.Aswe The left panel of Fig. 7 shows our results for the masses of increase λ the minimum increases, and hence also the the axial-vector states as a function of the parameter C0, masses of the scalar mesons. For λ < 0 the potential allows compared against the corresponding values in the linear us to describe very light states. This statement is supported soft wall model, i.e., λ ¼ 0. The right panel of the figure by the results displayed in Fig. 6, where we can see the shows how the masses depend on the parameter of the evolution of the meson with the parameter C0 (left panel) quark mass parameter c1. We see that in the axial sector the and c1 (right panel). Those results were obtained for masses increase with c1, which is the expected behavior λ ¼ −2 (solid lines) and λ ¼ 0 (dashed lines). for mesons. These results suggest the possibility of using Those results indicate the possibility of fixing the the mass of the first axial-vector meson a1ð1230Þ to fix the parameter C0 for a given λ, requiring the first eigenvalue parameter C0 for a given λ. Note, however, that the axial- of the Schrödinger equation to match the mass of the vector meson masses grow very fast and tend to diverge at a 5 scalar meson f0ð550 MeVÞ. However, the status of the finite value of c1. That value corresponds to the unphysical f0ð550 MeVÞ as a scalar meson is not established [29]. upper bound for the quark mass, described in the previous We follow a more conservative approach and consider subsection. f0ð980 MeVÞ as the first scalar meson, as in [32]. As the upper limit for the scalar mass in our model is the one 4. Spectrum of the pseudoscalar sector obtained for λ ¼ 0, ms ¼ 950 MeV (see Fig. 6), it is not The pseudoscalar sector is special because it is described possible to reach the state f0ð980 MeVÞ when λ < 0. It is worth mentioning that the behavior of the scalar by a coupled system of differential equations (A14) and (A15). After performing the Fourier transform π xμ;z → meson mass as a function of the quark mass, i.e., c1, ð Þ π μ φ μ → φ μ displayed in Fig. 6 is the opposite of that expected in QCD. ðk ;zÞ and ðx ;zÞ ðk ;zÞ in both equations, where □ → 2 Hence, the model with λ < 0 is pathological in the scalar we have set mπ, we get the following equations: sector. −AsþΦ As−Φ 2 2Asþ2 log v e ∂zðe ∂zφÞþg5e ðπ − φÞ¼0; ð55Þ 5 There were some attempts to obtain a light scalar meson in the 2 2 2 2 − ∂ φ Asþ log v∂ π 0 top-down approach; see, e.g., [30,31]. mπ z þ g5e z ¼ : ð56Þ

026013-10 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

1600 1400 1400 1200 1200 1000 1000 (MeV) (MeV)

s 800 s m m 600 800 400 600 200 –3–2–10123 0 100 200 300 400 500

C0 c1(MeV)

FIG. 6. Mass of the scalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the results for λ ¼ −2, 2 dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ð388 MeVÞ .

2500 2500 2000 2000 (MeV) (MeV) n n A 1500 A 1500 m m 1000 1000

–3 –2 –1 0 1 2 3 0 100 200 300 400 500

C0 c1(MeV)

FIG. 7. Masses of axial-vector mesons as functions of (left panel) C0 and (right panel) c1. Solid lines represent the results for λ ¼ −2, 2 dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ð388 MeVÞ .

We follow [33] and decouple this system of equations. The −∂2ψ ψ 2ψ z πn þ Vπ πn ¼ mπ πn ; ð58Þ decoupled equation is second order in the auxiliary field Π ¼ ∂zπn and takes the form with the potential given by

2 ∂ 2 − ∂2 β − ∂zΠ þ ∂zðΦ − As − ln βÞ∂zΠ Vπ ¼ð zBπÞ z Bπ þ : ð59Þ ∂2 Φ − − β − 2 β Π 0 þð z ð As ln Þ mπ þ Þ ¼ ; ð57Þ At this point it is interesting to display a plot of the potential (59). This is shown in the left panel of Fig. 8, 2 2 2 β As where we have introduced the function ðzÞ¼g5e v . where we observe potential wells emerging for different 2 − Φ β Defining the function Bπ ¼ As þ log and then values of λ. These potential wells, however, are not deep Π −Bπ ψ introducing the transformation ¼ e πn , we arrive at enough to allow a light state in the spectrum. This is related the Schrödinger equation to the fact that the soft wall backgrounds considered in this

8 8

6 6 6 6 ×10 ×10 4 =2 4 (z) V(z) V =0 =15 2 2 =–2 0 0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 z(MeV 1) z(MeV 1)

FIG. 8. (Left panel) Potential of the pseudoscalar Schrödinger equation in the NLSW model for λ ¼ 2 (blue line), λ ¼ 0 (dashed black 2 line), and λ ¼ −2 (red line). We have set ϕ∞ ¼ð388 MeVÞ . (Right panel) The potential of the Gherghetta-Kapusta-Kelley (GKK) model. This figure was obtained using the same parameters as in [27].

026013-11 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

3000 2500 2500 2000 2000 (MeV) (MeV) n n m 1500 m 1500

1000 1000 –3 –2 –1 0 1 2 3 0 100 200 300 400 500

C0 c1(MeV)

FIG. 9. Mass as a function of (left panel) C0 and (right panel) c1. Solid lines represent the results for λ ¼ −2, dashed lines for λ ¼ 0. 2 The results were obtained setting ϕ∞ ¼ð388 MeVÞ . work do not provide spontaneous symmetry breaking in the implies that we never reach the state f0ð550Þ, whose status chiral limit, and therefore we would not expect pseudo-NG is controversial anyway [29]. Meson mass increase with an modes. In the right panel of Fig. 8 we display the potential increasing quark mass is a behavior expected in QCD, so obtained in the holographic model investigated in [11], we conclude that the model with λ > 0 provides the best where the potential well allows a light state in the spectrum, scenario. It is worth mentioning that the masses of scalar as described in [27]. mesons do not vanish in the chiral limit. In Fig. 9 we show the evolution of the pseudoscalar meson masses with the parameter C0 (left panel) and c1 2. Spectrum of the axial-vector sector (right panel) compared to the results of the linear soft wall model, which are plotted with dashed lines. Note that the The Schrödinger equation for the axial-vector sector was masses increase with the quark mass parameter c1,as given in Eq. (53), and this time we consider the case of expected in QCD. However, since there is an unphysical λ > 0. Solving that equation, we find the axial-vector cutoff for the quark mass parameter c1 the pseudoscalar meson masses. In Fig. 11 we show the evolution of the meson masses grow very fast and tend to diverge when c1 masses as functions of the parameters C0 (left panel) and c1 reaches the cutoff. (right panel). The meson masses increase with the IR parameter C0 and the UV quark mass parameter c1. Meson E. Meson spectrum (λ > 0) mass increase with increasing quark mass is a behavior expected in QCD. In the chiral limit c1 → 0 the masses of As described in Sec. III D 1, the spectrum of vector axial-vector mesons and vector mesons become degenerate. mesons is insensitive to the nonlinear potential, so it is In Table III we present our results for the parameter identical to the linear case (λ ¼ 0). Below we describe the choice λ ¼ 7, C0 ¼ 0.3 fixed in the scalar sector. The spectrum of scalar, axial-vector, and pseudoscalar mesons λ results, labeled as NLSW, are compared with the linear soft for the case of positive , which corresponds to a Mexican wall model [7], the holographic model of [11], and hat potential in (18). experimental data [29]. We observe that our results are close to the results obtained in the linear soft wall model. 1. Spectrum of the scalar sector This is explained by the smallness of the tachyon field for The differential equation of the scalar sector, written in a Schrödinger form, was given in (51). The effect of going λ TABLE II. The masses of the scalar mesons (in MeV) obtained from negative to positive is displayed in Fig. 5.We in the nonlinear soft wall model with λ > 0 compared with the λ 0 concluded that states become heavier for > compared linear soft wall model [7], the holographic model of [11], and to the cases λ ¼ 0 and λ < 0. As described in the previous experimental data [29]. The values of the parameters are λ ¼ 7, 2 subsection, we consider the f0ð980Þ state as the first scalar C0 ¼ 0.3, and ϕ∞ ¼ð388 MeVÞ . state and use that value to fix the parameter C0 for a given λ. Our results for that parameter choice are displayed in n NLSW SW [7] GKK [11] f0 experimental [29] Table II, labeled as NLSW, compared against the linear soft 1 980 950 799 980 10 wall, the holographic model of [11], and experimental data. 2 1246 1227 1184 1350 150 1505 6 For λ ¼ 7 we obtain C0 ¼ 0.3 and find c1 ¼ 142.4 ðMeVÞ, 3 1466 1452 1466 which implies a large value for the quark mass. In Fig. 10 4 1657 1646 1699 1724 7 1992 16 we show the evolution of the scalar meson masses as 5 1829 1820 1903 6 1986 1978 2087 2103 8 functions of the parameters C0 (left panel) and c1 (right 7 2132 2125 2257 2314 25 panel). We observe that the masses increase with C0 and c1. 8 2268 2262 2414 The lower bound around 950 MeV for the first scalar meson

026013-12 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

1600 1500 1500 1400 1400 1300 1300

(MeV) (MeV) 1200

s 1200 s m 1100 m 1100 1000 1000 900 –1.0 – 0.5 0.0 0.5 1.0 0 100 200 300 400 500

C0 c1(MeV)

FIG. 10. Masses of the scalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for λ ¼ 2, 2 dashed lines for λ ¼ 0. The results were obtained by setting ϕ∞ ¼ð388 MeVÞ .

