PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.110.094008ARTICLESStrong InteractionsDilatonic dynamics of baryonic crystals, branes, and spheresDILATONIC DYNAMICS OF BARYONIC CRYSTALS, BRANES, …BERSINI, D’ALISE, SANNINO, AND TORREShttps://orcid.org/0000-0001-5176-915XBersiniJahmall1,*https://orcid.org/0000-0003-1249-782XD’AliseAlessandra2,3,4,†SanninoFrancesco2,3,4,‡TorresMatías2,3,5,§Kavli IPMU (WPI), UTIAShttps://ror.org/02chw6z69, The University of Tokyohttps://ror.org/057zh3y96, Kashiwa, Chiba 277-8583, JapanDepartment of Physics E. Pancini, Università di Napoli Federico IIhttps://ror.org/05290cv24, via Cintia, 80126 Napoli, ItalyINFN sezione di Napolihttps://ror.org/015kcdd40, via Cintia, 80126 Napoli, ItalyQuantum Theory Center (ℏQTC) at IMADA and D-IAS, Southern Denmark Universityhttps://ror.org/03yrrjy16, Campusvej 55, 5230 Odense M, DenmarkPhysique Théorique et Mathématique Université libre de Bruxelleshttps://ror.org/01r9htc13 and International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium
7November20241November2024110909400830January202422September2024Published by the American Physical Society2024authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
We systematically analyze the impact of dilatonic dynamics on Skyrme spheres, crystals, and branes. The effects of the dilatonic model parameters, encompassing different underlying near-conformal dynamics, on the macroscopic properties of skyrmions such as their mass and radius are discussed. For spheres and crystals we identify special values of the ratio of the decay constants for which the second order differential equations reduce to a solvable first order system. Additionally, in the case of the crystals, the dilaton presence spatially separates the baryon and isospin charge distributions. For branes, we show how the dilaton smooths out their configurations. Our results are expected to have wide implications from the study of near-conformal dynamics stemming from QCD-like theories to phenomenological investigations of nuclear matter in extreme regimes.
Agencia Nacional de Investigación y Desarrollo10.13039/5011000208847221039World Premier International Research Center Initiative Ministry of Education, Culture, Sports, Science and Technology10.13039/501100001700Japan Society for the Promotion of Science10.13039/501100001691JP23K19047Carlsbergfondet10.13039/501100002808CF22-0922INTRODUCTION
Strong dynamics is notoriously hard to tackle both analytically and numerically. Over several decades, methodologies have been devised to access different dynamical regimes of strongly interacting theories. At sufficiently low energies, chiral Lagrangians have been shown to faithfully describe the dynamics of specific strongly coupled underlying theories in terms of their Goldstone bosons. This approach has landed precious information on the dynamics and spectrum of various models of which quantum chromodynamics (QCD) is the time-honored example. The approach is also routinely employed to inform (and extract information from) first principle numerical simulations. Another remarkable property of chiral Lagrangians is that their nonperturbative solutions describe extended objects that, depending on the boundary conditions (BCs), can be identified with distinct physical configurations, from crystals to branes and spheres. The spherical case (the hedgehog solution) is the oldest example dating back to the pioneering work of Skyrme [1]. Once the topological sector of the theory is taken into account [2,3], the properly quantized hedgehog solutions describe half-integer states with topological charge identified as the baryon number. These extended states are taken to be the nucleons of the underlying theory. Astonishingly, the same model can simultaneously describe mesonic degrees of freedom and their scattering properties, as well as the spectrum and form factors of extended baryonic states, including the Goldstone boson scattering off them. Therefore, the same effective Lagrangian coefficients control both the pion dynamics as well as the baryon spectrum and form factors. The operators of the chiral effective Lagrangian can be organized in the number of derivative and pion masses. The solitonic solutions can be mapped into baryon states of the underlying QCD-like dynamics at large number of colors. The dependence on the number of colors is naturally encoded in the pion decay constant and depends on the underlying fermion representation with respect to the asymptotically free gauge group. The Skyrme model has also been enriched by adding at the Lagrangian level massive vector mesons that have been shown to play an important role, not only to give a deeper understanding of the origin of the Skyrme term but also for the associated phenomenological consequences [4–7]. Finally, a less explored avenue that still deserves much attention, especially after the renewed experimental interest in heavy baryon spectroscopy and transitions [8–10], is the study of heavy baryons as bound states of a Skyrme soliton and heavy mesons [11–23]. The original idea pioneered by Callan and Klebanov [11] and first tailored for baryons made by one strange quark and two light quarks could, however, account only for half of the physical states possible in QCD. The issue was resolved analytically in [24] by consistently introducing excited heavy mesons. When working with heavy quarks such as the charm and the bottom ones, the approach hugely benefits from the marriage between large number of color dynamics and the celebrated heavy quark limit of QCD [25–31].
Beyond the traditional phenomenological nuclear and particle physics applications, here we systematically investigate the solitonic dynamics when the underlying gauge-fermion theory is near conformal. The region in the number of flavors versus the number of colors of asymptotically free gauge theories where the infrared theory is conformal is dubbed “conformal window.” The phase diagram structure for QCD and QCD-like theories appeared in [32,33]. At the lower edge of the conformal window the theory undergoes a quantum phase transition from an infrared conformal field theory to a phase characterized by both conformal and chiral symmetry breaking [34,35]. When the transition is sufficiently smooth, close to the lower end of the conformal window, the theory exhibits a near-conformal phase characterized by the existence of a region of the renormalization group (RG) flow in which the coupling remains nearly constant signaling the occurrence of “walking dynamics” [36–38]. Walking behavior lies at the core of many phenomenological models of dynamical electroweak symmetry breaking, e.g., within the technicolor [32,33,38–43] and (fundamental partial) composite Goldstone Higgs scenarios [44–52].
The infrared dynamics of the near-conformal theory can be modeled by augmenting the standard chiral Lagrangian via the introduction of a new light scalar degree of freedom with the same quantum numbers of the vacuum which is commonly referred to as the “dilaton” or “radion.” In this scenario, the dilaton is the Goldstone boson stemming from the spontaneous breaking of scale invariance, whereas sources of explicit conformal breaking leading to the near-conformal phase can be encoded in the effective dilaton potential. The aforementioned phenomenological applications have motivated several investigations of the resulting dilaton effective field theory (EFT) [42,53–69].
In this work, we present several classes of extended objects emerging as solitons of the dilaton augmented EFT. We identify crystal, brane, and sphere phases constituted, respectively, by ordered arrays of hadronic tubes, layers, and spherical hedgehog solitonic solutions at nonvanishing baryon number.
Until very recently, it was considered hard to construct analytical solutions representing baryonic condensates in the low-energy limit of QCD. However, in Refs. [70–77] an exact method to build hadronic solitons has been introduced. These solutions disclose intriguing nonperturbative features of the finite density phase diagram. For these reasons, it is very interesting to analyze what happens when the dilatonic degree of freedom is taken into account.
Skyrmions under the effects of dilaton dynamics have already been employed to describe dense nuclear matter such as the core of neutron stars [78–86]. In this framework, conformal invariance is seen as an emergent hidden symmetry of real-world QCD at high densities as suggested by both theoretical models and neutron star data [80,86–88]. Starting from the pioneering work of Brown and Rho [89], who introduced the dilaton mode to derive a series of scaling relations among the values of decay constants and masses in vacuum and at finite density, the dilaton EFT is considered to be a key ingredient for studying skyrmion matter in extreme conditions. These relations are shown to be modified when considering generalized EFTs [90]. While earlier works considered a simplified version of the effective action, the construction has later been refined in [91] and since then the dilaton EFT has become one of the building blocks of the generalized nuclear effective field theory, which aims at the description of hadronic matter from low to compact star densities (see, e.g., [92,93] for recent reviews). However, to the best of our knowledge, no comprehensive analysis of the spherical skyrmion properties in near-conformal theories has been performed.
