aa r X i v : . [ nu c l - t h ] J a n Kaon Condensation in Baryonic Fermi Liquid at High Density
Won-Gi Paeng
Rare Isotope Science Project, Institute for Basic Science, Daejeon 305-811, Korea
Mannque Rho
Institut de Physique Th´eorique, CEA Saclay, 91191 Gif-sur-Yvette c´edex, France (June 15, 2018)
ABSTRACT
We formulate kaon condensation in dense baryonic matter with anti-kaons fluctuating fromthe Fermi-liquid fixed point. This entails that in the Wilsonian RG approach, the decimationis effectuated in the baryonic sector to the Fermi surface while in the meson sector to theorigin. In writing the kaon-baryon (KN) coupling, we will take a generalized hidden localsymmetry Lagrangian for the meson sector endowed with a “mended symmetry” that hasthe unbroken symmetry limit at high density in which the Goldstone π , scalar s , and vectors ρ (and ω ) and a become massless. The vector mesons ρ (and ω ) and a can be identified asemergent (hidden) local gauge fields and the scalar s as the dilaton field of the spontaneouslybroken scale invariance at chiral restoration. In matter-free space, when the vector mesonsand the scalar meson – whose masses are much greater than that of the pion – are integratedout, then the resulting KN coupling Lagrangian consists of the leading chiral order ( O ( p ))Weinberg-Tomozawa term and the next chiral order ( O ( p )) Σ KN term. In addressing kaoncondensation in dense nuclear matter in chiral perturbation theory (ChPT), one makes anexpansion in the “small” Fermi momentum k F . We argue that in the Wilsonian RG formalismwith the Fermi-liquid fixed point, the expansion is on the contrary in 1 /k F with the “large”Fermi momentum k F . The kaon-quasinucleon interaction resulting from integrating out themassive mesons consists of a “relevant” term from the scalar exchange (analog to the Σ KN term) and an “irrelevant” term from the vector-meson exchange (analog to the Weinberg-Tomozawa term). It is found that the critical density predicted by the latter approach,controlled by the relevant term with the irrelevant term suppressed, is three times less thanthat predicted by chiral perturbation theory. This would make kaon condensation take placeat a much lower density than previously estimated in chiral perturbation theory. Negatively charged kaons ( K − that will be referred to simply as K or kaon) are predictedin simple chiral Lagrangians to bose-condense in s-wave in dense baryonic matter at somehigh density [1]. Within the regime of strong interactions such as in relativistic heavy-ioncollisions, the critical density for the process – that we will refer to as kaon condensation – is e-mail: [email protected] e-mail: [email protected] n K ∼ > n where n is the ordinary nuclear matter density n ≈ . − . This process may be difficult to access in the laboratories for various reasons, amongwhich the high temperature expected, given that the temperature produced would melt thekaon condensate. It would also be compounded with other strange degrees of freedom such asstrange quark matter. In compact-star matter in weak and chemical equilibrium, mean-fieldapplications of chiral Lagrangians typically predict the critical density considerably lower, n K ∼ n [2, 3]. As reviewed in [3], this critical density is arrived at by three different startingpoints, one from the vacuum ( n = 0), the second from near nuclear matter density ( n ∼ n )and third from the vector manifestation fixed point n c at which the ρ -meson mass vanishes(in the chiral limit) and chiral symmetry gets restored. One common element in these threetreatments, stressed in [3], is the underlying symmetry that governs the strong interactions,namely, hidden gauge symmetry implementing chiral symmetry of QCD appropriate at thekinematic regime of density. In this way, both the boson and the fermion involved are treatedin one framework as we will elaborate below in a specific way in terms of a model Lagrangian.It is generally understood that if kaons condensed at a density as low as ∼ n , the EoSof the matter would be considerably softened from the one without kaons condensed. Thisproperty was exploited by Brown and Bethe [4] to argue that stars with mass ∼ > . M ⊙ wouldcollapse and give rise to a considerably larger number of light-mass black holes in the Universethan estimated before [3]. The recent observations of accurately measured ∼ ∼ > n K ∼ n discussed in [3] are based on the mean-field approximation of a given Lagrangianthat encodes chiral symmetry. What we will find is that if kaons condensed at as low adensity as a few times the nuclear matter density, then the treatment for the condensationso far performed with the given Lagrangian could not be reliable. There can be at leastone phase change, if not several, in the vicinity of the condensation phenomenon, most likelyaffecting the critical density importantly. One such phase change is of topological nature thatis seen in the soliton-crystal description of dense matter – on which a brief comment will begiven below.The second question is what happens, within the given Lagrangian framework, afterkaons condense in baryonic matter. We are interested in exploring whether under certaincircumstances, the baryonic matter which is described as a Fermi liquid could not turninto a non-Fermi liquid with modified thermodynamic properties. Such a transition wouldconsequently affect the EoS at a density relevant to the interior of compact stars. We aremotivated by a number of analogies in condensed matter systems that point to a need for
The variety of approaches giving a similar critical density found in the literature wherein the boson sectorand the fermion sector are treated separately are basically of the same approximation, although phrased indifferent ways. . These phenomena involve anomalous dimensions. Aparticularly intriguing case that insinuates an analogy to what we are interested is the space-time-dependent coupling between a fermion in Fermi liquid and a critical boson [8].