1600 1800

1400 1600 1400 1200 (MeV) (MeV) n n 1200 A A m 1000 m 1000 800 800 –1.0 –0.5 0.0 0.5 1.0 0 100 200 300 400 500

C0 c1(MeV)

FIG. 11. Masses of the axial-vector mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for 2 λ ¼ 2, dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ð388 MeVÞ . the parameters chosen to calculate the spectrum. For a 3. Spectrum of the pseudoscalar sector small tachyon field the last term in the potential (54) is The pseudoscalar sector is described by the coupled relatively small and the potential gets closer to the case of equations (55) and (56). The system was decoupled and the linear soft wall model. written in a Schrödinger form in Eq. (58), and this time we We point out that the spectrum of the vector and axial- consider λ > 0. Figure 12 shows our numerical results for vector mesons presented in Tables I and III is not the pseudoscalar meson masses as functions of the param- degenerate (the parameter c1 is outside the chiral limit). eters C0 (left panel) and c1 (right panel). Again, we observe In our model, the mass difference between the vector and that the masses increase with the quark mass parameter c1. axial-vector mesons at relatively large values of c1 is purely Note that in the chiral limit c1 → 0 the mass of the first a consequence of explicit chiral symmetry breaking, in pseudoscalar state has a finite value. This result can be contrast to QCD, where we expect another contribution interpreted as the absence of pseudo-NG bosons in the coming from spontaneous chiral symmetry breaking. spectrum and supports the background analysis leading to a vanishing chiral condensate in the chiral limit. TABLE III. The masses of the axial-vector mesons (in MeV) As described previously, the potential well in the obtained in the nonlinear soft wall model compared to the linear Schrödinger potential of the pseudoscalar sector, shown soft wall model [7], the holographic model of [11], and in the left panel of Fig. 8, is not deep enough to support experimental data [29]. The values of the parameters used are 2 the presence of a very light state. Therefore, the first state λ ¼ 7, C0 ¼ 0.3, and ϕ∞ ¼ð388 MeVÞ . behaves as a pion resonance instead of a true pseudo-NG n NLSW SW [7] GKK [11] a1 experimental [29] boson. This is in contrast to the Schrödinger potential of [11,27], shown in the right panel of Fig. 8, which is deep 1 897 891 1185 1230 40 2 1172 1168 1591 1647 22 enough to allow for a pseudo-NG boson. We will show in þ30 Sec. III F that all decay constants of the pseudoscalar 3 1398 1395 1900 1930−70 mesons in our model go to zero in the chiral limit c1 → 0, 4 1594 1592 2101 2096 122 2270þ55 characterizing them as resonances. 5 1770 1768 2279 −40 What new feature does the holographic model in [11] 6 1930 1928 7 2079 2077 have to allow for a pseudo-NG mode [27]? It is worth stressing that the main difference lies in the behavior of the

026013-13 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

1600 1500 1800 1400 1600 1300 (MeV) (MeV)

n 1400 n 1200 m m 1100 1200 1000 1000 900 –1.0 –0.5 0.0 0.5 1.0 0 100 200 300 400 500

C0 c1(MeV)

FIG. 12. Masses of the pseudoscalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for 2 λ ¼ 2, dashed lines for λ ¼ 0. The results were obtained by setting ϕ∞ ¼ð388 MeVÞ . dilaton field. In this model we consider a monotonically [11,27], and experimental data [29]. We also point out increasing positive dilaton, whereas the dilaton profile in that the masses of scalar and pseudoscalar mesons shown in [11] is negative in the UV and positive in the IR, with a Tables II and IV are not degenerate (the parameter c1 is minimum lying in the intermediate region; see Fig. 26 in outside the chiral limit). Appendix B. This difference in the dilaton profile has an We finish this subsection by showing in Fig. 13 the important consequence in the tachyon solution near the evolution of the parameters C0 (left panel) and c1 (right boundary. In the chiral limit the tachyon solution in our panel) when varying λ, matching the first scalar state with model vanishes, while the tachyon profile in [11,27] is still the meson f0ð980Þ. In the figure we observe why it is not nonzero. These two important differences have an effect on possible to get small values for c1 when λ is small. To get a the Schrödinger potential that allows for a pseudo-NG small quark mass, for example, mq ¼ 8 MeV, we would mode in [11,27]. We remark, however, that we have need c1 ¼ 2.21 MeV, and therefore the value of λ would be followed a more conservative approach of taking a positive very large, i.e., λ ≈ 3 × 104, with the corresponding value monotonically increasing dilaton to avoid instabilities in −3 for C0 ≈ 4.7 × 10 . Those results would have dramatic the meson spectrum. As described in Appendix B, the consequences in the spectrum because the IR parameter C0 dilaton profile in [11] leads to instabilities in the scalar would be so small that the contribution to the Schrödinger sector. It would be interesting if the model of [11] were to equations would be negligible and the spectrum of scalar allow for an extension that avoids those instabilities. and pseudoscalar mesons, as well as the vector and axial- We also note from Figs. 10 and 12 that in the chiral limit vector mesons, would be degenerate. c1 → 0 the masses of the pseudoscalar and scalar mesons reach the same values. We therefore conclude that the scalar and pseudoscalar mesons become degenerate in the chiral F. Decay constants limit. As explained previously, the vector and axial-vector In this subsection we calculate the decay constants of the mesons also become degenerate in the chiral limit. vector, axial-vector, and pseudoscalar mesons. In holo- The spectrum obtained using the parameters λ ¼ 7 and graphic QCD models the meson decay constants are related C0 ¼ 0.3, fixed in the scalar sector, is displayed in Table IV. to the normalization constants for the field perturbations; Our results, labeled as NLSW, are compared against the see, e.g., [34]. The normalization condition for the vector linear soft wall model [7], the holographic model of field is given by Z A −Φ TABLE IV. The masses of the pseudoscalar mesons (in MeV) dze s vmðzÞvnðzÞ¼δmn; ð60Þ obtained in the nonlinear soft wall model with λ > 0 compared to the linear soft wall model [7], the holographic model of [27], and λ 7 2 experimental data [29]. The values of the parameters are ¼ where As ¼ − ln z is the AdS warp factor, ΦðzÞ¼ϕ∞z is and C0 0.3. ¼ the dilaton, and vn is the vector meson mode related to the − wave function by v ðzÞ¼e BV ψ ðzÞ. In turn, the wave n NLSW SW [7] KBK [27] π experimental [29] n vn function ψ v satisfies the Schrödinger equation (47).As 1 …… 144 140 n described in Appendix A, the decay constants are related to 2 1100 951 1557 1300 100 3 1321 1227 1887 1816 14 4D conserved currents and they are defined through the 4 1518 1452 2090 2070 relations in Eq. (A21), 5 1697 1646 2270 2360 A −Φ 6 1861 1820 2434 e s 2 F ¼ −lim ∂ v ¼ N : ð61Þ 7 2013 1980 2586 vn ϵ→0 z n vn g5 z¼ϵ g5

026013-14 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

1.0 400 0.8 300

0 0.6 C (MeV)

1 200

0.4 c 0.2 100 0.0 0 0 5 10 15 20 0 5 10 15 20

FIG. 13. Evolution of (left panel) C0 as a function of λ and (right panel) c1 as a function of λ when the first scalar state is matched to the 2 meson f0ð980Þ. We remind the reader that ϕ∞ ¼ð388 MeVÞ fixed in the vector sector.

50 350 1 2 f 1 300 Fa1 40 250 f F12 30 2 200 a2 (MeV) (MeV) n 2 n / 150 12 20 f 3 a 1 f Fa3 F 100 10 50 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600

c1(MeV) c1(MeV)

FIG. 14. Decay constants of the (left panel) axial-vector mesons and (right panel) pseudoscalar mesons as a function of c1 obtained in 2 the NLSW model for ϕ∞ ¼ð388 MeVÞ and λ ¼ 2.