The overarching goal of the present work is to combine numerical and analytical methods to perform a systematic investigation of skyrmion solutions in the presence of dilatonic dynamics. Applications range from the dynamics of QCD-like theories close to the lower end of the conformal window to nuclear matter in extreme regimes.
We structure our work as follows. After introducing the dilaton augmented EFT in Sec. II, we present a systematic numerical analysis in Sec. III. More specifically, we employ the standard hedgehog ansatz [2] and determine relevant physical quantities such as the skyrmion profile, mass, and size as a function of the physical parameters responsible for the deviations from conformal dynamics. Concretely, these are the anomalous dimension of the quark mass operator, the dilaton mass, and the conformal dimension of the relevant operator triggering the RG flow away from the infrared fixed point. We unveil an intricate dependence of the solitonic properties on these parameters as a consequence of the competition of dilaton and pion mass terms. We discover that the dilaton decouples rapidly from the dynamics once its mass exceeds the mass of the Goldstone modes. Moreover, we show that the dilaton decouples faster when the anomalous dimension of the chiral condensate is large.
We further show the existence of a special value of the ratio of dilaton and pion decay constants where the coupled second order field equations of the dilaton augmented EFT reduce to a first order solvable system. Therefore, the construction of analytic solitonic solutions is achieved through the identification of a novel special point in the parameter space of the theory. Interestingly, such a point does not correspond to any bound on the energy. This is a surprising result since, without the dilaton, the field equations with the spherical hedgehog ansatz are not solvable. However, we find that the so-constructed solutions, which feature a negative topological charge B=-1, are singular at the origin and do not carry finite energy.
In Sec. IV, we study the impact of the dilaton on the skyrmion crystals investigated in [74–77]. After briefly reviewing the analytical solution in the absence of the dilaton, we show that even for the emerging hadronic tubes there exists a special parameter point where the coupled second order equation of motions (EOMs) reduce to a first order system [again without a Bogomol’nyi-Prasad-Sommerfield (BPS) bound on the energy]. The special parameter point is shown to be related to the one observed in the spherically symmetric case by mapping into each other the respective EOMs. We further notice that the presence of the dilaton spatially separates the baryon and isospin charge distributions similar to spin-charge separation discussed in [94,95].
Finally, it has been proposed that nuclear matter at low temperatures may exist in nonuniform branelike structures which may be realized in the crusts of neutron stars [96]. In fact, these configurations appear as ground states in numerical simulations where nucleons are treated as classical bodies with pairwise interactions (see, e.g., [97] for a recent review). Intriguingly, a class of analytic solutions of the Skyrme model describing solitonic brane configurations has been discovered in [70–73,77]. While it is tempting to identify these exact solutions with the results of the numerical studies, establishing a quantitative connection is not straightforward. In particular, a full quantization of the solitonic solutions appears to be extremely involved [73], and, therefore, their phenomenological relevance remains an interesting open problem that lies beyond the scope of the present work. Instead, inspired by the possible emergent scale symmetry in dense skyrmion matter, we study the impact of the dilaton on the analytic brane solutions in Sec. V. In particular, assuming that the fields depend at most on time and a single spatial coordinate, we find the general brane solution to the EOMs of the dilaton-dressed Skyrme model. We discover that, while the soliton profile is left unaltered by the presence of the dilaton, the latter acquires a homogeneous vacuum expectation value which is nontrivially determined by the soliton properties and the dilaton potential. For any given topological charge, we determine the lower bound on the dilaton mass such that the solutions exist. Finally, we discuss how the dilaton smooths out the brane structure at low values of the dilaton mass.
We offer our conclusion in Sec. VI. The Appendix summarizes our numerical results for the spherical skyrmion solutions in Tables I–VI.
DILATON DRESSING OF THE CHIRAL LAGRANGIAN
The Skyrme Lagrangian for Nf=2 massive flavors in four dimensions reads I[U]=∫d4x-gK2Tr(12RμRμ+λ16FμνFμν+mπ22(U+U†)),Rμ=U-1∇μU=Rμjtj,Fμν=[Rμ,Rν].Here U∈SU(2), ∇μ is the covariant derivative, and ta=iσa are the generators of the SU(2) algebra, being σa the Pauli matrices. Here mπ is the pion mass, while K is related to the meson decay constant via fπ=K. The above theory can describe low-energy quantum chromodynamics or any other underlying quantum field theory featuring the same pattern of chiral symmetry breaking.
The theory admits a conserved topological current Jμ given by Jμ=εμνλρTr(RνRλRρ).Integrating the zero component of Jμ on a time-slice surface Σ we obtain the topological charge B=124π2∫ΣεijkTr(RiRjRk).This quantity corresponds to the winding number associated with U and is interpreted as the baryon number. The classical equations of motion are obtained by performing the functional variation with respect to U, yielding ∇μ(Rμ+λ4[Rν,Fμν])-mπ22(U-U†)=0.These correspond to a set of three nonlinear coupled partial differential equations (PDEs).
In this work, we explore the solitonic dynamics in a quasiconformal regime by adding to the action a dilatonic degree of freedom. The latter is introduced to restore conformal invariance at the action level and provide a mechanism for breaking conformality in a controllable manner. The pion mass term acts as an independent and controllable parameter for explicit conformal symmetry breaking. We, therefore, dress every operator Ok of mass dimension k with the dilaton field σ as [98,99]Ok→e(k-4)σfOk,where under a scale transformation x→eαx the field σ transforms as σ→σ-αf.Here f is the length related to the spontaneous breaking of scale invariance. Possible sources of explicit conformal breaking can be modeled by perturbing the underlying conformal theory via δLO=λOO,with O an operator with dimension Δ≠4 and λO the associated coupling. The presence of such an operator generates the following effective dilaton potential [65,66,68,99–101]: V(σ)=f-4e-4σf∑n=0∞cne-n(Δ-4)fσ,where the coefficients cn depend on the microscopic theory and scale as cn∼λOn[66,68,99,101]. In the present work, we are interested in cases where the explicit conformal breaking is small which, in turn, can be realized when λO≪1 and/or Δ→4. In the first case, one can truncate the expansion (9) and obtain [61,66,68,99,101]V(σ)=mσ2e-4fσ4(4-Δ)f2(1-4Δe-(Δ-4)fσ)+O(λO2),where we introduced the dilaton mass mσ and required that the ground state is realized for σ=0. In the EFT spirit, the nature of the conformal breaking deformation and its conformal dimension Δ are left unspecified.
The potential (10) has also been related to the gluon condensate ⟨G2⟩ and the usual soft dilaton theorems [61,68]. However, it has recently been argued that the only value of Δ consistent with the soft theorems is Δ=2[102].
When Δ=2, Eq. (10) reduces to the usual ϕ4 Higgs-like potential, whereas when Δ vanishes the coefficient c1 becomes parametrically larger than c0 and the potential (10) diverges. In Sec. III, we will conduct our numerical investigations in the range 1≤Δ≤4. On the other hand, by subtracting the infinite constant
The potential in (10) is strictly defined for any Δ>0. One way to derive the Δ=0 limit from (10) is to perform an ad hoc subtraction that depends on the underlying theory and yields (11). This subtraction, however, does not affect the equations of motion.
appearing in the expansion of Eq. (10) around Δ=0, one obtains the dilaton potential considered in the classic work of Coleman [98], VΔ→0(2)(σ)=-mσ216f2(1-4fσ-e-4fσ).For the sake of completeness, we will also consider the traditional Coleman potential in the following analyses.