Suppose that we arrive at a highly reliable theoretical tool – which is yet to be found– for treating meson condensation in which we have strong confidence. If such a treatmentstill yielded a low kaon condensation density and made the EoS too soft to support ∼ > M ⊙ stars within the standard theory of gravity, then one could envisage a possibility that achange in fundamental physics is required, with the EoS of compact stars providing a seriousindication for a modified gravity. In fact, a suitably modified gravity such as for instance f ( R )gravity is argued to be able to accommodate the massive stars even with the EoS softenedby hyperons [9].In this paper, we treat kaon condensation for kaons fluctuating around the Fermi-liquidfixed point arrived at in Wilsonian RG approach [10]. In compact-star matter, one is dealingwith weak and chemical equilibrium, so the kaons figure via weak interactions. There theelectron chemical potential increases as a function of density, and provides a crucial “effectiveattraction” to drive the kaon mass to drop. For this process, the strong (attractive) KN-interaction and the electron chemical potential have to be treated self-consistently.Here we focus on nuclear matter in strong interactions without weak and chemical equi-librium, as in [11]. Hyperons will not figure since they will be suppressed in the large N limit – where, as defined precisely below, N ∝ k F where k F is the Fermi momentum. Oncewe understand what happens in nuclear matter, then we can address what could happen incompact-star matter. In compact stars, when the effective kaon mass drops sufficiently lowso as to equal the electron chemical potential in compact-star matter, the electrons will betadecay to kaons and the kaons will then bose-condense [2]. The kaon mass need not vanishfor the condensation to set in. This means that the information that one gains from kaons innuclear matter will give an upper bound for the critical density for condensation in compactstars.As a spin-off, there are other questions that call for answers. For instance, what happensto the baryonic matter in a Fermi liquid when the kaons condense? What form of state itis after condensation, specifically what is its EoS? What happens if a phase transition inthe baryonic matter – such as e.g., the skyrmion-half-skyrmion changeover to be mentionedbelow –takes place before or in the vicinity of the critical density at which kaons condense?These questions have never been addressed before. In this paper we make the first step toanswering the questions.It should be mentioned that the model we will analyze with the KN coupling defined byEqs. (2)-(5) given below was first studied in [12]. What differentiates this model from the In dense baryonic matter, a similar phenomenon is found to take place. It is found that the appearanceof a half-skyrmion phase in skyrmion crystal at density n / ∼ n drives a Fermi liquid to a state resemblingnon-Fermi liquid. More on this below. A possible implementation of this idea in our problem with the local density operator ψ † ( x ) ψ ( x ) figuringin the coupling constant will be mentioned in the discussion section. π , scalar s , and vectors ρ (and ω ) and a becomemassless. We shall call this “mended HLS.” The vector mesons ρ (and ω ) and a can beidentified as emergent (hidden) local gauge fields and the scalar s as the dilaton field of thespontaneously broken scale invariance at chiral restoration.The approach we take is in the notion of the double decimation explained in [14]. TheFermi-liquid matter to which kaons as pseudo-Goldstone bosons couple is the matter arrivedat by the “first RG decimation” from the chiral symmetry scale ∼ πf π ∼ h although it isirrelevant, and that at one loop order, in certain parameter space, the flow necessarily leadsto the vanishing of the “effective kaon mass” signaling kaon condensation. The crucial issuethere was the parameter space that delineates the critical line that leads to or away from thecondensation point. What happens to the Fermi liquid matter after the kaons are condensedwas not analyzed in detail there but the immediate prediction was that the fermion chemicalpotential µ F increased with a contribution proportional to h K i and since µ F = k F / m ∗ , fora given density (i.e., k F ), the effective fermion mass m ∗ decreased. The quadratic dispersionrelation used there that resulted in approximating the effect of the electron chemical potentialwas, however, not correct for a Goldstone boson in medium – which belongs to type-A [15],but the qualitative behavior was still valid.In this paper we address the issue with a correct linear dispersion relation relevant for thekaon (as a pseudo-Goldstone boson) and a relevant KN interaction that can arise when theHLS with mended symmetries is taken into account. We will find that the resulting theory,which is an expansion in 1 /k F , differs basically from chiral perturbation theory where theexpansion is in k F . The prediction one can make is that kaons could condense at a densityconsiderably lower than what has been predicted in ChPT. We start by defining the action for our system. In the standard approach anchored onchiral Lagrangians, one couples kaons as pseudo-Goldstone bosons to nucleons in baryonicmatter treated either in standard nuclear many-body approach or in the mean-field approx-imation [16, 3] of, or in chiral perturbation theory with, the effective Lagrangians that aredefined in the vacuum. Here we take a different starting point which is in line with the notion4f sliding vacuum [14] in which nuclear matter arises as a fixed point, i.e., Fermi-liquid fixedpoint. This naturally incorporates the notion of BR scaling [13] into the description of kaoncondensation. Here we take the nuclear equilibrium state as the Fermi-liquid fixed point andcouple (pseudo-) Goldstone mesons φ to quasiparticles ψ in the vicinity of the Fermi-liquidfixed point. We write the partition function in Euclidean space as Z = Z [ dφ ][ dφ ∗ ][ d Ψ][ d Ψ † ]e − S E (1) S E = S Eψ + S Eφ + S Eψφ (2) S Eψ = Z dτ d ~x Ψ † σ h ∂ τ + ǫ (cid:16) − i~ ∇ (cid:17) − ǫ F i Ψ σ − Z dτ d ~x λ Ψ † σ Ψ † σ ′ Ψ σ Ψ σ ′ (3) S Eφ = Z dτ d ~xφ ∗ (cid:16) − ∂ τ − ~ ∇ + m φ + · · · (cid:17) φ (4) S Eψφ = − Z dτ d ~x h φ ∗ φ Ψ † σ Ψ σ , (5)where the sum over spin projection σ (and σ ′ ) is implied.We prefer to work in momentum space, and so write Z = Z [ dφ ][ dφ ∗ ][ dψ ][ d ¯ ψ ] e − ˜ S E (6)with the actions˜ S E = ˜ S Eψ + ˜ S Eφ + ˜ S Eψφ (7)˜ S Eψ = Z dǫd ~k (2 π ) ¯ ψ σ n − iǫ + ( e ( ~k ) − e F ) o ψ σ − Z dǫd ~k (2 π ) ! λ ¯ ψ σ ¯ ψ σ ′ ψ σ ψ σ ′ δ ( ǫ, ~k ) (8)˜ S Eφ = Z dωd ~q (2 π ) { φ ∗ ( ω + q ) φ + m φ φ ∗ φ + · · · } (9)˜ S Eψφ = − Z dǫd ~k (2 π ) ! (cid:18) dωd ~q (2 π ) (cid:19) h φ ∗ φ ¯ ψ σ ψ σ δ ( ω, ǫ, ~q, ~k ) . (10)where ψ and ¯ ψ are the eigenvalues of Ψ and Ψ † acting on | ψ i and h ¯ ψ (cid:12)(cid:12) ,Ψ | ψ i = ψ | ψ i and h ¯ ψ (cid:12)(cid:12) Ψ † = h ¯ ψ (cid:12)(cid:12) ¯ ψ (11)called fermion coherent state. Although our notations for the fields