The normalization constant N appears as the UV coef- 1=2 ∼ 1 b 2 6 vn at large c1: approximately as Fa1 =c1, with b ¼ . , for ficient near the boundary of a vector mode, i.e., vnðzÞ¼ the first state. In Table V we display our numerical results 2 1 Nvn z þ, satisfying the normalization condition (60). for the ground state (n ¼ ) decay constants for the Thus, the decay constants are proportional to the normali- parameter choices λ ¼ 7 and C0 ¼ 0.3. zation constants of the field perturbations. We follow the For the pseudoscalar sector the normalization condition same procedure for the axial-vector sector, where the is given by [34] normalization condition is given by Z Z As−Φ 2 dze βðzÞð∂zπmÞð∂zπnÞ¼mπ δmn: ð64Þ As−Φ n dze amðzÞanðzÞ¼δmn; ð62Þ

− BA ψ There are different approaches to calculate the decay where amðzÞ¼e an ðzÞ is the axial-vector model sat- 2 constants of pseudoscalar mesons in the literature. First, isfying the UV behavior a ðzÞ¼N z þ and the m an we consider the prescription used in the hard wall model [4] normalization condition (62). N is the normalization an (see also [35,36]). The authors considered the massless constant and ψ a is the solution of the Schrödinger 2 n case, mπ ¼ 0. In that limit the pion decay constant can be equation (53). Thus, the decay constants are given by extracted from the axial current correlator by the relation

A −Φ e s 2 F ¼ −lim ∂ a ¼ N : ð63Þ an z n an TABLE V. The decay constants (in MeV) obtained in the ϵ→0 g5 ϵ g5 z¼ nonlinear soft wall model compared to the result obtained in The problem of finding decay constants has been reduced the linear soft wall model [7] and experimental results of PDG [29]. The results for λ > 0 were obtained by setting λ ¼ 7 and to the problem of calculating normalization constants of C0 ¼ 0.3. the field perturbations (or, equivalently, wave functions). In the left panel of Fig. 14 we display the results for the NLSW ðλ > 0Þ SW [7] Experimental [29] decay constants of axial-vector mesons as functions of the 1=2 Fρ 260.12 261 346.2 1.4 quark mass parameter c1. The decay constants do not vary 1=2 433 13 Fa1 215.37 261 significantly in the region of small c1, but they decrease fast

026013-15 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

A −Φ 2 e s hand, as the quark mass increases the pseudoscalar meson fπ ¼ −lim ∂ Að0; ϵÞ; ð65Þ ϵ→0 2 z decay constants display a nonmonotonic behavior and, in g5 particular, go to zero in the heavy quark limit c1 → ∞.Itis 0 ϵ worth mentioning that in the perturbative QCD approach where Að ; Þ is the non-normalizable solution for the pffiffiffiffiffi axial-vector field dual to the 4D axial-vector current. We for heavy quarks [39] one expects the behavior f ∼ 1= M point out that the prescription is valid only in the case of for meson decay constants (see also [40]), where M is the 2 mπ ¼ 0 and does not allow the investigation of pion mass of the heavy mesons. In our case, the numerical data resonances. displayed in the right panel of Fig. 14 indicate the ∼ 1 a 4 4 A prescription for calculating decay constants for the approximate behavior fπ1 =c1, with a ¼ . for the pion and their resonances was developed in [33]. Details on first pseudoscalar state. the derivation are given in Appendix A. The holographic Finally, we note in the right panel of Fig. 14 a crossing dictionary for the decay constant may be written in of the different curves. The hierarchy between the decay the form constants changes when going from the regime of small quark mass to the regime of heavy quark mass. Numerical A −Φ e s results are displayed in Table VI, for λ ¼ 7 and specific fπ ¼ −lim ∂ φ ðzÞ ; ð66Þ n ϵ→0 z n g5 z¼ϵ values of C0. We see that the decay constants increase with the radial number for C0 ¼ 0.4, corresponding to φ where nðzÞ is the normalized wave function satisfying the c1 ¼ 203.68 MeV, whereas for the case C0 ¼ 0.1, corre- normalization condition sponding to c1 ¼ 44.17 MeV, the decay constants decrease Z −Φ with the radial number. In the regime of small quark mass, eAs the hierarchy fπ >fπ >fπ was found in [33] for pion dz ð∂zφmÞð∂zφnÞ¼δmn: ð67Þ 1 2 3 βðzÞ resonances in the hard wall model. We have found the same hierarchy in the (nonlinear) soft wall model in the regime In terms of ∂ π the decay constant is obtained by plugging z n of small quark mass and, interestingly, we have found an Eq. (56) into Eq. (66): inversion of that hierarchy in the regime of heavy As−Φ quark mass. − e β ∂ π fπn ¼ lim 2 ðzÞ z nðzÞ : ð68Þ ϵ→0 g5mπ ϵ n z¼ IV. NONLINEAR SOFT WALL MODELS Hence, the normalization condition takes the form WITH RUNNING MASS Z Recent works in holographic soft wall models have −Φ 2 2 As β ∂ π ∂ π δ dze ðzÞð z mÞð z nÞ¼mπn mn: ð69Þ considered the possibility of a tachyon squared mass mX depending on the radial coordinate z [41,42]. The motivation Finally, in terms of the Schrödinger wave function defined for a tachyon running mass was to gain a nontrivial IR in Eq. (58), the decay constant takes the form contribution to the tachyon differential equation and therefore find a richer dynamics. In holographic QCD a 5D running ðAs−ΦÞ=2 e 1=2 mass for the tachyon would correspond to the anomalous fπ ¼ −lim β ðzÞψ π ðzÞ ; ð70Þ n ϵ→0 2 n dimension for the 4D quark mass operator [22,42]. g5mπ z ϵ n ¼ For the tachyon running mass we take the ansatz where the normalization condition is given by 2 −3 − ϕ 2 Z mXðzÞ¼ cz ; ð72Þ 2 dzψ π ðzÞψ π ðzÞ¼mπ δ : ð71Þ m n n mn with ϕC > 0. The tachyon differential equation now becomes In the end, the decay constant depends only on the λ normalization constant. Thus, the procedure above allows 2 2 2 2 3 ½z ∂z − ð3 þ 2ϕ∞z Þz∂z þ 3 þ ϕcz v − v ¼ 0: ð73Þ us to calculate decay constants of the fundamental state and 2 its resonances. In our calculations for the pseudoscalar mesons we have used the three formulas described above to TABLE VI. The decay constants (in MeV) obtained in the show the consistency of our numerical results. In the right nonlinear soft wall model. The results for λ ¼ 7 were obtained by setting C0 ¼ 0.4 ðc1 ¼ 203.68 MeVÞ, C0 ¼ 0.1 ðc1 ¼ 44.17 MeVÞ, panel of Fig. 14 we plot the decay constants as a function 2 and ϕ∞ ¼ð388 MeVÞ . of the parameter c1 of the first three pseudoscalar mesons. In the chiral limit the decay constants of all pseudoscalar fπ1 fπ2 fπ3 states go to zero linearly, i.e., fπ ∼ c1, which agrees with n NLSW ðC0 ¼ 0.4Þ 27.20 40.84 47.36 the observed result for pion resonances in the hard wall NLSW ðC0 ¼ 0.1Þ 19.63 17.25 15.25 model ([34]) and in QCD ([37]; see also [38]). On the other

026013-16 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

IR α A. Asymptotic analysis v ðyÞ¼C0y : ð77Þ In the UV we consider again the Frobenius ansatz Plugging this ansatz into Eq. (76), we obtain the poly- 3 3 5 5 nomial equation vðzÞ¼c1z þ d3z ln z þ c3z þ d5z ln z þ c5z þ …: ð74Þ λ α α2 4α 3 α−2 2αϕ ϕ − 3 3α 0 C0y ð þ þ ÞþC0y ð ∞ þ cÞ 2 C0y ¼ : Plugging this ansatz into Eq. (73), we find the UV ð78Þ coefficients α −1 α −1 1 Again, we distinguish among three cases: > , ¼ , 2λ 4ϕ − 2ϕ and α < −1. The last case is trivial because it leads d3 ¼ 4 c1ðc1 þ ∞ cÞ; to C0 ¼ 0. 1 2 2 In the case α > −1 the second term dominates and we d5 ¼ c1ð−c1λ − 4ϕ∞ þ 2ϕ Þð−c1λ − 12ϕ∞ þ 2ϕ Þ; 64 c c find 1 5 2 3 3 2 c5 ¼ ½−9c1λ − ð−24c1ϕc þ 56c1ϕ∞ þ 48c1c3Þλ 256 ϕc α − ; ϕ < 2ϕ∞: 79 2 2 ¼ 2ϕ c ð Þ − 12c1ϕc þ 64c1ϕcϕ∞ − 80c1ϕ∞ ∞

− 32c3ϕ − 192c3ϕ∞…: ð75Þ c This is a natural deformation of the regular solution found in the previous section. For 0 > α > −1 the solution is 2λ 4ϕ − 2ϕ 0 In the special case c1 þ ∞ c ¼ we have d3 ¼ actually divergent and admits the expansion 1 d5 ¼ 0 and c5 ¼ 4 ϕcc3, so we do not expect logarithmic ϕ terms, but there may be another solution besides the linear IR α β c v ðyÞ¼y ðC0 þ C2y þ …Þ; β ¼ 2 − ; ð80Þ solution vðzÞ¼c1z due to a nonzero c3. ϕ∞ In the IR we work with the variable y ¼ 1=z. Equation (73) becomes with λ 2 −2 −2 3 3λ ½ðy∂ Þ þ 2ð2 þ ϕ∞y Þðy∂ Þþð3 þ ϕ y Þv − v ¼ 0: C0 y y c 2 C2 ¼ − : ð81Þ 4ðϕc − 2ϕ∞Þ ð76Þ

We will focus on the case ϕc ¼ ϕ∞ where, in terms of z, the We take again the power ansatz tachyon solution reads

3 10

2 5 1 1 3 c c 0 0 –1/2 –3/2

–1 –5 –2

–3 –10 –50 0 50 –50 0 50

C0 C0 pffiffiffiffiffiffiffi FIG. 15. Numerical results for (left panel) c1 and (right panel) c3 as functions of C0 in units of ϕ∞. The blue and red solid lines (dashed lines) correspond to λ ¼ 2 and λ ¼ 5 (λ ¼ −2 and λ ¼ −5). The black dot-dashed line represents the case λ ¼ 0 (linear soft wall model with running mass).