In the limit Δ→4 we have that O becomes a near marginal operator and it is legitimate to expand Eq. (9) in powers of Δ-4 as V(σ)=-mσ2e-4fσ16f2(1+4fσ)+O((Δ-4)2).The form of the potential in (12) agrees with the counting scheme proposed in [57,58,60,62,64]. Moreover, at the considered order in the EFT expansion, Eq. (12) coincides with the potential derived in [67] starting from the partially conserved dilatation current relation and has often been employed to describe dense skyrmion matter, see, e.g., [78,79,82,85,89]. Crucially, since Eq. (12) can be seen as a subcase (the Δ→4 limit) of Eq. (10), in this work we will consider the generic form of the dilaton potential appearing in Eq. (10) to encompass all the proposals considered in the literature so as to keep our analysis as general as possible without committing to a specific model.
We can now also take into account the explicit breaking of conformal symmetry stemming from the presence of quark masses. We achieve this by assuming the mass term to have dimension y=3-γ, where 0<γ<2 is the anomalous dimension of the chiral condensate and its range is limited by the unitarity bound. Moreover, since for γ≃1 the underlying four-fermion operator becomes nearly marginal, we will focus on the interval 0<γ<1.
We, therefore, arrive at the following dilaton augmented chiral Lagrangian: I[U,σ]=∫d4x-g[K2Tr(12RμRμe-2σf+λ16FμνFμν+mπ22(U+U†)e-yσf)-12e-2σf∇μσ∇μσ-V(σ)].Although we restricted the number of flavors to two, we stress that conformality can be realized with such a small number of flavors in a number of theories, such as the ones with fermions in the two index symmetric or antisymmetric representation of the gauge group as shown first in [32] and generalized in [33]. In any case, here we focus on the impact of the novel dilatonic dynamics on time-honored solitons first discussed for the two and three flavor cases.
In order to investigate these extended objects, one starts with the equations of motion that read ∇μ(Rμe-2σf+λ4[Rν,Fμν])-mπ22(U-U†)e-yσf=0,∇μ(e-2σf∇μσ)+f∇μσ∇μσe-2σf-∂σV(σ)-K2fTrRμRμe-2σf-K4yfmπ2Tr(U+U†)e-yσf=0,while the stress-energy tensor Tμν=-2-g∂-gL∂gμν is given by Tμν=-K2Tr[(RμRν-12gμνRαRα)e-2σf+λ4(gαβFμαFνβ-14gμνFαβFαβ)-mπ22gμν(U+U†)e-yσf]-(-∇μσ∇νσ+12gμν(∇σ)2)e-2σf-gμνV(σ).The presence of the mass term shifts the value of the minimum of the dilaton potential from σ=0 to a finite value that we indicate with ⟨σ⟩=σf. The latter is determined by the following equation: ∂σV(σ)|σ=σf+Kyfmπ2e-yfσf=0.This leads to new values for the decay constants and the masses of all the states appearing in the Lagrangian satisfying certain scaling relations [62,64–66]. Equipped with the above, we are now ready to investigate several classes of solitonic configurations starting with the celebrated hedgehog solution.
DILATON AUGMENTED HEDGEHOG SKYRMIONSSetup
In this section, we investigate the modification to the time-honored hedgehog solution in the presence of dilatonic dynamics. Let us start with a general parametrization for U(x), U±1(xμ)=cos(α)12×2±sin(α)nata,n1=sin(Θ)cos(Φ),n2=sin(Θ)sin(Φ),n3=cos(Θ),where 12×2 is the 2×2 identity matrix and α(x), Θ(x), and Φ(x) are the three scalar degrees of freedom of SU(2). The hedgehog ansatz for SU(2) in spherical coordinates {r,θ,ϕ} reads α=α(r),Θ=θ,Φ=ϕ.The energy and topological charge densities are, respectively, given by ε=T00=12e-2fσ(2Ksin2(α)r2+Kα′2+σ′2)+λKα′2sin2(α)r2+λKsin4(α)2r4-Kmπ2cos(α)e-fyσ+V(σ),ρB=J0=-12α′sin2(α)r2,where the prime denotes the derivative with respect to the radial coordinate. The baryon mass MS and charge B are, respectively, obtained by integrating the energy and charge densities of the solution over the whole spacetime, that is Ms=4π∫0∞r2εdr,B=16π∫0∞r2ρBdr.A measure of the size of the solitonic object is given by the root-mean-square radius of the baryon charge ⟨r2⟩1/2 defined as ⟨r2⟩1/2≡(16π∫0∞ρBr4dr)1/2=-2π∫0∞r2sin2(α)α′dr.
Inserting the hedgehog ansatz in (5) and (15) we obtain the following set of coupled second order differential equations: (2λe2fσsin2(α)+r2)α′′-2rα′(frσ′-1)+λe2fσsin(2α)(r2α′2-sin2(α))r2-sin(2α)-mπ2r2sin(α)e-f(y-2)σ=0,σ′′-fσ′2+2rσ′-e2fσ∂σV(σ)+fKα′2+2r2fKsin2(α)-Kfmπ2ycos(α)ef(2-y)σ=0.In the next section, we will numerically solve the above equations to investigate the dependence of the skyrmion properties on the parameters Δ,mσ,y encoding the explicit breaking of conformal invariance.
Numerical solutions
To study the impact of dilatonic dynamics, we numerically solve the EOMs (24) and (25) in the sector of topological charge B=1. We measure all the dimensionful quantities in units of K (in QCD K=93MeV) and as a starting point we consider the following QCD-inspired values [2,66,78]: f=0.29,λ=14.752,mπ=14093.We then solve the EOMs for different values of the parameters Δ, mσ, y. In particular, we consider mσ=1,2,…,10 and Δ=1,1.2,1.4,…,3.8. Moreover, we study the limiting cases Δ→4 and Δ→0 corresponding to the dilaton potentials (12) and (11), respectively. For every value of Δ and mσ, we determine the profile functions α(r) and σ(r), the baryon mass MS, and the root-mean-square radius of the baryon charge ⟨r2⟩1/2. Since the topologically trivial vacuum exhibits a divergent zero-point energy, we define MS as the energy difference between the solutions with B=1 and B=0. To simplify the presentation, we momentarily set y=5/2 and postpone to Sec. III B 2 the discussion of the dependence on the anomalous dimension of the chiral condensate. However, as we shall see, all the conclusions below apply to any value of y in the range 2<y<3.
To set the boundary conditions, we impose that for r→∞ the fields approach the topologically trivial vacuum. We recall that at fixed values of time, the field U(r) defines a map from the spatial manifold R3 to the isospin manifold S3 with the boundary condition that U(r) goes to its trivial vacuum for asymptotically large distances. In other words, we have U(r→∞)=1⇒α(r→∞)=0,U(0)=-1⇒α(0)=π,where the latter condition fixes the baryon charge to unity [2]. Analogously, we impose σ(r→∞)→σf with σf given in Eq. (17). Since σf depends on the values of the parameters we vary, this can be seen as a fixed BC for the normalized variable χ(r)≡e-f(σ(r)-σf),namely, χ(r→∞)→1. We note that one of the main differences with respect to earlier numerical studies [78,79] is that we do consider the dependence of σf on the pion mass. Finally, we impose χ′(0)=0 in line with previous investigations [78,79].