are general as they canapply to other systems like pions/nucleons and electrons/phonons, we should keep in mindthat we are specializing to the K − field for the boson and the proton and neutron for thefermion. We note that the action (7) is essentially the same model studied in [12] except for5he bosonic action. The bosonic action used there is different from (9) in that the systemconsidered there was assumed to be a compact-star matter in weak and chemical equilibrium(with the boson approximated to satisfy a quadratic dispersion relation – which as mentioned,is most likely incorrect for kaons) whereas here we are dealing with relativistic bosons whichwill be tuned to criticality by attractive KN interactions. It is interesting that a same-typeaction figures in condensed matter physics. In fact, (7) is quite similar to the action studiedfor quantum critical metals in [6] from which we shall borrow various scaling properties forthe renormalization group.For later discussions, it is useful to identify the constant h in (10) with what is in thestandard chiral Lagrangian. There are two terms contributing to it. In terms of the chiralcounting, the leading term (Ø( p )) is WT term and gives, for the s-wave kaon, h W T ∝ q /f π (12)where q is the fourth component of the kaon 4-momentum and the next chiral order ( O ( p ))term is the KN Σ-term h Σ ∝ Σ KN /f π . (13)As written, the action for the fermion system (8) is for a Fermi liquid with marginal four-Fermi interactions [10]. So the energy-momentum of the quasiparticle (fermion) is measuredwith respect to the Fermi energy e F and Fermi momentum k F . The bosonic action is formassive Klein-Gordon field, which later will be associated with a pseudo-Goldstone field,with higher-field terms ignored. In contrast to that of the quasi-particle, the boson energy-momentum will be measured from zero. We would like to look at meson-field fluctuationson the background of a fermionic matter given by the Fermi-liquid theory. The Fermi-liquidstructure is expected to be valid as long as N ≡ k F / ¯Λ ≫ − k F where Λ is the(momentum) cutoff scale from which mode decimation – in the sense of Wilsonian – is made.For nucleon systems, the action (8) gives the nuclear matter stabilized at the fixed point,Landau Fermi-liquid fixed point, with corrections suppressed by 1 /N [10]. The four-Fermiinteractions and the effective mass of the quasiparticle are the fixed-point parameters. Weexpect that as long as N ≫
1, the Fermi-liquid action can be trusted even at higher densitiesthan n provided of course there are no phase transitions that can destroy the Fermi-liquidstructure.For simplicity, we shall assume a spherically symmetric Fermi surface in which case wecan set for the quasiparticle momentum ~k = ~k F + ~l ≈ ˆΩ( k F + l ) . (14)Then for ¯Λ ≪ k F , we have e ( k ) − e F ≈ ~v F · ~l + O ( l ) (15)where v F is the Fermi velocity v F = k F /m ⋆ with m ⋆ the effective quasiparticle mass. Being an effective action, we need to set the cutoff scale Λ at which the classical action hasbare parameters. Quantum effects are calculated by doing the mode elimination (“Wilsonian6 (cid:112)(cid:112)(cid:112) (cid:146)(cid:146) (cid:154)(cid:147)(cid:147) (cid:154) (cid:71)(cid:71) (cid:111)(cid:111) (cid:47)(cid:111)(cid:47) (cid:83) (cid:83) (cid:147) (cid:38) (cid:112)(cid:112) (cid:146) (cid:38) (cid:71) (cid:109) (cid:146) (cid:38) (cid:11) (cid:12) (cid:47)(cid:14)(cid:58)(cid:32) (cid:109) (cid:146)(cid:146) (cid:7)(cid:38)
Figure 1: Only the orthogonal component of the momentum to the fermi surface scales as l → s l , where δ~k k is the parallel component of the momentum to the fermi surface.decimation”) by lowering the cutoff from Λ to s Λ with s <
1. We shall do this closelyfollowing the procedure given by [6]. The important point to note is that the boson andfermion fields have different kinematics. While the boson momentum is measured from theorigin – and hence the momentum cutoff is Λ , the fermion momentum is measured from theFermi momentum k F . Therefore the mode elimination for the fermion involves lowering thefermion momentum from ¯Λ = Λ − k F to s ¯Λ. As noted above, the strategy in the fermionsector is to take the large N limit where N ≡ k F / ¯Λ. We define the scaling laws of fields and other quantities like the delta functions by requiringthat the kinetic energy terms of the fermion and the boson be invariant under the scaling ǫ → sǫ, ω → sω, l → sl, ~q → s~q. (16)As is seen from Fig. (1), only the fermion momentum orthogonal to the Fermi surface scalesin the way the fermion energy and all components of the boson momentum do. Since thequasiparticle mass m ⋆ is a fixed-point quantity, the Fermi velocity does not scale. Thereforewe have the fields scaling as φ → s − φ,ψ → s − / ψ , (17)for which we denote the scaling dimensions as [ φ ] = − ψ ] = − / and the meson Note that the larger the k F or density, the more ¯Λ shrinks, which would mean that the large N ar-gument would hold better in the fermion sector as the density increases. This suggests that the mean fieldapproximation with effective Lagrangians – chiral Lagrangian or hidden local symmetry Lagrangian – wouldget better the higher the density. A caveat to this is that the argument would be invalid if there intervenednon-perturbative phenomena such as the topology change suggested in the skyrmion crystal model of densematter [17]. This notation will be used in what follows. Z dωd q m φ φ ∗ φ → Z dωd q m φ s − φ ∗ φ = Z dωd q m ′ φ φ ∗ φ, (18)with m ′ φ ≡ s − m φ . This shows the well-known fact that the meson mass term is “relevant.”Using the procedure of scaling toward the Fermi surface, we have[ dǫd k ] = 2 , [ δ ( ǫ, k )] = − . (19)This confirms that the four-Fermi interaction term in (8) is marginal. The scaling of the coupling term (10) is more subtle. Using the scaling dimensions[ φ ] = − , [ ψ ] = − / , [ dǫd ~k ] = 2 , [ dωd ~q ] = 4 , (20)we find the scaling of the integrand I ψφ of the action (10) written as R I ψφ is[ I ψφ ] = 3 + [ h ] + [ δ ( ω, ǫ, ~q, ~k )] , (21)where the bare coupling constant h will have the scaling dimension [ h W T ] = [ q ] = 1 and[ h Σ ] = 0. The ψφ coupling will be “relevant” if [ I ψφ ] <
0, marginal if = 0 and “irrelevant” if >
0. It is thus the scaling of the delta function that determines the scaling of the coupling h .In Appendix A, we suggest that depending upon the kinematics of the nucleon momentumparallel to the Fermi surface, there can be two scaling possibilities[ δ ( ω, ǫ, ~q, ~k )] = − , (22)and [ δ ( ω, ǫ, ~q, ~k )] = − . (23)As in Appendix A, we refer to (22) as “case 1” and (23) as “case 2”. From (21), we get forthe two cases of possible scaling:1. Case 1 [ I ψφ ] h WT = 2 , [ I ψφ ] h Σ = 1 . (24)Both couplings are irrelevant.2. Case 2 [ I ψφ ] h WT = 0 , [ I ψφ ] h Σ = − . (25)Here the Weinberg-Tomozawa coupling is marginal, while the Σ-term coupling is rele-vant. For two or three spatial dimensions, this applies to only forward scattering and BCS-type scattering.Others are marginal.