026013-17 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020) 10 ffiffiffi 2λ 3 4λ2 − 10ϕ p 1 C0 −1 C0 ∞ −2 v ¼ C0 z þ z þ 2 z þ : 4ϕ∞ 32ϕ∞ 8 ð82Þ

This is a divergent solution depending on only one 6 pffiffiffi 3 ∝ c parameter, C0. The IR leading behavior v z was considered as an IR constraint in a previous approach [43]. –3/2 4 In the special case α ¼ −1 the first term in Eq. (78) vanishes, whereas the second and third terms lead to the 2 condition −C0λ þ 2ϕc − 4ϕ∞ ¼ 0. This linear solution, 2 i.e., vðzÞ¼C0z, is valid only for ðλ < 0; ϕc < 2ϕ∞Þ or (λ > 0; ϕc > 2ϕ∞). 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1. Running mass with λ =0 –1/2 c1 We finish this subsection by pointing out that the divergent solution (80) survives in the case λ ¼ 0. This FIG. 16. VEV parameter ffiffiffiffiffiffiffic3 as a function of the source pϕ λ corresponds to the linear soft wall model with running parameter c1, in units of ∞, for different values of . The ϕ ϕ blue and red solid lines (dashed lines) correspond to λ ¼ 2 and mass. In the case c ¼ ∞ this solution has the following λ ¼ 5 (λ ¼ −2 and λ ¼ −5). The black dot-dashed line represents UV and IR behavior: the case λ ¼ 0 (linear soft wall model with running mass). ϕ UV c1 ∞ 3 3 0 v ðzÞ¼c1z þ 2 z ln z þ c3z þ ; ð83Þ ffiffiffi 5 IR p −2 v ðzÞ¼C0 z 1 − z þ : ð84Þ 16ϕ∞ 4

B. Numerical solution 3

6 The numerical results for the nonlinear soft wall model =2 in the presence of a tachyon running mass are qualitatively

)×10 2 z

( similar to the case without the running mass. The main s V =0 effect of the running mass will be extending the range for 1 λ 0 =–2 the IR parameter C0. In particular, for the case > the 2 upper bound C0λ < 6 found in the previous section is not 0 present anymore. 0.000 0.002 0.004 0.006 0.008 0.010 0.012 We present numerical results for the case ϕ ¼ ϕ∞ z(MeV–1) c corresponding to the IR behavior (82). Figure 15 displays FIG. 17. Potential of the Schrödinger equation in the scalar the UV parameters c1 and c3 as functions of the IR 2 λ λ 0 sector for ϕc ¼ ϕ∞ ¼ð388 MeVÞ and three different values of parameter C0 for different values of . The case ¼ , the parameter λ. represented by black dot-dashed lines, corresponds to the

1400 1400 1200 1200 1000 1000 (MeV) (MeV) s s m m 800 800

600 600 –100 –50 0 50 100 150 200 250 300 350

C0 c1(MeV)

FIG. 18. Mass of the scalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for λ ¼ −2, 2 dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ .

026013-18 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

3000 2500 2500 2000 2000 (MeV) (MeV) n n A 1500 A 1500 m m 1000 1000

–60 –40 –20 0 20 40 60 0 50 100 150 200 250 300 350

C0 c1(MeV)

FIG. 19. Masses of axial-vector mesons as functions of (left panel) C0 and (right panel) c1. Solid lines represent the results for λ ¼ −2, 2 dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ .

2500 2500

2000 2000 (MeV) (MeV) n 1500 n

m m 1500

1000 1000 –60 – 40 – 20 0 20 40 60 150 200 250 300 350

C0 c1(MeV)

FIG. 20. Mass of the pseudoscalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for 2 λ ¼ −2, dashed lines for λ ¼ 0. The results were obtained by setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ .

1600 1500 1400 1400 1300 1200 1200 (MeV) (MeV) s s m

m 1100 1000 1000 900 800 –20 –10 0 10 20 0 200 400 600 800 100012001400

C0 c1(MeV)

FIG. 21. Masses of scalar mesons as functions of (left panel) C0 and (right panel) c1. Solid lines represent the result for λ ¼ 2, dashed 2 lines for λ ¼ 0. The results were obtained by setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ .

(linear) soft wall model with a running mass for the tachyon. Figure 16 displays c3 as a function of c1, which TABLE VII. Masses of the scalar mesons (in MeV) obtained in can be interpreted in terms of the 4D chiral condensate as a the nonlinear soft wall model with running mass compared to the function of the quark mass, as described in Sec. III C.For results of Refs. [11,42] and the experimental results of PDG [29]. the case λ < 0 we note a decrease in the range of c1 and c3 The values of the parameters used are λ ¼ 7 and C0 ¼ 7.6. compared to the case without running mass described in the previous section. For the case λ > 0, despite having a n NLSW-RM FLZ A [42] GKK [11] f0 experimental [29] bigger C0 range, we do not notice a significant difference in 1 980 586 799 980 10 c3 vs c1 compared to the case without the running mass. 2 1238 1346 1184 1350 150 Again, we conclude that the case λ > 0 provides the more 3 1455 1466 1505 6 realistic scenario for chiral symmetry breaking. 4 1645 1743 1699 1724 7 1992 16 One of the motivations for considering a 5D running 5 1816 2232 1903 6 1973 2420 2087 2103 8 mass for the tachyon was to gain a nontrivial dynamics in 2314 25 ϕ 7 2118 2257 the IR depending on the parameter c in Eq. (72). However, 8 2255 2414 we have found that the numerical results were very similar,

026013-19 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

2000 2000

1500 (MeV) (MeV) 1500 n n A A m m 1000 1000

–20 –10 0 10 20 0 200 400 600 800 100012001400

C0 c1(MeV)

FIG. 22. Mass of the axial-vector mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the results for λ ¼ 2, 2 dashed lines for λ ¼ 0. The results were obtained setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ . despite having this time a tachyon solution divergent in the light state, just as in the model without running mass; see IR. This may be related to the fact that the divergence is of Sec. III D 2. However, as in that case, we follow a more α the form z with 0 < α < 1. As in the case without running conservative approach and consider f0ð980Þ as the first mass, in the chiral limit c1 → 0 all the parameters go to scalar meson. zero. This is a negative result for nonlinear soft wall models In Fig. 18 we show the mass of the scalar mesons as a because they do not describe spontaneous symmetry function of the IR parameter C0 (left panel). From this figure breaking in the chiral limit, in constrast to QCD. we observe that the scalar meson mass decreases with C0 (solid lines), suggesting the possibility of a light state in the spectrum. The right panel of Fig. 18 shows the evolution of C. Meson spectrum (λ < 0) the scalar meson masses with the quark mass, i.e., c1 ∝ mq. As in the case without running mass, the vector meson is All of the masses decrease as the quark masses increase, insensitive to the tachyon dynamics, and therefore we focus which is not expected in QCD. This pathology of the case on the scalar, axial-vector, and pseudoscalar sectors. λ < 0 had arisen previously in the model without running mass and may be related to the absence of a minimum in the 1. Spectrum of the scalar sector Higgs potential (18) when λ < 0. The equation of the scalar sector is again Eq. (50), 2 2. Spectrum of the axial-vector sector but this time considering a running mass term mXðzÞ¼ 2 −3 − ϕ z . We focus on the case ϕ∞ ϕ , where the The Schrödinger equation describing the axial-vector c ¼ c pffiffiffi tachyon solution diverges in the IR region as v ∼ z. sector is the same as Eq. (53), this time with a running mass 2 −3 − ϕ 2 We expect a different behavior of the potential in the term mXðzÞ¼ cz . We display the results of the Schrödinger equation, i.e., Eq. (51), and consequently a evolution of the mass (for the first three states) as a function of the parameter C0 in the left panel of Fig. 19 with solid different spectrum. A plot of the potential is displayed in lines, while dashed lines represent the results for λ ¼ 0. Fig. 17 for different values of λ; we observe the main From this figure we see that the mass increases as the difference between the models with negative, zero, or λ λ 0 parameter C0 increases. The right panel of Fig. 19 displays positive . In the case < it is possible to find a very the axial-vector meson masses as functions of the quark mass parameter c1. The masses of the axial-vector mesons TABLE VIII. Masses of the axial-vector mesons (in MeV) initially increase slowly with the quark masses but then obtained in the nonlinear soft wall model with running mass all grow rapidly becoming divergent as the quark mass compared to the results of GKK and Fang-Liang-Zhang (FLZ) of parameter reaches its upper bound. As described previ- Refs. [11,42] and the experimental results [29]. The values of the ously, an upper bound for the quark mass is not expected in parameters used are λ ¼ 7, C0 ¼ 7.6, and QCD. In our model this upper bound arises as a saturation 2 ϕ∞ ¼ ϕc ¼ð388 MeVÞ . effect due to a negative quartic coupling λ for the Higgs potential (18). n NLSW-RM FLZ A [42] GKK [11] a1 experimental [29] 1 1147 1121 1185 1230 40 3. Spectrum of the pseudoscalar sector 2 1359 1608 1591 1647 22 þ30 The coupled equations of the pseudoscalar mesons are 3 1547 1922 1900 1930−70 2096 122 the same as Eqs. (55) and (56). Combining those equations, 4 1718 2156 2101 we reduced them into a Schrödinger form in Eq. (58). 2270þ55 5 1876 2352 2279 −40 Using the same numerical procedure applied in the case 6 2023 2526 without running mass, we are able to find the masses of 7 2161 pseudoscalar masses as functions of the parameters C0 and