We study the dilaton potential (10) for 1≤Δ<4 and its Δ→4 limit (12) for Δ=4. We do not consider values of Δ smaller than 1 being the potential (10) ill defined in the Δ→0 limit.
Figure 1 shows the dilaton profile χ(r) for Δ=1–3 and mσ=1,…,10. The general trend displayed in the figures can be summarized as follows.
At fixed Δ the dilaton profile flattens toward its asymptotic value χ=1 as mσ increases.
The case mσ=1 constitutes an exception, since in this case the dilaton profile stays below the one obtained for mσ=2 only for small values of the radius.
At fixed mσ the dilaton profile flattens toward its asymptotic value χ=1 as Δ decreases.
The profile α(r)/π is depicted in Fig. 2 for the same choice of the parameters. As mσ increases, the dilaton decouples from the dynamics and α(r) converges rapidly to the solution obtained in the absence of the dilaton. In general, the solutions exhibit little dependence on Δ.
110.1103/PhysRevD.110.094008.f1
Dilaton profile χ(r) for Δ=1–3 and mσ=1–5 (left) and mσ=6–10 (right).
210.1103/PhysRevD.110.094008.f2
Skyrmion profile α(r) for Δ=1–3 and mσ=1–5 (left) and mσ=6–10 (right).
Our results for MS are collected in Appendix A 1 and illustrated in Fig. 3 for mσ=3, 5, 7, 10. For all the values of the parameters, the relative numerical error is less than 0.5%, where such a value should be seen as a conservative upper limit. One can see that at fixed values of mσ the baryon mass gets smaller as Δ increases. On the other hand, the dependence of MS on mσ at fixed Δ is not monotonic. In fact, MS first decreases with increasing mσ until it reaches its minimum for mσ∼5, after which it climbs toward its Skyrme model value MSkyrme=15.67. This behavior is illustrated in Fig. 4. A change in the behavior of the solutions around mσ=5 can be also seen by analyzing the values of ⟨r2⟩1/2, which are given in Tables II and IV in Appendix A 1. In fact, ⟨r2⟩1/2 increases with increasing mσ until mσ∼5 and then starts getting smaller for higher values of the dilaton mass. Moreover, while ⟨r2⟩1/2 grows monotonically with Δ for mσ=1,2,…,5, 6, the opposite holds for mσ=7–10 as can be seen from Fig. 3.
However, for mσ=6, the root-mean-square radius shows almost no dependence on Δ.
310.1103/PhysRevD.110.094008.f3
MS (left) and ⟨r2⟩1/2 (right) as a function of Δ for mσ=3, 5, 7, 10.
410.1103/PhysRevD.110.094008.f4
The skyrmion mass MS as a function of the dilaton mass mσ for Δ=1 (purple), Δ=2 (red), Δ=3 (orange), and Δ=4 (yellow). The dashed line marks the value of the skyrmion mass in the absence of the dilaton (MSkyrme=15.67 in units of K).
Finally, it is important to stress that the dependence of the skyrmion properties on both Δ and mσ becomes quickly milder as we increase mσ, implying that the dilaton decouples rapidly from the dynamics once its mass exceeds mπ. For instance, while for mσ=1 both MS and ⟨r2⟩1/2 differ more than 20% from their values in the absence of the dilaton, such a difference reduces to less than 3% for mσ≥3.
Skyrmions in the <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> limit
As previously discussed, the potential (10) diverges in the Δ→0 limit. For completeness, we now discuss the solitonic solution in the presence of the potential (11), which can be seen as a regularized version of the Δ=0 case. To this end, we use the values of the parameters considered in the previous section.
The profiles of the fields are displayed in Fig. 5. Their behavior as we vary mσ is similar to the one discussed in the 1≤Δ≤4 case with the dilaton profile becoming more flat and converging faster to χ=1 as mσ increases, while α(r) converges quickly to the Skyrme model profile. However, note the exception in the dilaton profile for small values of mσ, with the mσ=1 profile staying below the mσ=2 one only for small values of r. The behavior of the skyrmion mass and the root-mean-square radius as a function of mσ is illustrated in Fig. 6 and mimics the one found in the previous section. In fact, MS decreases, reaches its minimum around mσ∼5, and then grows toward the Skyrme model value. At the same time, ⟨r2⟩1/2 mirrors such a behavior. Our results for MS and ⟨r2⟩1/2 are listed in Appendix A 2.
510.1103/PhysRevD.110.094008.f5
Dilaton profile χ(r) (left) and α(r)/π (right) for the dilaton potential (11) and mσ=1–10.
610.1103/PhysRevD.110.094008.f6
The skyrmion mass MS (left) and ⟨r2⟩1/2 (right) as a function of the dilaton mass mσ. The dashed line marks the values obtained in the absence of the dilaton MSkyrme=15.67 and ⟨r2⟩Skyrme1/2=0.215.
The role of the anomalous dimension of the chiral condensate
The time is ripe to discuss the impact of the anomalous dimension of the chiral condensate on the skyrmion properties, i.e., the dependence on y. First, we note that all the conclusions drawn for the y=5/2 case in the previous sections hold for every 2<y<3. The dilaton profile χ(r) becomes flatter and converges earlier to its asymptotic value as y decreases. At the same time, the skyrmion profile α(r) approaches the profile in the absence of the dilaton as y decreases, signaling the decoupling of σ(r) from the dynamics. To illustrate this behavior, we show the profiles obtained for Δ=1 and mσ=2 and different values of y in Fig. 7.
710.1103/PhysRevD.110.094008.f7
Dilaton profile χ(r) (left) and α(r)/π (right) for mσ=2 and Δ=1.
MS and ⟨r2⟩1/2 get, respectively, larger and smaller as y varies from y=2 to y=3, as shown in Fig. 8. Moreover, the dependence on y gets weaker as the dilaton decouples for larger mσ. Finally, our results for MS and ⟨r2⟩1/2 confirm that the dilaton decouples faster for smaller y. For instance, this can be seen by looking at how the dependence of MS (and ⟨r2⟩1/2) on both Δ and mσ becomes softer as y decreases. This last point is exemplified in Fig. 9.
810.1103/PhysRevD.110.094008.f8
MS (left) and ⟨r2⟩1/2 (right) as a function of y for Δ=1 and mσ=1–5.
910.1103/PhysRevD.110.094008.f9
The skyrmion mass MS as a function of Δ for mσ=2 and y=2.01, 2.1, 2.3, 2.5, 2.7, 2.9, 2.99 (left) and as a function of mσ for Δ=1 and y=2.01, 2.1, 2.3, 2.5, 2.7, 2.9, 2.99 (right).
Numerical solutions in the chiral limit
We conclude our numerical analysis with a brief discussion of the solitonic solution in the chiral limit mπ=0. Here the topological trivial vacuum state occurs for U=1 and σ=0. Therefore, the BC for χ(r) at infinity no longer depends on the value of mσ and Δ. Keeping the same numerical values adopted in Sec. III B we solve the EOMs for Δ=1,1.2,…,4 and mσ=0–4. The first observation is that all the solitonic properties are nearly independent on Δ for any value of mσ. This is a direct consequence of the fact that the dilaton ground state value vanishes for any Δ. The dilaton χ(r) and α(r)/π profiles are displayed in Fig. 10 for Δ=1 and mσ=0–4. Analogously to mπ≠0 case, the dilaton profile progressively flattens as we increase its mass, exhibiting a faster convergence to unity. At the same time, the associated α(r)/π profile shows almost no dependence on mσ. The baryon mass MS is shown in Fig. 11. The dependence on Δ is similar (MS gets smaller as Δ increases) but much weaker than in the mπ≠0 case for all values of mσ. For instance, for mσ=1 the relative variation of the solitonic mass along the whole range of values of Δ is only ∼0.07%, to be compared with ∼15% obtained for mπ=140/93 in Sec. III B. Moreover, the dependence on mσ of both MS and ⟨r2⟩1/2 is such that now the solitonic mass increases and the radius decreases toward the dilaton-decoupled limit as shown in Fig. 12. In the case of a nonvanishing mπ, it is the solitonic mass that decreases while the radius increases.