Here we present an argument that hidden local symmetry supplemented with scale invariancecould lead to the scaling (25).When the non-linear sigma model, with chiral perturbation theory as the flagship effectivefield theory for nuclear dynamics, is elevated to hidden local symmetry theory with thevector mesons (i.e., ρ and ω for two-flavor case) figuring as local gauge fields [18], it acquiresthe power, absent in the non-linear sigma model, to make certain unique predictions forthe properties of hadrons in the environment in which the vacuum is strongly modified bytemperature or density. One striking feature – which however remains neither confirmed norfalsified by experiments – is that when the quark condensate vanishes in the chiral limit asexpected at high temperature or density, what is referred to as “vector manifestation (VM)”fixed point is approached with the vector meson masses vanishing. If one assumes U (2) flavorsymmetry for the vector mesons, this then means both ρ and ω would be massless at the VM.In addition, if a scalar degree of freedom corresponding to the (pseudo-)Goldstone boson ofbroken scale symmetry, say, “dilaton”, is implemented, then the argument based on “mendedsymmetries” [19] would imply that the dilaton mass will also approach zero at the chiralrestoration point. Let us call this “mended hidden local symmetry (mHLS)” and the approachto the vanishing masses “mended vector manifestation (mVM).”In mHLS, one can think of the WT-type term and the Σ-type term in (10) arising inthe vacuum when the vector mesons and the scalar are, respectively, integrated out, therebygiving local interactions. This localization is clearly justified for low-energy dynamics inthe vacuum given the heavy masses involved. In dense medium, however, this localizationwould become problematic if the density (or temperature) were in the vicinity of the chiralrestoration point where the VM phenomenon sets in.What can we say about the RG flow of dense nuclear matter in mHLS?It was found in [20, 21] that the flavor U (2) symmetry for the vector mesons ( ρ, ω ) – whichis a fairly good symmetry in the vacuum – is strongly broken in medium. As a consequence, These terms in HLS implemented with dilaton, i.e., mHLS, have the same forms as the WT and Σ termsof the chiral Lagrangian, so we loosely call them by the same names. ρ meson is to decrease in accordance with the approach to the VM fixedpoint, the ω mass must remain more or less unaffected by the density except, perhaps, nearthe chiral restoration point. Though not firm, this feature is not inconsistent with what isobserved with the EoS of compact-star matter [22]. This means that the ω meson-exchangecontribution to the Weinberg-Tomozawa term in medium will be the same as the local formin the vacuum. As for the ρ contribution, the dropping ρ mass could make the local forma suspect at high density but its contribution is proportional to ( n p − n n ) – compared to( n p + n n ) for the ω exchange – where n n,p is (proton, neutron) number density. For n n > n p as in compact stars, the contribution is repulsive. Therefore hidden local fields are expectednot to modify the RG properties of the coupling (10) in medium.The situation with the Σ-like term contribution, however, can be quite different if thescalar dilaton implemented into HLS Lagrangian drops in mass as is indicated in [20]. Itis not clear whether the behavior of the scalar dilaton can be incorporated into a mVMstructure. But “mended symmetries” [19] for the π , ρ , scalar (denoted ǫ in [19]) and a dosuggest that the scalar mass could also drop significantly at high density. We shall refer tothe approach to mended symmetries as “mended-symmetry limit.”Implementing satisfactorily the mVM properties mentioned above would require a consis-tent treatment of both hidden local symmetry and conformal symmetry which are intricatelylinked to each other. We have at present no systematic treatment of the two symmetriesin dense medium but it is possible to gain insight into what could be going on by tree-levelconsideration of the Feynman diagrams in mHLS.The argument is quite simple. The crucial point to note is that in the one- φ exchangetree diagrams, the KN coupling giving rise to (10) differs from that leading to the four-Fermiinteraction (8) by that one of the two ψψφ vertices in the former is replaced by a KKφ vertex. Within the scaling rule we adopt, the ψψφ vertex is marginal but the
KKφ vertexis relevant. All other quantities are the same. Since in our approach, we want the four-Fermi interaction to remain marginal so that the baryonic matter remains in Fermi liquid,we need to enforce that the presence of the φ field leave unaffected the marginal four-Fermiinteraction. As shown in Appendix B, this is achieved by assigning a definite scaling propertyto the φ propagator. By imposing this scaling condition on the φ propagator that figuresin the KN-KN interaction, one sees immediately that the Σ-term type KN coupling in (10)must be relevant. This is simply because the KKφ vertex is relevant. One can verify thisresult by an explicit counting of the scalings involved in the Feynman diagrams as shown inAppendix B.Now using the same simple counting rule, it is easy to see that the WT term shouldbe marginal, i.e., it is in the case 2 (25). For this, simply replace the ψψφ vertex of thescalar-exchange term by a ψψω coupling and the
KKφ vertex by a q KKω vertex, whichamounts to replacing a marginal coupling by a marginal coupling and a relevant coupling bya marginal coupling. The net result is marginal because q KKω vertex is marginal.