026013-20 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

2200 2000 2000 1800 1600 (MeV) (MeV) n 1400 n 1500 m 1200 m 1000 1000 800 –20 –10 0 10 20 0 200 400 600 800 100012001400

C0 c1(MeV)

FIG. 23. Mass of the pseudoscalar mesons as a function of (left panel) C0 and (right panel) c1. Solid lines represent the result for λ ¼ 2, 2 dashed lines for λ ¼ 0. The results were obtained by setting ϕ∞ ¼ ϕc ¼ð388 MeVÞ . c1, We display the numerical results in Fig. 20. The left compatible with experimental results, this does not guar- panel of the figure shows the variation of the masses as antee a physical value for the quark mass. For example, the functions of C0 with solid lines, while the results for λ ¼ 0 parameter choice λ ¼ 7 and C0 ¼ 7.6 corresponds to a very are represented with dashed lines; we see that the masses of large value for the quark mass parameter c1, even larger pseudoscalar mesons always increase with C0. We also than the value obtained in the case without a running mass. point out that the mass increases faster close to C0 ¼ 0 and more slowly for large values of C0. The right panel of 2. Spectrum of the axial-vector sector Fig. 20 displays the evolution of the masses as a function of the quark mass parameter c1. We observe that the masses The axial-vector sector is described by the same equa- of pseudoscalar mesons increase slowly in the region of tions as were used in Sec. III D 3, this time with running 2 −3 − ϕ 2 small quark mass and faster in the intermediate and large mass term mXðzÞ¼ cz . The evolution of the quark mass region. Again, it seems that the masses diverge masses as functions of the parameter C0 (c1) is displayed when the quark mass parameter c1 reaches its upper bound. in the left panel (right panel) of Fig. 22. We see that the masses of axial-vector mesons increase slowly with c1 and λ D. Meson spectrum ( > 0) seem to reach asymptotic finite values in the limit c1 → ∞. In Table VIII we display the results for the parameter choice 1. Spectrum of the scalar sector ðλ ¼ 7;C0 ¼ 7.6Þ, which were fixed previously in the The potential of the Schrödinger equation in the scalar scalar sector. sector was displayed in Fig. 17 for λ ¼ 2. Our numerical It is worth mentioning that the spectrum of axial-vector results of the masses as functions of the parameter C0 (c1) mesons is very different from the spectrum of the vector are displayed in the left panel (right panel) of Fig. 21. mesons (see Table I), which means that neither sector is We see that the scalar meson masses increase with both degenerate. This nondegeneracy of the spectrum is parameters. This allows us to use the same strategy enhanced by the tachyon running mass because the tachyon implemented in Sec. III E 1 to fix the parameters; i.e., 980 field becomes divergent in the IR. This is also a signal that we use the state f0ð Þ. Having fixed the parameters, we chiral symmetry is never restored in the axial-vector excited calculate the spectrum, which is displayed in Table VII as states. NLSW-RM. Although we are able to find a spectrum 3. Spectrum of the pseudoscalar sector TABLE IX. Masses of the pseudoscalar mesons (in MeV) The pseudoscalar mesons are described by the same obtained in the nonlinear soft wall model with running mass equations as in Sec. III D 4, this time with running mass compared to the results of Kelley-Bartz-Kapusta (KBK) and 2 −3 − ϕ 2 FLZ of Refs. [27,42] and the experimental results [29]. The term mXðzÞ¼ cz . The evolution of the pseudosca- values of the parameters used are λ ¼ 7, C0 ¼ 7.6, and lar meson masses as functions of the parameter C0 (c1)is 2 ϕ∞ ¼ ϕc ¼ð388 MeVÞ . displayed in the left panel (right panel) of Fig. 23. As in the axial-vector sector, the masses in the pseudoscalar sector π n NLSW-RM FLZ A [42] KBK [27] experimental [29] increase slowly with c1 and seem to reach asymptotic 1 … 139.6 144 140 finite values in the limit c1 → ∞. Again, in the chiral limit 2 1301 1269 1557 1300 100 c1 → 0 the mass of the lightest state does not vanish. This 3 1479 1753 1887 1816 14 means that we do not have a pseudo-NG boson in the 4 1642 2051 2090 2070 spectrum, and the pseudoscalar mesons behave simply as 5 1796 2277 2270 2360 pion resonances. 6 1942 2467 2434 Our numerical results for the parameter choice λ ¼ 7 and 7 2081 2586 C0 ¼ 7.6 are displayed in Table IX as NLSW-RM. Since a

026013-21 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

60 350 f 300 1 2 50 1 Fa1 250 40 1 2 f 2 Fa 200 2 30 (MeV) (MeV) n 2 n 150 1 2 f 3 a 1 f Fa3 20 F 100 50 10 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600

c1(MeV) c1(MeV)

FIG. 24. Decay constants of the (left panel) axial-vector mesons and (right panel) pseudoscalar mesons as a function of c1 obtained in 2 the NLSW model with running masses for ϕ∞ ¼ ϕc ¼ð388 MeVÞ and λ ¼ 2.

pseudoscalar mesons are displayed in the right panel of TABLE X. The decay constants (in MeV) obtained in the Fig. 24. Again, the results are qualitatively similar to the nonlinear soft wall model with running mass compared to the results of Ref. [42] and the experimental results of PDG [29]. case without a running mass, which is displayed in the right The results were obtained setting λ ¼ 7, C0 ¼ 7.6, and panel of Fig. 14. In particular, we find again an inversion of 2 ϕ∞ ¼ ϕc ¼ð388 MeVÞ . hierarchy for the pseudoscalar meson decay constants. Considering the specific parameter choice λ ¼ 7, C0 ¼ 7.6 NLSW-RM ðλ > 0Þ FLZ A [42] Experimental [29] that fixes the mass of the first scalar state, we may calculate 1=2 Fρ 260.12 296 346.2 1.4 the decay constants of the first state of the vector and axial- 1=2 433 13 vector mesons. These results are displayed in Table X as Fa1 152.78 389 NLSW-RM. In Table XI we display the decay constants for the first λ 7 pseudo-NG boson is missing in the spectrum, the model at three pseudoscalar mesons in the case ¼ and specific hand allows us to calculate only the spectrum of pion values for C0. These results show explicitly the inversion of resonances. We also compare our results against the results hierarchy for the pseudoscalar decay constants when going available in Refs. [27,42] and experimental data [29]. from the regime of small quark mass to the regime of heavy quark mass; see the end of Sec. III F.

E. Decay constants V. DISCUSSION AND CONCLUSIONS In this subsection we calculate the decay constants of the vector, axial, and pseudoscalar mesons in the nonlinear soft In this work we investigated a nonlinear realization for wall model with running mass. As explained in Sec. III F, chiral symmetry breaking in soft wall models based on a the decay constants are related to the normalization con- Higgs potential. Soft wall models allow for a more realistic dition on the field perturbations. Again, we investigate the description of the meson spectrum because a positive evolution of the decay constants of axial-vector and quadratic dilaton in the IR guarantees linear Regge trajec- pseudoscalar mesons as a function of the quark mass tories. Solving the nonlinear differential equation of the parameter c1. In the left panel of Fig. 24 we display the tachyon, we found that the tachyon solution in the IR evolution of the decay constants of the first three axial- depends on only one parameter C0, Integrating numerically vector mesons; the results are qualitatively similar to the the tachyon differential equation, we found nontrivial case without running mass (see left panel of Fig. 14). In relations between the UV parameters c1 and c3 and the turn, the evolution of the decay constants of the first three IR parameter C0. Moreover, implementing the procedure of holographic renormalization, we obtained a dictionary for the 4D chiral condensate in terms of the VEV parameter c3. TABLE XI. Decay constants of the first three pseudoscalar This allowed us to find the evolution of the 4D chiral mesons (in MeV) obtained in the nonlinear soft wall model with condensate with the quark mass. For the case λ > 0, running mass. The results were obtained by setting λ ¼ 7, corresponding to a Mexican hat Higgs potential, we found C0 ¼ 7.6 ðc1 ¼ 237.2 MeVÞ, C0 ¼ 0.2 ðc1 ¼ 4.25 MeVÞ, and ϕ ϕ 388 2 that the chiral condensate grows nonlinearly with the quark ∞ ¼ c ¼ð MeVÞ . mass, as expected in QCD. We found, however, that a nonlinear Higgs potential for the tachyon is not sufficient to fπ1 fπ2 fπ3 provide spontaneous symmetry breaking in the chiral limit NLSW-RM ðC0 ¼ 7.6Þ 21.81 37.58 48.11 because the chiral condensate vanishes as the quark mass NLSW-RM ðC0 ¼ 0.2Þ 2.45 1.93 1.63 goes to zero.