1010.1103/PhysRevD.110.094008.f10
Dilaton χ(r) (left) and α(r)/π (right) profiles for Δ=1 and mσ=0–4.
1110.1103/PhysRevD.110.094008.f11
Skyrmion mass MS as a function of Δ for mσ=1–4 (left) and a detail of the mσ=1 line (right).
1210.1103/PhysRevD.110.094008.f12
Skyrmion mass MS and root-mean-square radius ⟨r2⟩1/2 normalized at their values in the absence of the dilaton (MSkyrme=15.49 and ⟨r2⟩Skyrme1/2=0.213) as a function of mσ.
Analytical solution in the massless case
In the case of the chiral Lagrangian alone, the spherical hedgehog ansatz leads to a differential equation that can only be solved using numerical methods. Quite remarkably, when the coupling with the dilaton is not neglected, it is possible to find an analytical solution for a special value of the couplings f and K due to an interesting phenomenon, namely, “BPS-like equations without a BPS bound.” In fact, when f2K=7/8, the second order field equations can be reduced to a first order system even though there is no BPS bound on the energy of the system. The equations of motion (14) and (15) in the absence of mass and Skyrme term reduce to two nonlinear coupled ordinary differential equations (ODEs) r2α′′-2rα′(frσ′-1)-sin(2α)=0,r2σ′′+r(fKrα′2+σ′(2-frσ′))+2fKsin2(α)=0.For later convenience, we change variables as ρ=logr and ∂σ∂ρ→∂σ∂ρ+12f and rewrite the EOMs as α¨-2fα˙σ˙-sin(2α)=0,σ¨-fσ˙2+fKα˙2+2fKsin2(α)+14f=0,where the dot denotes the derivative with respect to ρ. Noticeably, when f2K=78 we can find a first order system implying the EOMs, α′=2rcos(α),σ′=-2frsin(α)+12fr.We can therefore solve for the analytical solution of the EOMs that read α±(r)=nπ+2tan-1(1-2c1c1±(rr0)2),n∈Z,σ±(r)=1f{c2+12log(rr0)+log[cos(2tan-1(1-2c1c1±(rr0)2))]},where the plus and minus signs apply for c1>0 and c1<0, respectively. The topological charge associated with this solution is B=-1. Conventional solitonic solutions of the chiral Lagrangian with negative winding numbers are called “antiskyrmions.” The energy density of the solution reads ε=e-2c28c12f2r2(rr0)-22-1[(2+4)c14-(2-4)(rr0)42]and diverges at the origin as r-22-3 due to the dilaton profile χ(r) being singular in r=0. Moreover, even excluding the singularity at r=0, the total energy would still diverge since the energy density does not decrease fast enough at large r.
DILATON AUGMENTED CRYSTALS
In this section, we first review the strategies to find analytical solutions describing arrays of tubes of baryons with nontrivial topological charge on flat spacetime [74–77] and then we will generalize the solutions in the presence of the dilaton. The qualitative behavior of these exact solutions suggests they may be relevant for describing nuclear pasta states [96,97].
Crystals without the dilaton
We consider the action (2) in the absence of the Skyrme and mass terms and retain the same general parametrization for U(x) given in (18). The equations of motion (5) constitute the following set of three nonlinear coupled PDEs, -□α+sin(α)cos(α)(∇μΘ∇μΘ+sin2(Θ)∇μΦ∇μΦ)=0,sin2(α)□Θ+2sin(α)cos(α)∇μα∇μΘ-sin2(α)sin(Θ)cos(Θ)∇μΦ∇μΦ=0,sin2(α)sin2(Θ)□Φ+2sin(α)cos(α)sin2(Θ)∇μα∇μΦ+2sin2(α)sin(Θ)cos(Θ)∇μΘ∇μΦ=0,where we have not yet made any ansatz for the parameters of U in terms of the coordinates. Here the topological charge density ρB≡J0 takes the following form: ρB=12sin2(α)sin(Θ)dα∧dΘ∧dΦ.From this expression, we observe that in order to have dα∧dΘ∧dΦ≠0 the fields α, Θ, and Φ must be independent. Additionally, in order to decouple the previous set of equations, we impose ∇μΦ∇μα=∇μα∇μΘ=∇μΦ∇μΦ=∇μΘ∇μΦ=0.We proceed by confining the system into a box described by the following line element: ds2=-dt2+Lx2dx2+Ly2dy2+Lz2dz2,where {x,y,z} are dimensionless coordinates having the range 0≤x≤2π,0≤y≤π,0≤z≤2π.The condition (42) is then realized by considering the following ansatz: α=α(x),Θ=qy,Φ=p(tLz-z),q=2v+1,v,p∈Z,such that the equations of motion reduce to a single integrable ODE for the profile α(x), ∂x[12(α′)2-V(α)-E0]=0,V(α)=-q2Lx22Ly2cos(2α),where E0 is an integration constant that depends on the boundary conditions for α. The explicit solution can be written in terms of a certain Jacobi elliptic function. The corresponding topological charge density reads ρB=3qpsin(qy)LxLyLz∂x(2α-sin(2α)).By considering the boundary conditions for α, α(2π)-α(0)=nπ,n∈Z,then the topological charge is given by B=np. Therefore, the baryon charge can be an arbitrary integer number. This solution describes stationary soliton crystals with the shape of ordered arrays of baryonic tubes carrying topological charge. Here the integer p can be interpreted as the baryonic charge per unit of length in the z direction per tube. This can be seen by looking at the energy density of the solution ε≡T00, ε=12K(α′2Lx2+sin2(α)(q2Ly2+2p2sin2(qy)Lz2)),which is depicted in Fig. 13.
1310.1103/PhysRevD.110.094008.f13
Energy density of the solution (46) for p=q=n=1 (left) and p=q=n=3 (right). In both cases, we have assumed K=Lx=Lz=1 and Ly=2.
Crystals with the dilaton
Here we study the effect of the dilaton on the solutions considered in the previous section. In particular, we consider U given by Eqs. (18) and (45) and we assume that the dilaton field depends only on the x coordinate, i.e., σ=σ(x). Consequently, the field equations reduce to two coupled nonlinear ODEs for α(x) and σ(x), α′′-2fα′σ′-Lx22Ly2q2sin(2α)-Lx2mπ2sin(α)e-f(y-2)σ=0,σ′′+fKα′2-fσ′2+Lx2Ly2fKq2sin2(α)-Lx2e2fσ∂σV(σ)-KfLx2mπ2ycos(α)e-f(y-2)σ=0.The topological charge is given by Eq. (47), while the energy density evaluated on this ansatz reads ε=12Ke-2fσ(sin2(α)(q2Ly2+2p2sin2(qy)Lz2)+α′2Lx2)+e-2fσσ′22Lx2-Kmπ2cos(α)e-fyσ+V(σ)and reduces to Eq. (49) for σ=mπ=0. We proceed by focusing on the massless case mπ=mσ=0, which unlike the massive case can be partially addressed analytically. In fact, it is quite remarkable that, at the special parameter point f2K=1, the field equations (50) and (51) can be reduced to a system of first order ODEs given by α′=qcos(α),σ′=-qfsin(α),where for the sake of simplicity we have chosen Lx=Ly. This can also be seen by noting that the EOMs (50) and (51) coincide with the EOMs (31) and (32) for the hedgehog ansatz up to a constant term 14f in Eq. (32). The latter shifts the special combination of parameters from f2K=7/8 to f2K=1 reducing the second order equations of motion to the above set of first order ones, yielding the solutions α(x)=2arctan(tanh(12(c1+qx))),σ(x)=-1f[c2+2arctanh(tanh2(12(c1+qx)))],where c1 and c2 are the integration constants. The behavior of these solutions is shown in Fig. 14. We can restore all the integration constants stemming from the second order original EOMs and write the solution as α(x)=nπ+arctan(d1eqx+d2e-qx),n∈Z,σ(x)=-1f(c2+12log((d1eqx+d2e-qx)2+1)).