We are interested in the flows of the kaon mass and the kaon-nucleon coupling and the effectof kaon condensation on the Fermi liquid structure of the baryonic system. Since the WT10oupling in its local form is irrelevant at the tree level for the case 1 and at one-loop level forthe case 2, we focus on the Σ-term coupling. Depending on the two cases of the delta functionscaling, its tree order scaling is either irrelevant or relevant. What matters crucially is thenthe loop correction, so we first look at one-loop contributions with the Σ-term coupling.Two-loop corrections should be suppressed for the large N limit.One-loop corrections to the beta function for the WT coupling are given in the nextsubsection. Σ -like coupling As shown above, under the scale transformation the coupling, h in the action scales h ′ = s − a h h , (26)where a h = − m ′ φ = s − m φ . (27) (cid:11)(cid:68)(cid:12)(cid:3) (cid:11)(cid:69)(cid:12)(cid:3) (cid:152) (cid:38) (cid:83) (cid:90) (cid:146) (cid:38) (cid:83) (cid:72) (cid:152) (cid:38) (cid:83) (cid:90) (cid:146) (cid:38) (cid:83) (cid:72) (cid:152) (cid:99)(cid:99)(cid:99)(cid:99) (cid:38) (cid:83) (cid:90) (cid:146) (cid:99)(cid:99)(cid:99)(cid:99) (cid:38) (cid:83) (cid:72) (cid:152) (cid:99)(cid:99)(cid:99)(cid:99) (cid:38) (cid:83) (cid:90) (cid:146) (cid:99)(cid:99)(cid:99)(cid:99) (cid:38) (cid:83) (cid:72) Figure 2: One-loop graphs contributing to h . The process corresponds to f(ermion) ( ǫ, ~k )+m(eson) ( ω, ~q ) → f(ermion) ( ǫ ′′ , ~k ′′ )+ m(eson) ( ω ′′ , ~q ′′ ). fermion meson Figure 3: One-loop graph contributing to m φ .11he one-loop graphs contributing to h and m φ are given, respectively, in Figs. 2 and3. What is significant for our analysis is that the one-loop graphs Figs. 2 turn out notto contribute to h Σ . Since this result is nontrivial, we sketch here the calculation for thevanishing contribution to h Σ with the details relegated to an appendix. The calculation for h W T – which does not vanish – is given in the next subsection.For the kinematics involved, the figures (a) and (b) of Fig.2 give the same result, so wefocus on figure (a). Denote the loop energy-momentum of the fermion and of the meson,respectively, as ( e ′ , ~k ′ ) and ( ω ′ , ~q ′ ). Then the energy-momentum conservation at each vertexleads to ω ′ = ǫ ′ − Q , (28) ~q ′ = Ω (cid:0) K F + l ′ (cid:1) − Ω ( K F + l ) , (29)where Q ≡ ǫ − ω and Ω ( K F + l ) ≡ ~k − ~q . Note that Q is the external energy differencebetween the fermion and the meson satisfying the dispersion relations, ǫ = ~k m ⋆ − ǫ F , (30) ω = ~q + m φ . (31)Then the one-loop contribution to the action (10) from Fig. 2(a) – detailed in Appendix B– is given by δS ( a ) ≡ Z (cid:0) dωd q (cid:1) (cid:0) dǫd k (cid:1) (2 π ) h φ ∗ < φ < ¯ ψ < ψ < δ (cid:16) ǫ, ω, ~k, ~q (cid:17) × Z dǫ ′ d k ′ (2 π ) − iǫ ′ − ~v F · ~l ′ ǫ ′ − Q ) + ( l ′ − l ) + 2 k F (1 − cos θ ) + m φ (32)where fields with < denote low-frequency modes retained after integrating out high-frequencymodes as defined in Appendix B. Since the Renormalization Group Equation(RGE) shouldnot depend on the external kinematics, i.e., Q and l , one can set Q = l = 0 to simplify theintegral. One can easily convince oneself that the integral vanishes, as shown explicitly inAppendix C. Thus there is no one-loop contribution to the β function for h Σ .The one-loop graph Fig. 3 contributing to the φ mass is easier to evaluate. By decimatingfrom ¯Λ to s ¯Λ in the nucleon loop, we get Z dωd ~q (2 π ) φ ∗ φs − " − m φ + γhk F π Z s ¯Λ < | l ′ | < ¯Λ dl ′ (cid:12)(cid:12) sgn (cid:0) l ′ (cid:1)(cid:12)(cid:12) , (33)where γ is the degeneracy factor for flavor (=2) and spin (=2): γ = 4 for nuclear matter and γ = 2 for neutron matter. It follows from above that δ (cid:2) m φ (cid:3) = s − (cid:20) m φ − γh ¯Λ k F π (1 − s ) (cid:21) − m φ . (34)12 .1.3 Flow analysis Although the case 2 (25) is favored by out approach based on mHLS, we treat both cases forcomparison.From what’s discussed above, it is straightforward to write down the RGEs. Setting t ≡ − ln s , d m φ dt = 2 m φ − A h , (35) d hdt = a h h − B h , (36)where A = γk F ¯Λ π , (37) B = 0 . (38)It is easy to get the analytic solutions for m φ and hm φ ( t ) = (cid:18) m φ (0) − h (0) A − a h (cid:19) e t + h (0) A − a h e a h t , (39) h ( t ) = h (0)e a h t , (40)where a h = ± m φ and h flow as t increases ( s decreases)for given values of A , h (0) and m φ (0). From Eqs. (39) and (40), we have the relation, m φ ( t ) − A h ( t )2 − a h [ h ( t )] /a h = m φ (0) − A h (0)2 − a h [ h (0)] /a h , (41)which is satisfied for any value of t . It is convenient to introduce the new parameter c a h which is independent of t , c a h ≡ m φ (0) − A h (0)2 − a h [ h (0)] /a h . (42)The quantities at t = 0, i.e., m φ (0) and h (0), are given at s = 1, that is, at the scalefrom which the decimation starts for a given k F . These parameters in the bare action –and the parameter c a h – depend only on density. Since the flow depends on c a h , the RGproperties of dense medium depend on the density dependence of the parameters of the barechiral Lagrangian at the scale Λ . This is equivalent to the BR scaling [13] that is obtainedin relativistic mean field treatment of effective Lagrangians of the HLS-type which is againequivalent to Landau Fermi-liquid theory. Now using Eqs. (41) and (42), we readily obtain the formulas for m φ ( t ) for the two cases a h = ± m φ ( t ) = c − h ( t ) + A h ( t ) for a h = − , (43)= c +1 h ( t ) + A h ( t ) for a h = +1 . (44)13igure 4: RG flows of the parameters, m φ ( t ) and h ( t ), are shown in the parameter space.The lines are drawn with the fixed value of A and the different values of c a h . The black solidline stands for the “critical line (or surface)” that delineates the parameter spaces.The flows m φ ( t ) vs. h ( t ) are plotted in Fig. 4 for both a h = − a h = 1 (rightpanel) for various values of c a h that depend on k F / ¯Λ. We note that m φ ( t ) flows toward zeroand becomes negative for c − ≤ c +1 <