026013-22 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

We calculated the spectrum of the scalar, vector, axial- breaking in the chiral limit. Running mass models were vector, and pseudoscalar mesons. The spectrum of vector considered in [41,42,45] as an attempt to allow for sponta- mesons decoupled from the other sectors and their solu- neous symmetry breaking in the chiral limit. We found, tions was the same as in the linear soft wall model. We however, that in a consistent description of nonlinear soft concluded that the case λ > 0, corresponding to a Mexican wall models based on a Higgs potential, the tachyon running hat Higgs potential, provides the most realistic scenario for mass does not solve the lack of spontaneous symmetry the meson spectrum. The spectrum of the scalar mesons breaking in the chiral limit. An interesting approach to the presented a pathology in the case λ < 0 because the masses problem of chiral symmetry breaking is to consider a decreased with the quark mass. In the case λ > 0 we found negative profile for the dilaton [9]. A negative dilaton profile that it was possible to match the scalar state f0ð980Þ state has some issues; see, for instance, [46] for a discussion. For by fixing appropriately the parameters λ and C0. The example, it would violate the null energy condition in the analysis of the spectrum showed us that nonlinear soft gravitational background [47]. Nevertheless, Chelabi et al. wall models led to a nondegeneracy between vector and [9] proposed that the profile of the dilaton in the UV may be axial-vector mesons and also between scalar and pseudo- negative, while in the IR it must be positive in order to scalar mesons. However, we found that it was not possible guarantee confinement and Regge-like behavior. They claim to fit the meson spectrum with a small quark mass that in this way it is possible to describe spontaneous chiral parameter; see the end of Sec. III E 3. We calculated the symmetry breaking in the chiral limit. decay constants of vector, axial-vector, and pseudoscalar In conclusion, nonlinear soft wall models based on a mesons. The vector meson decay constants were insensitive Higgs potential, with or without a tachyon running mass and to the tachyon dynamics and hence the quark mass. The a positive quadratic dilaton, do not provide spontaneous axial-vector meson decay constants decreased with the chiral symmetry breaking in the chiral limit. Consequently, quark mass, whereas the pseudoscalar meson decay con- there are no pseudo-NG bosons in the spectrum of pseudo- stants presented a nonmonotonic behavior consistent with scalar mesons. This conclusion is supported by the study of pion resonances. We found, in particular, that all of the masses and decay constants in the region of small quark pseudoscalar decay constants vanished in the chiral limit, mass. We found, however, in the case of λ > 0 avery indicating the absence of pseudo-NG bosons. A separate reasonable behavior for the chiral condensate in the regime comment deserves mention concerning the decreasing of large quark mass, similar to the behavior expected in λ 0 behavior of the pseudoscalar decay constants in the regime QCD. For the case > we also found a reasonable of heavy quark mass, which agrees qualitatively with the behavior for all meson masses as growing functions of perturbative QCD prediction; see the end of Sec. III F. the quark mass. Interestingly, we found that the decay As an attempt to allow for spontaneous symmetry constants of axial-vector mesons and pseudoscalar mesons breaking in the chiral limit, we also investigated nonlinear decreased in the regime of heavy quark mass. This behavior soft wall models with a tachyonic running mass. The is also expected in QCD and encourages us to continue the consequence of the running mass was to increase the investigation of nonlinear soft wall models. Finally, a natural extension of this work would be the investigation of back- interval of C0 in relation to the model without a running mass. However, we realized that the results were qualita- reacted Einstein-dilaton backgrounds, where the confine- tively similar to the case without a running mass. Again, we ment criterion is satisfied. This would allow for a consistent found that λ > 0 provides the most realistic scenario for the description of confinement and chiral symmetry breaking in meson spectrum and decay constants, and we were able to a minimal holographic setup. again match the f0ð980Þ state in the scalar sector by fixing appropriately the parameters. Again, we found that fixing ACKNOWLEDGMENTS the parameter to describe the meson spectrum led to a very The authors would like to thank Carlisson Miller for large value for the quark mass. We found that the non- the stimulating discussions during the early stages of this degeneracy between vector and axial-vector mesons was project. The authors would also like to acknowledge the very enhanced by the running mass, and the same was true for useful conversations with Saulo Diles, Diego Rodrigues, the scalar and pseudoscalar mesons. Jonathan Shock, and Dimitrios Zoakos during the develop- Let us discuss briefly some other approaches to the ment of this work. A. B.-B. would like to thank the problem of chiral symmetry in holographic QCD. The organizers of WONPAQCD 2019 for providing a stimulating model proposed in [10] (see also [44]), which implemented environment. The work of A. B.-B. is partially funded by the ideas of [21], claimed that the tachyon must blow up in Conselho Nacional de Desenvolvimento Científico e the IR when the background has confinement properties. Tecnológico (CNPq) Grants No. 306528/2018-5 and This statement may be related to the Coleman-Witten No. 434523/2018-6. L. A. H. M. has financial support from theorem [1]. As in our case, the IR tachyon solution in Coordenação de Aperfeiçoamento do Pessoal de Nível [10] depends on only one parameter. In contrast to our case, Superior—Programa Nacional de Pós-Doutorado [(PNPD/ the model in [10] led to spontaneous chiral symmetry CAPES) Brazil].

026013-23 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

m ml ml m APPENDIX A: EQUATIONS OF MOTION where PS , PV;a, PA;a, and Pπ;a are the conjugate momenta AND DECAY CONSTANTS associated with the scalar, vector, axial-vector, and pseu- In this Appendix we write details about the derivation doscalar fields, respectively. Explicitly, the conjugate of the equations of motion used to calculate the meson momenta are given by spectrum as well as the holographic dictionary for meson ∂L ffiffiffiffiffiffi decay constants. We will expand the action (19) up to m 2 −4 −ϕp− ∂m PS ¼ ¼ e g S; second order on the fields and take the 5D metric as ∂ð∂mSÞ in Eq. (1). −ϕ ∂L2 e pffiffiffiffiffiffi Pml − −gvml; We follow [34] (see also [48]). For simplicity, we take V;a ¼ ∂ ∂ a ¼ 2 a ð mVl Þ g5 Nf ¼ 2 and assume isospin symmetry (mu ¼ md). First, we −ϕ decompose the bifundamental field X in the form ∂L2 e pffiffiffiffiffiffi Pml − −gaml; A;a ¼ ∂ ∂ a ¼ 2 a ð mAl Þ g5 1 2 πa a ∂L ffiffiffiffiffiffi X ¼ e i T vðzÞþS ; ðA1Þ m 2 − −ϕp− 2 ∂mπa − m;a 2 Pπ;a ¼ a ¼ e gv ðzÞð A Þ: ðA6Þ ∂ð∂mπ Þ πa μ a where ðx ;zÞ is the pseudoscalar field, T are the In turn, the derivatives of the Lagrangian are generators of SUð2Þ, and Sðxμ;zÞ is the scalar fluctuation related to the scalar mesons. We also rewrite the fluctuation ∂L2 pffiffiffiffiffiffi ∂L2 for the gauge fields as −e−ϕ −g 4m2 − 6v2 z S; −0; ∂ ¼ ½ X ð Þ ∂ a ¼ S Vl ðL=RÞ ∂L2 −ϕpffiffiffiffiffiffi 2 ∂L2 Am ¼ Vm Am; ðA2Þ e −gv z ∂lπa − Al;a ; 0: A7 ∂ a ¼ ð Þð Þ ∂πa ¼ ð Þ Al m m a m m a where V ¼ Va T and A ¼ Aa T are the vector and axial fields, respectively. Plugging Eqs. (A1) and (A2) into From these results we find that the equation of motion of a m m the action (19) and expanding on the fields π , Va , and Aa the vector sector takes the form up to second order, we obtain S ¼ S0 þ S2 þ, with S0 ϕ ffiffiffiffiffiffi the effective 1D action for the background vðzÞ and e −ϕp ml pffiffiffiffiffiffi ∂mðe −gva Þ¼0: ðA8Þ Z −g 5pffiffiffiffiffiffi −Φ 2 2 2 2 2 S2 ¼ − dx −ge 2ð∂mSÞ þ 2m S þ 3λv ðzÞS X The Abelian field strength was defined in Eq. (A4). 1 1 Working in the axial gauge Va 0, we get mn a mn a z ¼ þ 2 va vmn þ 2 aa amn 4g5 4g5 − Φ −Φ Asþ ∂ As ∂ □ 0 1 e zðe zVν;aÞþ Vν;a ¼ : ðA9Þ 2 ∂ π − 2 þ 2 v ðzÞð m a Am;aÞ ðA3Þ The equation of motion of the scalar sector takes the form the action describing the kinetic terms for the 5D field eϕ ffiffiffiffiffiffi 3 fluctuations πa, Vm, and Am. We have defined the Abelian ffiffiffiffiffiffi∂ −ϕp− mn∂ − 2 λ 2 0 a a p− mðe gg nSÞ mX þ2 v S ¼ ; ðA10Þ tensors g

a a a a a a which may be written in the form vmn ¼ ∂mVn − ∂nVm;amn ¼ ∂mAn − ∂nAm: ðA4Þ −3 Φ 3 −Φ 2 2 3 2 To obtainR the Euler-Lagrange equations, we write the action e Asþ ∂ e As ∂ S □S − e As m λv S 0: 5 zð z Þþ X þ 2 ¼ as S2 ¼ dx L2. The variation takes the form Z ðA11Þ ∂L ∂L 5 2 m 2 ml a δS2 ¼ d x − ∂ P δS þ − ∂ P δV ∂S m S ∂Va m V;a l l The remaining equations of motion are ∂L2 ∂L2 ml a m a ϕ þ − ∂mP δA þ − ∂mPπ;a δπ ffiffiffiffiffiffi ∂Aa A;a l ∂πa e −ϕp ml 2 2 l a l;a l pffiffiffiffiffiffi ∂ ðe −ga Þþv ðzÞg5ð∂ π − A Þ¼0; Z −g m a 5∂ mδ ml δ a ml δ a m δπa ffiffiffiffiffiffi þ dx mðPS S þ PV;a Vl þ PA;a Al þ Pπ;a Þ; −ϕp 2 2 m a m;a ∂m½e −gv ðzÞg5ð∂ π − A Þ ¼ 0: ðA5Þ ðA12Þ