1410.1103/PhysRevD.110.094008.f14
The profiles α(x) (left) and σ(x) (right) for q=1, 3, 5.
When the parameter p is an integer it is possible to have an integer topological charge B only by considering a box that has an infinite length in the x direction. In such a case we have B=p, but the total energy carried by the solution diverges. This is in net contrast with the solution in the absence of the dilaton field, where for p∈Z one can confine an arbitrary amount of topological charge in a finite-size box and the corresponding energy is finite. On the other hand, we can relax the condition p∈Z and determine the value of p such that a certain amount of topological charge is confined in a finite-size box. Without losing generality, we consider 0≤x≤2π and impose BCs α(0)=0 and α(2π)=s with 0<s<π/2. The topological charge reads B=-2∫α(0)α(2π)pπsin2(α)dα.Hence B∈Z can be achieved when p=2πBsin(2s)-2s.
We conclude this section by numerically investigating the behavior of the solution for different values of X≡f2K. The corresponding energy density is shown in Fig. 15. At X=0 the EOMs decouple; the dilaton solution is σ(x)=c1-log(c2+x), {c1,c2}∈R, while α(x) solves Eq. (46) yielding a field configuration with localized energy and integer topological charge. On the other hand, for X=1 we have the analytical solution (54), whose energy grows indefinitely for increasing x. The physical impact of this solution will be discussed below.
1510.1103/PhysRevD.110.094008.f15
Contour plot of the energy density in the x-y plane for mπ=mσ=0 varying X=f2K. From left to right and top to bottom we have X=0, 0.3, 0.6, 0.9, 1. As X goes from 0 to 1 the results interpolate between the analytical solution found in [74–77] in the absence of the dilaton field and the solution (54).
Stability analysis
In this section, we consider a nontrivial test of the stability of the crystal solutions in the presence of the dilaton. In many situations when the hedgehog property holds, i.e., the equations of motion reduce to a single equation for the soliton profile, the most “dangerous” perturbations are those that keep the structure of the ansatz since they are likely to lead to a lower-energy state [103,104]. In the presence of the dilaton, these perturbations are of the following form: α=α0(r)+ϵu(r),σ=σ0(r)+ϵv(r),where α0 and σ0 solve Eq. (53) and 0≤ϵ≪1. These perturbations do not involve the isospin degrees of freedom Θ and Φ. A straightforward computation shows that the linearized EOMs for u(r) and v(r) always have a zero mode u(r)=∇rα0(r) and v(r)=∇rσ0(r). The contribution of this configuration to the density energy is given by δE=2KqL2ϵe-2fσ0(r)sin(α0(r))(p2sin2(θq)+q2)≥0.Therefore, the present solutions are stable under the above potentially dangerous perturbations.
Isospin-charge separation
A further intriguing feature of the effective chiral Lagrangian dressed with the dilaton is the following. In the solutions without dilaton [74–77], the peaks corresponding to the local maxima of the topological charge density are in the same locations as the peaks corresponding to the local maxima of the isospin charge density (as one can also check via a direct computation). On the other hand, the isospin current [which is the Noether current associated with the SU(2) isospin rotations of the theory] is deformed by the presence of the dilaton, while the definition of the topological charge density does not change (being the usual baryon charge). This implies that the local maxima of these two relevant quantities no longer coincide. Indeed, by setting for simplicity Lx=Ly=Lz≡L, the topological charge density reads ρB=12pqL3sin(yq)α′(x)sin2(α(x)),while the isospin charge density is ρI=KTr(R0τ3)e-2fσ(x)=2KpLe-2fσ(x)sin2(yq)sin2(α(x)).In Fig. 16 we compare the two charge densities to illustrate the different positions of the peaks.
1610.1103/PhysRevD.110.094008.f16
Topological- (left) and isospin- (right) charge densities for L=B=q=1 and α(2π)=s=π4.
This “separation” of the peaks of baryonic and isospin charge densities could open the interesting possibility of defining the analog of the well-known phenomenon in condensed matter physics called spin-charge separation (see, e.g., [94,95] and references therein). In the condensed matter version of this phenomenon, due to the presence of strong correlations, spin and charge may cease to be tied generating independent branches of excitations. In the present case, it could be possible to observe isospin-baryon charge separation thanks to the presence of the dilaton which can support excitations that change directly the isospin and only indirectly the baryon charge density.
DILATON AUGMENTED BRANES
In [70–73,77], another family of analytical solutions of the massless chiral Lagrangian has been discovered. These are again obtained by confining the theory in a box according to the metric equation (43) and considering the following ansatz: U=e12pyτ3e14xτ2e12τ3F(t,z),p∈Z.The above represents a particular instance of Euler angle parametrization for SU(2) which implies the following range for x and y[105,106]: 0≤y≤π,0≤x≤2π.Without losing generality, we can also consider 0≤z≤2π.The topological charge density is given by ρB=3p4LxLyLzsin(x2)∂zF.The periodicity of physical observables along with the properties of the Euler angles parametrization fixes the boundary condition satisfied by F(t,z) as F(t,z=0)-F(t,z=2π)=±8πq,q∈Z.As a consequence, the topological charge reads B=-p8π(F(t,z=0)-F(t,z=2π))=pq.Noticeably, for mπ=0 the EOMs (5) evaluated on the ansatz (65) reduce to the field equation of a free massless scalar in d=2 dimensions [73], (∂t2-1Lz2∂z2)F(t,z)=0.The general solution to the previous equations can be decomposed as the sum of two independent modes F=F+(tLz+z)+F-(tLz-z). The latter can be generically written using the following representation: F+=z+0+v+(tLz+z)+∑n≠0(an+sin[n(tLz+z)]+bn+cos[n(tLz+z)]),F-=z-0+v-(tLz-z)+∑n≠0(an-sin[n(tLz-z)]+bn-cos[n(tLz-z)]),with coefficients z±0,v±,an±,bn±. In this way, the topological charge (70) is nonzero when v+-v-≠0. Additionally, the coefficients v± have to be chosen such that F(t,z=0)=z+0+z-0+(v++v-)tLz⇒v++v-=0,F(t,z=2π)=z+0+z-0+(v+-v-)2π⇒v+-v-=4q.The energy density of the solution reads ε=K8[14(1Lx2+4p2Ly2)+(∂tF)2+1Lz2(∂zF)2]and is illustrated in Fig. 17. One can see that this solution describes modulated layers of nuclear matter.
1710.1103/PhysRevD.110.094008.f17
Energy density of a solution of the EOMs (71) exhibiting a layer structure. We choose F=F+(tLz+z)=cos(tLz+z).
The inclusion of the Skyrme term yields an additional equation (∂tF)2-1Lz2(∂zF)2=(∂tF-1Lz∂zF)(∂tF+1Lz∂zF)=0,with solution F(t,z)=F(z±tLz). On the other hand, the equations of motion do not admit any solution of the form (65) when mπ≠0.