0. This signals the instability toward kaoncondensation. What happens after the condensation is a matter that goes beyond the model.It could be stabilized by terms higher order in the kaon field not taken into account in ouranalysis.The conclusion is that only the m φ (0) and h (0) that satisfy m φ (0) ≤ A h (0) , for a h = − m φ (0) < Ah (0) for a h = +1 (46)will be in the parameter space for the condensation to take place. The value c ± = 0 definesthe critical line in the h ( t ) − m φ ( t ) plane, any point on which the parameters flow toward(for a h = −
1) or from (for a h = +1) the origin satisfying m φ ( t ) = A h ( t ) for a h = − , (47) m φ ( t ) = A h ( t ) for a h = +1 . (48)Note that in the RG flows of m φ ( t ) and h ( t ), as t goes up, the attractive KN interaction getsweaker if a h = − a h = +1. We are unable to precisely pin down the critical density for kaon condensation from the one-loop RG analysis given above. We can, however, use the critical line to get the “parameterwindow” that signals the instability toward the condensation.14t some density, according to the values of c a h , the RG flows of m φ and h ( t ) will followcertain of the flow lines in Fig. 4. If the value of c a h is changed by changing k F to k ′ F , theRG flow of m φ and h ( t ) follows from the line (in k F ) to the line (in k ′ F ). By locating the signchange of m φ (0) − A h (0)2 − a h , (49)one can then locate the density at which the phase is moved to the region that signals theinstability toward the kaon condensation. We use Eqs. (45) and (46) to make this estimate.Note that this does not pinpoint the critical condensation density. The sign change of c a h takes place when n ≡ γ k F π = N (2 − a h )6 m φ (0) h (0) . (50)If one takes h (0) = Σ KN /f from chiral Lagrangian, we get n ≈ (2 − a h ) N m K f Σ KN (51)where f ≈ f π . For H ≡ (2 − a h ) N ≈
1, this result reproduces the tree-order critical densityin chiral perturbation theory [23]. Just for illustration, let us take m φ (0) ≈
500 MeV and h (0) ≈ Σ KN /f π ≈
250 MeV / (93 MeV) . Then, by increasing k F , we move from the flowlines with c a h > c a h <
0. In this way, we obtain the parameterwindow for the condensation density n K ∼ n for (2 − a h ) N ≈
1. In the density above n K , wesatisfy the conditions Eqs. (45) and (46), and m φ ( t ) will flow to zero as t increases.As mentioned, the RG analysis anchored on the Fermi-liquid fixed point relies on N = k F / ¯Λ being large. Thus the higher the density, the better the one-loop approximation be-comes, with higher-loop terms suppressed for N ≫
1. The quantity
H ≡ (2 − a h ) N ≈ N ∼ a h = +1, whereas for the irrele-vant case with a h = − N ∼
2. This implies that a relevant coupling (which we suggestedis possible in HLS theory supplemented with a dilaton) would make the large-N argumentswork more powerfully for arriving at a low condensation density.The upshot of the above remark is that the relevance of the Σ-term scaling brings a largereduction of the critical density, say, by a factor of 3, from the naive irrelevant couplingexpected in the absence of mHLS. Equally importantly, if we consider kaon condensation incompact-star matter which is in weak and chemical equilibrium, incorporating µ e will reducefurther the critical density from what is estimated in nuclear matter. In compact-star matter,chemical equilibrium relates µ K to µ e which can be as large as ∼ m K /
2. The considerationof a large kaon chemical potential might even modify the dispersion relation of the kaon fieldas noted in [12]. This is a future work to be done.
The Weinberg-Tomozawa-type coupling has the meson fourth momentum q appearing in h .This makes the one-loop graph contribution to h W T non-vanishing contrary to what happened
The value for Σ KN ≈
250 MeV is the upper bound obtained in lattice QCD calculations [24]. S EW T = − Z dǫd ~k (2 π ) ! (cid:18) dωd ~q (2 π ) (cid:19) ih W T (cid:0) ω + ω ′′ (cid:1) φ ∗ ( ω, ~q ) φ ( ω ′′ , ~q ′′ ) ¯ ψ σ ψ σ δ ( ω, ǫ, ~q, ~k ) . (52)In doing the loop calculation similarly to the diagrams in Fig. 2, we get δS ≡ Z (cid:0) dωd q (cid:1) (cid:0) dǫd k (cid:1) (2 π ) h W T ( ω + ω ′ ) φ ∗ < φ < ¯ ψ < ψ < δ (cid:16) ǫ, ω, ~k, ~q (cid:17) × Z dǫ ′ d k ′ (2 π ) − iǫ ′ − ~v F · ~l ′ ǫ ′ ǫ ′ + l ′ + 2 k F (1 − cos θ ) + m φ , (53) ≈ Z (cid:0) dωd q (cid:1) (cid:0) dǫd k (cid:1) (2 π ) h W T ( ω + ω ′ ) φ ∗ < φ < ¯ ψ < ψ < δ (cid:16) ǫ, ω, ~k, ~q (cid:17) × Z dǫ ′ k F dl ′ d Ω(2 π ) − iǫ ′ − ~v F · ~l ′ ǫ ′ ǫ ′ + l ′ + m ′ φ , (54)which gives − Z dǫd ~k (2 π ) ! (cid:18) dωd ~q (2 π ) (cid:19) ih W T (cid:0) ω + ω ′′ (cid:1) φ ∗ ( ω, ~q ) φ ( ω ′′ , ~q ′′ ) ¯ ψψδ ( ω, ǫ, ~q, ~k ) + 4 δS. (55)After doing the integration, we have for the case 1 (case 2) d h W T dt = − h W T − k F B ′ N (2 π ) h W T , (56)where B ′ = Z d Ω 1 v F ¯Λ + q ¯Λ + m φ + 2 k F (1 − cos θ ) . (57)Although h W T has the non-vanishing one-loop contribution, suppressed by 1 /N , it makes h W T more irrelevant for the case 1. As for the case 2 that we argue to be the correctscenario, since the tree-order coupling is marginal and the one-loop correction is irrelevant,it is marginally irrelevant . Note also that B ′ is non-negative, so there is no non-trivial fixedpoint in h W T . We remark below what the implication of this marginally irrelevant term is inkaon condensation is.