026013-24 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

a We work in the axial gauge, Az ¼ 0, and decompose the Plugging Eq. (A18) into the vector and axial-vector μ μ ∂μφ l gauge field as Ab ¼ Ab⊥ þ b, where Ab⊥ is the trans- currents (A17), we obtain the expansions μ verse part and ∂ φb the longitudinal part. The tranverse X As−Φ part leads to the equation of motion for the axial-vector μ − e ∂ ˆ μ hJV;ai¼ zva;nðzÞ Va;nðxÞ; sector g5 n X A −Φ −Φ −Φ μ;a μ;a 2 2 2 μ;a μ e s μ As ∂ As ∂ □ − As 0 − ∂ ˆ e zðe zA⊥ Þþ A⊥ v ðzÞg5e A⊥ ¼ ; hJA;ai¼ zaa;nðzÞ Aa;nðxÞ n g5 ðA13Þ X A −Φ e s μ þ − ∂zφa;nðzÞ ∂ πˆ a;nðxÞ: ðA19Þ For the pseudoscalar sector we find the coupled equations n g5

As−Φ As−Φ a 2 2 2As a a e ∂zðe ∂zφ Þþv ðzÞg5e ðπ − φ Þ¼0; ðA14Þ On the other hand, the meson decay constant are defined by the following relations: 2 2 2 −∂ □φa þ v ðzÞg e As ∂ πa ¼ 0: ðA15Þ z 5 z 0 μ b;m λ −ip·xϵμ λ δab h jJV;aðxÞjV ðp; Þi ¼ Fva;m e ðp; Þ ; 0 μ b;m λ −ip·xϵμ λ δab The dictionary for the decay constants is obtained from the h jJA;aðxÞjA ðp; Þi ¼ Faa;m e ðp; Þ ; holographic currents. The latter arise in the surface term of μ b;m −ip·x ab 0 J x π p fπa;m e δ : A20 Eq. (A5), which may be written as h j A;að Þj ð Þi ¼ ð Þ Z The quantities Fva;m , Faa;m , and fπa;m are the decay constants δ Bdy − 4 δ μ δ a S2 ¼ dx ðhJsið SÞþhJV;aið VμÞ of the vector, axial-vector, and pseudoscalar mesons. Comparing Eqs. (A19) and (A20), we arrive at the holo- μ δ a δπa þhJA;aið AμÞþhJπ;aið ÞÞz¼ϵ: ðA16Þ graphic dictionary for meson decay constants

−Φ −Φ The VEV of 4D operators appearing in Eq. (A16) is eAs eAs Fv ¼ − ∂zva;nðzÞ;Fa ¼ − ∂zaa;nðzÞ defined by a;n g5 a;n g5 −Φ ffiffiffiffiffiffi eAs z −ϕp z − ∂ φ hJsi¼Ps ¼ −4e −g∂ S; fπ ¼ z a;nðzÞ: ðA21Þ a;n g5 −Φ μ μ e ffiffiffiffiffiffi z − p− zμ hJV;ai¼PV;a ¼ 2 gv ; g5 −Φ APPENDIX B: THE GKK MODEL: A REVIEW μ μ e ffiffiffiffiffiffi z − p− zμ hJA;ai¼PA;a ¼ 2 ga ; g5 In this section we summarize the model investigated ffiffiffiffiffiffi z −Φp 2 a z;a in [11], known as the GKK model (see also [27]). This hJπ i¼Pπ ¼ −e −gv ðzÞð∂π − A Þ: ðA17Þ ;a ;a model was motivated by the original soft wall model [7], μ μ which considers a quadratic dilaton from the UV to the IR. We identify hJ i and hJ i as the holographic vector V;a A;a In turn, the GKK model proposes reconstructing the dilaton and axial currents leading to the meson decay constants. profile by rewriting the tachyon differential equation as The next step is to decompose the fields into their Kaluza- Klein modes as follows:

X∞ ˆ Sðx; zÞ¼ snðzÞSnðxÞ; n¼0 X∞ μ ˆ μ Vaðx; zÞ¼g5 va;nðzÞVa;nðxÞ; n¼0 X∞ μ;⊥ ˆ μ Aa ðx; zÞ¼g5 aa;nðzÞAa;nðxÞ; n¼0 X∞ πaðx; zÞ¼g5 πa;nðzÞπˆ a;nðxÞ; n¼0 X∞ FIG. 25. Profile of the tachyon. To get this figure, we fix φ φ πˆ aðx; zÞ¼g5 a;nðzÞ a;nðxÞ: ðA18Þ the parameters as follows: κ ¼ 15, mq ¼ 9.75 MeV, Σ ¼ 3 2 n¼0 ð204.5 MeVÞ and ϕ∞ ¼ 0.1831 GeV .

026013-25 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

FIG. 26. (Left panel) Profile of the dilaton field. (Right panel) Profile of the derivative of the dilaton. Both results were obtained by 3 2 setting κ ¼ 15, mq ¼ 9.75 MeV, Σ ¼ð404.5 MeVÞ , and ϕ∞ ¼ 0.1831 GeV .

FIG. 27. (Left panel) Potential of the Schrödinger equation associated with the scalar mesons. (Right panel) First wave functions 3 associated with the scalar mesons; see Table XII. Both results were obtained by setting κ ¼ 15, mq ¼ 9.75 MeV, Σ ¼ð204.5 MeVÞ , 2 and ϕ∞ ¼ 0.1831 GeV .

3 2 vðzÞ¼Az þ BCz þ; ðB4Þ ∂zv 2 2 κ 2 v ∂ Φ − AsðzÞ − 3∂ z ðzÞ¼∂ e mX 2v ∂ þ zAsðzÞ: ðB1Þ zv zv where A ∝ mq and BC ∝ Σ. In the way the parameters were defined in Eq. (B3), we can see that in the chiral limit, i.e., We observe that this equation depends on the tachyon mq → 0, Σ ≠ 0, which means that the spontaneous chiral field. Thus, to solve this equation, we must know the symmetry breaking is nonzero. In the IR the tachyon tachyon. On the other hand, the asymptotic expansion of reduces to the tachyon field close to the boundary takes the form rffiffiffiffiffiffiffi 3 v ¼ c1z þ c3z . In turn, in the IR the tachyon is linearly ϕ∞ v z A B z 2 z: divergent, v ∼ z. Additional constraints were imposed by ð Þ¼ð þ Þ ¼ κ ðB5Þ the phenomenology; see [11] for details. Thus, the following interpolation function recovers the asymptotic behavior A plot of the tachyon field is shown in Fig. 25. In turn, the in the UV and IR: dilaton field and its derivative are displayed in the left and right panels of Fig. 26, respectively. As we can see, we notice 2 that the dilaton field becomes negative in a small region vðzÞ¼zðA þ B tanh ðCz ÞÞ; ðB2Þ close to the boundary. However, regarding the analysis developed in [47], a negative dilaton violates the null energy where the parameters of the model are defined as condition in the gravitational side, where the dilaton must Φ0 0 6 pffiffiffiffiffiffi rffiffiffiffiffiffiffi pffiffiffiffiffiffi rise monotonically so that ðzÞ > . Hence, perhaps this N m ϕ∞ N m 2πΣ A ¼ c q ;B¼ 2 − c q ;C¼ pffiffiffiffiffiffi : 2π κ 2π 6 NcB It is worth mentioning that the analysis done by Kiritsis and ðB3Þ Nitti in [47] states that the dilaton field should be monotonically increasing. However, as the soft wall model is not backreacted, perhaps the null energy condition could be “relaxed” in some Close to the UV Eq. (B2) takes the form manner, at least locally.