Equipped with the above, we now investigate the impact of the dilaton on this class of topologically nontrivial solutions of the chiral and Skyrme models in the chiral limit mπ=0. Assuming σ=σ(t,z), the EOMs are ∂t2F-1Lz2∂z2F=0,(∂tF)2-1Lz2(∂zF)2=0,∂zF∂zσ-Lz2∂tF∂tσ=0,∂t2σ-1Lz2∂z2σ-f((∂tσ)2-1Lz2(∂zσ)2)-fk16(4p2Ly2+1Lx2)+e2fσ∂σV(σ)=0.The baryon charge does not depend on the dilaton and, therefore, it is still given by Eq. (70). The energy density reads ϵ=Ke-2fσ(4Lx2Ly2(Lz2∂tF2+∂zF2)+Lz2(Ly2+4Lz2p2))32Lx2Ly2Lz2+e-2fσ(Lz2∂tσ2+∂zσ2)2Lz2+λK((Lz2∂tF2+∂zF2)(Ly2+4Lx2p2sin2(r2))+Lz2p2)128Lx2Ly2Lz2+V(σ).
Branes in the conformal limit
We start by setting to zero both the Skyrme term and the dilaton potential. In this case, the EOMs for the brane ansatz (65) admits an analytical solution carrying topological charge where the U and σ fields are, respectively, static and homogeneous. It reads F(z)=c1+c2z,σ(t)=1f{d1-log|cos(K4c22Ly2Lx2+Lz2(Ly2+4Lx2p2)(d2+ft)4LyLxLz)|},where c1, c2, d1, and d2 are arbitrary integration constants. The boundary conditions (69) for F(z) impose c2=4q, whereas the energy density evaluated on the solution is both homogeneous and static, ε=K32Ly2Lx2Lz2e-2d1[Ly2(64Lx2q2+Lz2)+4Lx2Lz2p2].Moreover, since the U field is static, the isospin charge density of this solution vanishes identically. The time evolution of the dilaton field is depicted in Fig. 18. Note that this solution is nonanalytic in time due to a series of branch-cut singularities lying at ft=2LyLxLzπ(2n+1)KLy2(64Lx2q2+Lz2)+4Lx2Lz2p2-d2,n∈Z.
1810.1103/PhysRevD.110.094008.f18
Time evolution of the dilaton solution (83).
It is possible to construct another analytical solution to the EOMs of the massless chiral Lagrangian where, akin to the case without the dilaton, F(z,t) can be expressed as a particular linear combination of left- and right-moving modes F(t,z)=c1+c2[(z+tLz)-A(z-tLz)],σ(t,z)k=d1+d2[(z+tLz)+A(z-tLz)],where A=Lz2(Ly2+4Lx2p2)16Lt2Lx2(c22+4d22).However, for d2≠0 the energy of this solution either grows or decays exponentially in time depending on the boundary conditions. Noticeably, when d2=0 the solution for the dilaton is a constant and the energy of the solution is exactly twice that of (84).
There are no nontrivial solutions with U given by Eq. (65) for mπ≠0. Moreover, when mσ=0 there are no solutions of the EOMs when adding the Skyrme term together with dilatonic dynamics. However, as we will see in the next section, for nonvanishing dilaton mass we have been able to find analytical solutions with the ansatz (65) including the Skyrme term.
Branes in the near-conformal Skyrme model
Our goal in this section is to study the most general solution of the EOMs (78) of the dilaton augmented Skyrme model including the Skyrme term. This is of the form F(t,z)≡F(tLz±z),σ(t,z)=c,where for a dilaton potential of the form (10) the dilaton vacuum expectation value (VEV) c is determined by the following equation: χ0Δ-2+β(Δ-4)mσ2-χ02=0,χ0≡e-cf,where β=116f2K(4p2Ly2+1Lx2).The properties of the soliton affect the dilaton VEV through the value of the parameter p which is related to the topological charge via Eq. (70). For mσ→∞, the solution is c=0. On the other hand, when Δ→4 or mσ→0 the VEV becomes undetermined since the potential vanishes. In the special cases Δ→0, 4 and Δ=2 the dilaton VEV reads 2fc={sinh-1(2βmσ2)Δ→0-log(1-2βmσ2)Δ=2-W(-2βmσ2)Δ→4,where W(x) denotes the Lambert W function. Only in the Δ=0 case is c real for arbitrary values of the dilaton mass. For Δ=2 the VEV exhibits a branch-cut singularity at mσ2=2β and the solution is complex for smaller masses. Analogously, in the Δ→4 case the solution is real only for mσ2>2eβ.
The energy density carried by the solution for a generic Δ is ε=K32e-2cf(8F′(tLz+ϕ)2Lz2+4p2Ly2+1Lx2)+Kλ(2F′(tLz+ϕ)2(Ly2+4Lx2p2sin2(r2))+Lz2p2)128Lx2Ly2Lz2+V(c).As shown in Fig. 19, this solution describes ordered layers of matter analogous to the case without dilaton previously described. The dilaton VEV suppresses the energy density of the U field while adding a homogeneous contribution proportional to mσ2. As a consequence, the overall effect of the dilaton is to smooth out the layer structure.
1910.1103/PhysRevD.110.094008.f19
Energy density of the solution (88) for Δ→0 and different values of the dilaton mass. We consider mσ/K=0.001 (top-left), 0.05 (top-right), 1 (bottom-left), and 20000 (bottom-right). As we increase mσ the dilaton decouples from the dynamics as can be seen by comparing the mσ/K=20000 case to Fig. 17. We choose F(tLz+z)=cos(tLz+z).
Analogous solutions exist in the absence of the Skyrme term. In fact, we can now look for solutions of the form F(t,z)≡F+(tLz+z)+F-(tLz-z),σ(t,z)=c.Note that, different from Eq. (88), this is not the most general solution to the EOM. By plugging the above into the EOMs (78) with λ=0 we have fKLz2(Ly2+4Lx2p2)-16Ly2Lx2[Lz2e2cf∂V(σ)∂σ|σ=c-fKF+′(tLz+ϕ)F-′(ϕ-tLz)]=0,which admits solutions if and only if
either F+ or F- vanishes;
both F+ and F- are linear functions of their argument.
In the first case, the solution is equal to the one in the presence of the Skyrme term with the dilaton VEV given by Eq. (89). In the second case, we have F(t,z)=a0+a1(tLz+ϕ)+a2(tLz-ϕ),and the dilaton VEV is again determined by Eq. (89) with β now given by β=f2K(16a1a2Ly2Lx2+Lz2(Ly2+4Lx2p2))16Ly2Lx2Lz2.For mσ=0 this reduces to the solution (86) with d2=0.
CONCLUSIONS
We fused numerical and analytical methods to perform a thorough investigation of skyrmion solutions featuring dilatonic dynamics. Employing the dilaton-dressed chiral Lagrangian we addressed the impact on relevant physical quantities such as the traditional skyrmion profile, mass, and size stemming from varying the near-conformal controlling parameters. We discovered an intriguing dependence of these physical properties on the dilaton mass. A notable result is the observation of a quick dilaton decoupling once its mass surpasses the pion one. The decoupling is further enhanced at large values of the anomalous dimension of the chiral condensate. Another interesting feature is the occurrence of a special value of the ratio of the dilaton to pion decay constants where the coupled second order field equations of the dilaton augmented EFT reduce to a first order solvable system. In this case, one arrives at analytic solitonic solutions even though this phenomenon is not related to a BPS bound. The resulting special solutions have negative topological charge B=-1 and are singular at the origin. Because in several instances in the literature, nuclear matter, at the core of compact stars, has been argued to feature near-conformal dynamics [80,86–88], we employed QCD-inspired values of the parameters to represent our findings. Our setup can be further extended to investigate face-centered-cubic skyrmion crystals featuring dilaton dynamics which provide a description of nuclear matter and its equation of state [107,108].