We briefly recapitulate what we have found in this work. When the kaon condensation istreated on the background of Fermi liquid varying with density in the framework of chiral La-grangians, the Weinberg-Tomozawa-like term that plays a predominant role for kaon-nuclearinteractions at low energy associated with the Λ(1405) resonance is found to be at bestmarginally irrelevant in the RG sense at high density and hence unimportant for bringing16he system to kaon condensation. It is the Σ KN -like term which is subleading in chiral per-turbation theory, interpreted in terms of the dilaton scalar figuring in BR scaling, that cantrigger kaon condensation at densities relevant to compact stars, i.e. n K ∼ < n . That theΛ(1405) resonance – which is driven by the WT term in chiral effective field theory – playsno important role in the condensation per se was already foreseen many years ago even instandard chiral perturbation calculations starting with a chiral Lagrangian defined in thevacuum [23]. The RG analysis of this paper confirms what was found there in terms of thederivative coupling that is controlled by chiral symmetry. As in low-order chiral perturbationtheory, the critical density goes like ∼ / Σ KN in our RG analysis of the parameter win-dow. Though the analysis does not allow us to pin down the critical condensation density n K , it clearly indicates that the Fermi-liquid fixed point structure as arising in the mended-symmetry-implemented model gives the relevant RG coupling for the Σ KN -type interaction.This is how the RG approach anchored on the Fermi liquid structure would predict a con-siderably lower, say, a factor of ≃
3, critical density than that of ChPT. Furthermore asthe scalar mass approaches the mended-symmetry limit, the critical density proportional to m s (scalar mass squared) given in mHLS – or f in the eq. (51) – will be lower than thedensity without mended symmetries.The apparently insignificant role that the leading-chiral-order (Weinberg-Tomozawa) termfor KN interactions plays for the condensation process per se raises a tension with what isfound in anti-kaon nuclear interactions extensively studied both experimentally and theo-retically. It has been shown quantitatively [25] that the coupled-channel approach with thedriving term given by the Weinberg-Tomozawa term captures well kaon-nuclear interactionsat low energy. In this approach, the Λ(1405) resonance controlled by the WT term plays a cru-cial role. In fact, this is the approach adopted in most of the theoretical works on low-energykaon-nuclear physics found in the literature. Furthermore when K − ’s are present, bound, indense medium, the symmetry energy, an important ingredient for the EoS of compact stars,of the system in chiral perturbation theory at tree order is found to have a significant addi-tional term proportional to the square of the bound-kaon wave function determined more orless entirely by the WT term [11]. In both cases mentioned above, there is no visible directrole for the Σ-term coupling. What transpires here then is that the critical density is chieflygoverned by the chiral symmetry breaking term while low-energy kaon-nuclear dynamics andthe kaonic effect on the EoS of dense baryonic matter are controlled by the chirally symmetricterm. This could be better understood if the WT term, although playing no (significant) rolein determining the critical density n K , does figure in the EoS at the condensation point. Aclear understanding of this state of matter could be gained when the generalized HLS modelis treated self-consistently in weak and chemical equilibrium appropriate for compact stars.We return to this matter in a future publication.It has been recently suggested, based on a skyrmion-crystal description of dense mat-ter [17], that there can be a topological phase change at a density n ∼ > n which wouldmake the popular mean-field approximation used to describe the EoS of dense matter break-down [21]. Since the mean-field treatment of nuclear effective Lagrangians is equivalent toLandau-Migdal Fermi liquid theory [26], this would imply that the Fermi-liquid structureassumed for kaon condensation could be breaking down at a density corresponding to theonset of the topological phase change. An important question is then if there is a topologicalphase change at a density lower than a possible condensation density, how can the kaon con-17ensation process be described from such a non-Fermi liquid state? Another question tiedto the above is: Would kaon condensation, triggered by the Σ-term type interaction, inducethe onset of a non-Fermi liquid state of the type discussed in condensed matter physics? Ifso, what happens to the EoS in that phase? We have not yet succeeded to answer thesequestions fully and we will come back to them in the future.The RG analysis made here suggests that the WT term, playing an insignificant role inkaon condensation, would also be irrelevant in triggering the possible Fermi-liquid-to-non-Fermi liquid (FLNFL) transition. This follows from that the kaon is a (pseudo-)Goldstoneparticle and the WT term is the leading derivative coupling in kaon-Fermi-liquid interac-tions. And the derivative Goldstone-fermion coupling is known to prevent the FLNFL tran-sition [27]. It is the non-derivative coupling due to chiral symmetry breaking associated withthe spontaneous breaking of scale symmetry that could potentially provide a loophole to theGoldstone-boson’s “no-go theorem” for the FLNFL transition. Such loopholes are known toexist in condensed matter physics [27].A potentially promising possibility that is perhaps closely related to the role of the dilatonmentioned above is the density dependence on the coupling h suggested by BR scaling. Asshown by Song [28], the density-dependence in nuclear effective Lagrangians can be madeconsistent with the thermodynamic consistency only if it is given as a functional of the baryondensity local operator ψ † ψ . It seems possible in the spirit of [8] to have h Σ develop the scaling[ h Σ ] = − ψ † ψ dependence so that the action becomes relevant like [ I ψφ ] h Σ in Eq.(25).This is analogous to what happens in the presence of BR scaing. In chiral Lagrangians, theΣ-term goes as ∼ f − π which increases as the pion decay constant drops at increasing density.To conclude, we stress the basic difference between chiral perturbation theory where theexpansion is made in the “small parameter” k F whereas in the RG approach anchored onthe Fermi-liquid fixed point, the expansion is in the “small parameter” 1 /k F . This switchesthe importance of the WT-like term and the Σ-like term between the two approaches. Whatappears to be subleading in chiral counting in ChPT is relevant – and more important – in theRG sense in the effective model built on the Fermi-liquid fixed point. As the result, the latterpredicts that kaons could condense at a density considerably lower than in the former. If thisis correct, it will be necessary that kaon condensation be taken into account in calculatingthe EoS in stars. The question then is: If kaons can condense at a relatively low density,what about hyperons? The other question is: How can one reconcile with the ∼ Acknowledgments