026013-26 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

2 TABLE XII. The masses (in MeV) obtained in the modified λ < 0. We fix the parameter ϕ∞ ¼ð388 MeVÞ . Then we version of the soft wall model, including the quartic interaction solve the differential equation (20) numerically while term, compared to the results of [11] and the experimental results considering a family of two parametric solutions in the [29]. The values of the parameters used are κ ¼ 15, 9 75 Σ 404 5 3 ϕ 0 1831 2 UV and a family of one parametric solution in the IR. We mq ¼ . MeV, ¼ð . MeVÞ , and ∞ ¼ . GeV . use as the boundary condition the asymptotic solution in n Model GKK [11] f0 experimental [29] the IR (26). In the following analysis, we focus in the 250 energy scale in the UV such that the energy belongs to the 1 748 i 799 550þ −150 103; 106 2 799 1184 980 10 interval ½ MeV, which is equivalent to the interval ∈ 10−6 10−3 −1 3 1184 1466 1350 150 of the holographic coordinate z ½ ; MeV , 4 1465 1699 1505 6 which lies close to the boundary. Thus, the problem was 5 1698 1903 1724 7 reduced to solving a one parameter family of solutions in 6 1902 2087 1992 16 the IR and a two parameter family of solutions in the UV. 7 2087 2257 2103 8 What is expected is a nontrivial relationship between these 8 2256 2414 2314 25 parameters, which is obtained by numerically solving Eq. (20). Our numerical results for c3 as a function of c1 are displayed in Fig. 28 for different values of λ. From this figure, we observe that there are solutions aside from pathological behavior will have consequences in the ≠ 0 spectrum. the trivial solution with c3 in the chiral limit, i.e., c1 → 0, which corresponds to the limit of spontaneous chiral symmetry breaking. 1. Scalar sector We point out that, from the set of solutions showed in We have a special interest in the spectrum of the scalar Fig. 28, the physical solutions are those for which the sector of this model. Now let us compute the spectrum. For tachyon field is a monotonic increasing function. The doing that, we must rewrite the perturbation equation in the corresponding solutions for c1 as a function of C0 are Schrödinger form and solve it using a shooting method, for displayed in Fig. 29, whereas the solutions for c3 as a example. A plot of the potential is shown in the left panel of function of C0 are displayed in Fig. 30. However, when Fig. 27. The results of the spectrum are displayed in the first computing the spectrum of the vector mesons, for example, column of Table XII, where we see that the first state has an we obtain an inconsistency when we calculate the potential 2 imaginary mass, which indicates an instability, i.e., ms < 0. of the Schrödinger equation, which is given by It is worth mentioning that this instability was reported previously in [43] (see also [49]). To guarantee that this 15 2 2 V ¼ þ ϕ∞z þ 2ϕ∞: ðC1Þ state is, in fact, a solution of the Schrödinger equation, we V 4z2 plot the first wave functions in the right panel of Fig. 27.All of these results were obtained using the same parameters as 10−6 −1 At zmin ¼ MeV , the first term of the last equation is in [11]. 10 2 leading such that VV ∼ ð10 MeVÞ . On the other hand, at 10−3 −1 zmax ¼ MeV , the first term is still the leading APPENDIX C: NUMERICAL ANALYSIS— 3 2 V ∼ ð10 MeVÞ , meaning that the potential is a mono- NONLINEAR SOFT WALL MODEL V tonic decreasing function well. We also realized that the Here we write some details of our numerical results convergence of the asymptotic solution (26) is not obtained investigating the tachyon field in the model for guaranteed.

FIG. 28. Numerical results of the nonlinear soft wall model. The corresponding parameters are (left panel) λ ¼ −1 and (right panel) λ ¼ −20.

026013-27 ALFONSO BALLON-BAYONA and LUIS A. H. MAMANI PHYS. REV. D 102, 026013 (2020)

4000 1000 2000 500 0 0 (MeV) (MeV) 1 1 c c –2000 –500 –4000 –1000 –10 –5 0 5 10 –3 –2 –1 0 1 2 3

C0 C0

FIG. 29. Numerical results for the nonlinear soft wall model. The corresponding parameters are (left panel) λ ¼ −1 and (right panel) λ ¼ −20.

1.0 1.5

11 1.0 11 0.5 0.5 )×10 )×10 3 0.0 3 0.0 –0.5 (MeV (MeV

3 –1.0 3 –0.5 c c –1.5 –1.0 –10 –5 0 5 10 –3–2–10123

C0 C0

FIG. 30. Numerical results for the nonlinear soft wall model. The corresponding parameters are (left panel) λ ¼ −1 and (right panel) λ ¼ −20.

In conclusion, the above analysis shows us that there is spontaneous chiral symmetry breaking in the nonlinear soft wall model for λ < 0. However, it arises in the UV. This conclusion may be justified since the confinement scale introduced by the dilaton field is “fake” because the dilaton is introduced by hand and the backreaction on the metric neglected.

[1] S. R. Coleman and E. Witten, Phys. Rev. Lett. 45, 100 [11] T. Gherghetta, J. I. Kapusta, and T. M. Kelley, Phys. Rev. D (1980). 79, 076003 (2009). [2] Y. Kinar, E. Schreiber, and J. Sonnenschein, Nucl. Phys. [12] H. Boschi-Filho and N. R. F. Braga, J. High Energy Phys. 05 B566, 103 (2000). (2003) 009. [3] J. M. Maldacena, Phys. Rev. Lett. 80, 4859 (1998). [13] A. Cherman, T. D. Cohen, and E. S. Werbos, Phys. Rev. C [4] J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, Phys. 79, 045203 (2009). Rev. Lett. 95, 261602 (2005). [14] C. Csaki and M. Reece, J. High Energy Phys. 05 (2007) 062. [5] L. Da Rold and A. Pomarol, Nucl. Phys. B721,79 [15] U. Gursoy, E. Kiritsis, and F. Nitti, J. High Energy Phys. 02 (2005). (2008) 019. [6] J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 [16] A. Ballon-Bayona, H. Boschi-Filho, L. A. H. Mamani, A. S. (2002). Miranda, and V. T. Zanchin, Phys. Rev. D 97, 046001 [7] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys. (2018). Rev. D 74, 015005 (2006). [17] K. Ghoroku, N. Maru, M. Tachibana, and M. Yahiro, Phys. [8] C. McNeile, A. Bazavov, C. T. H. Davies, R. J. Dowdall, K. Lett. B 633, 602 (2006). Hornbostel, G. P. Lepage, and H. D. Trottier, Phys. Rev. D [18] P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau, and S. 87, 034503 (2013). Nicotri, Phys. Rev. D 78, 055009 (2008). [9] K. Chelabi, Z. Fang, M. Huang, D. Li, and Y. L. Wu, J. High [19] S. J. Brodsky, G. F. de Teramond, H. G. Dosch, and J. Energy Phys. 04 (2016) 036. Erlich, Phys. Rep. 584, 1 (2015). [10] I. Iatrakis, E. Kiritsis, and A. Paredes, J. High Energy Phys. [20] O. Bergman, S. Seki, and J. Sonnenschein, J. High Energy 11 (2010) 123. Phys. 12 (2007) 037.

026013-28 NONLINEAR REALIZATION OF CHIRAL SYMMETRY BREAKING … PHYS. REV. D 102, 026013 (2020)

[21] R. Casero, E. Kiritsis, and A. Paredes, Nucl. Phys. B787,98 [35] H. J. Kwee and R. F. Lebed, J. High Energy Phys. 01 (2008) (2007). 027. [22] M. Jarvinen and E. Kiritsis, J. High Energy Phys. 03 (2012) [36] H. J. Kwee and R. F. Lebed, Phys. Rev. D 77, 115007 (2008). 002. [37] A. Holl, A. Krassnigg, and C. D. Roberts, Phys. Rev. C 70, [23] D. Arean, I. Iatrakis, M. Järvinen, and E. Kiritsis, Phys. Lett. 042203 (2004). B 720, 219 (2013). [38] A. Krassnigg, C. D. Roberts, and S. V. Wright, Int. J. Mod. [24] M. Jarvinen, J. High Energy Phys. 07 (2015) 033. Phys. A 22, 424 (2007). [25] D. Areán, I. Iatrakis, M. Järvinen, and E. Kiritsis, J. High [39] A. V. Manohar and M. B. Wise, Heavy Quark Physics Energy Phys. 11 (2013) 068. (Cambridge University Press, Cambridge, England, [26] A. Karch, A. O’Bannon, and K. Skenderis, J. High Energy 2009), https://doi.org/10.1017/CBO9780511529351. Phys. 04 (2006) 015. [40] P. Maris and P. C. Tandy, Nucl. Phys. B, Proc. Suppl. 161, [27] T. M. Kelley, S. P. Bartz, and J. I. Kapusta, Phys. Rev. D 83, 136 (2006). 016002 (2011). [41] A. Vega and I. Schmidt, Phys. Rev. D 82, 115023 (2010). [28] C. P. Herzog, Phys. Rev. Lett. 98, 091601 (2007). [42] Z. Fang, Y. L. Wu, and L. Zhang, Phys. Lett. B 762,86 [29] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, (2016). 030001 (2018). [43] Y. Q. Sui, Y. L. Wu, Z. F. Xie, and Y. B. Yang, Phys. Rev. D [30] V. Kaplunovsky and J. Sonnenschein, J. High Energy Phys. 81, 014024 (2010). 05 (2011) 058. [44] I. Iatrakis, E. Kiritsis, and A. Paredes, Phys. Rev. D 81, [31] M. Ihl, M. A. C. Torres, H. Boschi-Filho, and C. A. B. 115004 (2010). Bayona, J. High Energy Phys. 09 (2011) 026. [45] L. X. Cui, Z. Fang, and Y. L. Wu, Eur. Phys. J. C 76,22 [32] L. Da Rold and A. Pomarol, J. High Energy Phys. 01 (2006) (2016). 157. [46] A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, J. High [33] A. Ballon-Bayona, G. Krein, and C. Miller, Phys. Rev. D 91, Energy Phys. 04 (2011) 066. 065024 (2015). [47] E. Kiritsis and F. Nitti, Nucl. Phys. B772, 67 (2007). [34] A. Ballon-Bayona, G. Krein, and C. Miller, Phys. Rev. D 96, [48] Z. Abidin and C. E. Carlson, Phys. Rev. D 80, 115010 (2009). 014017 (2017). [49] D. Li and M. Huang, J. High Energy Phys. 11 (2013) 088.

026013-29