We further presented the impact of dilatonic dynamics on skyrmion crystals and have identified a special parameter point, linked to the case of the traditional skyrmion above, for which the second order differential equations also reduces to a set of first order ones. Here another observed feature, stemming from the presence of the dilaton, is that the latter spatially separates the baryon and isospin charge distributions similar to the spin-charge separation phenomenon.
Additionally, we considered dilaton-modified solitonic branes and showed that the soliton profile is unaltered by the presence of the dilaton while the latter acquires a homogeneous vacuum expectation value. We have also shown how the dilaton smooths out the brane configurations. One of the original appeals of the brane solutions, without the dilaton, is that they can be analytically derived from the equations of motion. Our results show that it is still possible to obtain analytic solutions in the presence of dilaton dynamics. Attempts to connect brane solutions to phenomenological applications were discussed in [73] where the authors compared the thermodynamic properties of the solutions to lattice QCD calculations.
More generally, our results have broad implications ranging from the near-conformal dynamics of QCD-like theories to nuclear matter in extreme regimes. For example, for the near-conformal dynamics in vacuum, one can now compare our results to the ones from lattice investigations of the nucleon spectrum close to the lower end of the conformal window.
ACKNOWLEDGMENTS
We are indebted to Fabrizio Canfora for a fruitful collaboration at the early stage of the project and precious suggestions. M. T. was supported by Agencia Nacional de Investigación y Desarrollo (ANID) Grant No. 7221039. The work of J. B. was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and also supported in part by the JSPS KAKENHI Grant No. JP23K19047. The work of F. S. is partially supported by the Carlsberg Foundation, Grant No. CF22-0922.
TABLES OF THE SKYRMION MASS AND THE ROOT-MEAN-SQUARE RADIUS OF THE BARYON CHARGE
In this appendix, we list in a series of tables our results for the skyrmion mass MS and the root-mean-square radius of the baryon charge ⟨r2⟩1/2 in the single spherical skyrmion case studied in Sec. III. We estimated an error not larger than the 0.5% for all the entries. All the dimensionful quantities are measured in units of K. MSkyrme=15.67 and ⟨r2⟩Skyrme1/2=0.215 denote the skyrmion mass and the root-mean-square radius in the absence of the dilaton, respectively.
General caseI10.1103/PhysRevD.110.094008.t1
Numerical values of the skyrmion mass for different values of the conformal dimension Δ and dilaton mass mσ=1–5. MSkyrme=15.67 denotes the skyrmion mass in the absence of the dilaton.
Numerical values of the root-mean-square radius ⟨r2⟩1/2 of the baryon charge for different values of the conformal dimension Δ and dilaton mass mσ=1–5. ⟨r2⟩Skyrme1/2=0.215 denotes the root-mean-square radius of the baryon charge for the Skyrme model in the absence of the dilaton.
Root-mean-square radius of the baryon charge⟨r2⟩1/2 in units of 10-1⟨r2⟩1/2/⟨r2⟩Skyrme1/2 in units of 10-1Δmσ=1mσ=2mσ=3mσ=4mσ=5mσ=1mσ=2mσ=3mσ=4mσ=511.502.042.142.1582.1616.999.479.9410.0310.041.21.522.042.142.1582.1617.089.489.9410.0310.041.41.542.042.142.1582.1617.179.499.9410.0310.041.61.562.042.142.1582.1617.259.509.9410.0310.041.81.582.052.142.1582.1617.339.519.9410.0310.0421.602.052.142.1592.1617.439.529.9510.0310.042.21.612.052.142.1592.1617.509.539.9510.0310.042.41.632.052.142.1592.1617.579.549.9510.0310.052.61.642.052.142.1592.1617.649.549.9510.0310.052.81.662.062.142.1592.1627.709.559.9510.0410.0531.672.062.142.1592.1627.769.569.9510.0410.053.21.682.062.142.1602.1627.829.579.9610.0410.053.41.702.062.142.1602.1627.889.579.9610.0410.053.61.712.062.142.1602.1627.939.589.9610.0410.053.81.722.062.142.1602.1627.989.599.9610.0410.0541.732.072.142.1602.1628.059.619.9610.0410.05III10.1103/PhysRevD.110.094008.t3
Numerical values of the skyrmion mass for different values of the conformal dimension Δ and dilaton mass mσ=6–10. MSkyrme=15.67 denotes the skyrmion mass in the absence of the dilaton.
Numerical values of the root-mean-square radius ⟨r2⟩1/2 of the baryon charge for different values of the conformal dimension Δ and dilaton mass mσ=6–10. ⟨r2⟩Skyrme1/2=0.215 denotes the root-mean-square radius of the baryon charge for the Skyrme model in the absence of the dilaton.
Root-mean-square radius of the baryon charge⟨r2⟩1/2 in units of 10-1⟨r2⟩1/2/⟨r2⟩Skyrme1/2 in units of 10-1Δmσ=6mσ=7mσ=8mσ=9mσ=10mσ=6mσ=7mσ=8mσ=9mσ=1012.1602.1592.1592.1592.1591.0041.0041.0031.0031.0031.22.1602.1592.1592.1592.1591.0041.0041.0031.0031.0031.42.1602.1592.1592.1592.1591.0041.0041.0031.0031.0031.62.1602.1592.1592.1582.1591.0041.0041.0031.0031.0031.82.1602.1592.1592.1582.1591.0041.0031.0031.0031.00322.1602.1592.1582.1582.1591.0041.0031.0031.0031.0032.22.1602.1592.1582.1582.1591.0041.0031.0031.0031.0032.42.1602.1592.1582.1582.1591.0041.0031.0031.0031.0032.62.1602.1592.1582.1582.1591.0041.0031.0031.0031.0032.82.1602.1592.1582.1582.1591.0041.0031.0031.0031.00332.1602.1592.1582.1582.1581.0041.0031.0031.0031.0033.22.1602.1592.1582.1582.1581.0041.0031.0031.0031.0033.42.1602.1592.1582.1582.1471.0041.0031.0031.0031.0033.62.1602.1591.0032.1582.1581.0041.0031.0031.0031.0033.82.1602.1592.1582.1582.1581.0041.0031.0031.0031.00342.1602.1592.1582.1582.1581.0041.0031.0031.0031.003<inline-formula><mml:math display="inline"><mml:msub><mml:mi>M</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mo stretchy="false">⟩</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> in the <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> limitV10.1103/PhysRevD.110.094008.t5
Numerical values of the skyrmion mass MS and the root-mean-square radius of the baryon charge ⟨r2⟩1/2 for mσ=1–5.
Skyrmion massMSMS/MSkyrmemσ=124.481.56mσ=216.381.05mσ=315.450.99mσ=415.270.97mσ=515.250.97Root-mean-square radius of the baryon charge⟨r2⟩1/2⟨r2⟩1/2/⟨r2⟩Skyrme1/2mσ=10.1400.65mσ=20.2030.94mσ=30.2140.99mσ=40.2161.00mσ=50.2161.00VI10.1103/PhysRevD.110.094008.t6
Numerical values of the skyrmion mass and the root-mean-square radius ⟨r2⟩1/2 of the baryon charge for different values of the dilaton mass measured in units of K.
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