We are grateful for discussions and comments from Chang-Hwan Lee and Hyun Kyu Lee. Oneof us (MR) would like to thank Youngman Kim for arranging the visit to the theory groupof Rare Isotope Science Project of Institute for Basic Science which made this collaborationpossible. The work of WGP was supported by the Rare Isotope Science Project of Institutefor Basic Science funded by Ministry of Science, ICT and Future Planning and NationalResearch Foundation of Korea (2013M7A1A1075766).18
PPENDIX
A Scaling of the Delta Function
We evaluate the scaling of the delta function following the reasoning of Polchinski [10]. Whenthe momenta of ψ and φ are given as in Fig. 5, we have the Dirac delta function in themomentum space as δ (3) (cid:0) ~p + ~q − ~p ′ − ~q ′ (cid:1) . (A.1)We decompose ~p and ~p ′ as ~p = ~k F + ~l p + δ~p k , (A.2) ~q = ~l q + δ~q k , (A.3) ~p ′ = ~k F + ~l p ′ + δ~p ′k , (A.4) ~q ′ = ~l q ′ + δ~q ′k , (A.5)the momenta can be expressed as in Fig. 5, where l ’s are perpendicular to the Fermi surfaceand the vectors with the subscript k are parallel to the Fermi surface. Note that the l ’s are (cid:89)(cid:3) k x k y k z p l ! // p ! " q ! q p l // p F K F K Fqqpp Kl // ,,, Figure 5: The kinematic configuration in the momentum space.independent of the parallel components. That is, δ (3) (cid:0) ~p + ~q − ~p ′ − ~q ′ (cid:1) = δ (2) (cid:16) δ~p k + δ~q k − δ~p ′k − δ~q ′k (cid:17) δ (1) (cid:16) ~l p + ~l q − ~l p ′ − ~l q ′ (cid:17) . (A.6)When the high frequency modes are integrated out up to s ¯Λ, the l ’s scale as l → sl and thedelta function constrains l’s as δ (1) (cid:16) ~l p + ~l q − ~l p ′ − ~l q ′ (cid:17) → δ (1) (cid:16) s~l p + s~l q − s~l p ′ − s~l q ′ (cid:17) = s − δ (1) (cid:16) ~l p + ~l q − ~l p ′ − ~l q ′ (cid:17) . (A.7)On the other hand, for the parallel components of the delta function, we should consider twopossible cases for the nucleon momentum: 19. In the case 1, we have δ (2) (cid:16) δ~p k + δ~q k − δ~p ′k − δ~q ′k (cid:17) → δ (2) (cid:16) δ~p k + sδ~q k − δ~p ′k − sδ~q ′k (cid:17) (A.8) ≈ δ (2) (cid:16) δ~p k − δ~p ′k (cid:17) . (A.9)So the delta function for the parallel components does not scale. In this case, we have[ δ ( p, q, p ′ , q ′ )] = − . (A.10)2. In the case 2, we have δ (2) (cid:16) δ~p k + δ~q k − δ~p ′k − δ~q ′k (cid:17) → δ (2) (cid:16) sδ~q k − sδ~q ′k (cid:17) (A.11)= (cid:0) s − (cid:1) δ (2) (cid:16) δ~q k − δ~q ′k (cid:17) , (A.12)where we imposed δ~p k = δ~p ′k . Then, the delta function for the parallel componentsscales as δ → s − δ . In this case,[ δ ( p, q, p ′ , q ′ )] = − . (A.13)This is the scaling of the delta function implied by the generalized hidden local sym-metry approach discussed in Section 2.3. B Scaling of One-Meson Exchange Diagrams
Here we give the details of how one-meson-exchange graphs scale. We use the recipes forscaling given in Section 2.2 and in [6]. The ambiguity in scaling of the delta function[ δ ( p, q, p ′ , q ′ )] = − − φ and K ) momentum. (cid:80)(cid:79)(cid:152) (cid:73) (cid:80)(cid:79) (cid:88) (cid:151) (cid:99)(cid:92) (cid:80)(cid:79) (cid:89) (cid:151) (cid:99)(cid:92) (cid:80)(cid:79) (cid:88) (cid:151) (cid:92) (cid:80)(cid:79) (cid:89) (cid:151) (cid:92) (cid:80)(cid:79)(cid:146)(cid:114) (cid:99) (cid:80)(cid:79) (cid:88) (cid:151) (cid:99)(cid:92) (cid:80)(cid:79)(cid:152) (cid:73) (cid:80)(cid:79) (cid:88) (cid:151) (cid:92) (cid:80)(cid:79)(cid:146)(cid:114) (cid:80)(cid:79)(cid:136) (cid:80)(cid:79)(cid:137) Figure 6: One- φ -exchange graphs for (a) the four-Fermi interaction and (b) the KN-KNinteraction. 20he graphs we consider are given in Fig. 6. First look at the four-Fermi interactionFig. 6(a). Written in full detail, it has the form I fermi = Z ( d p )( d p )( d p ′ )( d p ′ )( d q ) ¯ ψ ( p ′ ) ¯ ψ ( p ′ ) ψ ( p ) ψ ( p ) δ ( p − p ′ − q ) δ ( p − p ′ + q ) × q + m φ . (B.14)One may be tempted to integrate over the φ momentum using the delta function but weshould eschew doing this. Defining δ ≡ δ ( p − p ′ − q ) and δ ≡ δ ( p − p ′ + q ), we express I fermi in a compact form I fermi = Z ( d p ) ( d q ) ¯ ψ ¯ ψψψ × δ δ × q + m φ . (B.15)Now using the counting rules given in Section 2.2 and in [6] (for the ψψφ vertex), we have[ d p ] = 2, [ d q ] = 4, [ δ ] = [ δ ] = −
2, and [ ψ ] = [ ¯ ψ ] = − /
2. Thus we get[ I fermi ] = 2 + [“1 / ( q + m φ )”] . (B.16)In order to assure that the incorporation of φ , one of the mHLS degrees of freedom, leaves theFermi-liquid structure intact, we have to impose that I fermi be marginal, i.e., [ I fermi ] = 0.Therefore in the form given where the φ integration is left undone , we have the constraint[“1 / ( q + m φ )”] = − . (B.17)Note the quotation mark representing a mnemonic. One can see that this is equivalent tothe scaling [ d qφ ] = − I KN = Z ( d p ) ( d k ) ( d q ) ¯ ψψ ¯ KK × δ δ K × q + m φ . (B.18)Here two quasiparticle fields on the right vertex of Fig. 6(a) are replaced by two kaon fieldswith the corresponding integration measures and the delta function δ ≡ δ ( p − p ′ + q ) isreplaced by δ K ≡ δ ( k − k ′ + q ). Since we have [ d kK ] = 1 and [ δ K ]=-4, we can immediatelyread off the scaling [ I KN ] = 1 + [“1 / ( q + m φ )”] . (B.19)Given that the structure of the graph is identical to that of the four-Fermi interaction, wecan impose the scaling (B.17). This then gives [ I KN ] = −
1, i.e., the nonlocal form of Σ-termtype interaction is relevant. This discussion does not depend on what the mass of φ is, so wecan take it to ∞ , arriving at the local interaction (10) in the class 2 (25). What happens inthe “mended-symmetry limit” with m φ → Calculation of One-Loop Graph for h Σ In this appendix, we give the details of calculating the one-loop graph of Fig. 2 for the Σ-coupling. The details for the Weinberg-Tomozawa coupling are the same but the result willbe different.The action for the renormalization of h term that we are concerned with here is given atone loop by − Z dǫd ~k (2 π ) ! (cid:18) dωd ~q (2 π ) (cid:19) h φ ∗ φ ¯ ψψδ ( ω, ǫ, ~q, ~k ) − δS , (C.20)where δS = Z (cid:18) dωd q (2 π ) (cid:19) (cid:18) dǫd k (2 π ) (cid:19) h h φ ∗ φ ¯ ψψδ ( ω, ǫ, ~q, ~k ) i . (C.21)Decomposing the fields into ‘ < ’ for low frequency mode and ‘ > ’ for high frequency mode, andeliminating the high-frequency modes, there are four terms containing low-frequency modescontributing to h which can be equated to the two graphs given in Fig. 2. For the kinematicswe are interested in, the graphs (a) and (b) give the same results, so we will focus on (a). Ifwe set the loop energy-momentum of the fermion and the meson respectively as ( E ′ , ~k ′ ) and( ω ′ , ~q ′ ), and imposing the energy-momentum conservation E − ω + ω ′ − E ′ = 0 , (C.22) ~k − ~q + ~q ′ − ~k ′ = 0 , (C.23)we have ω ′ − E ′ = ω − E ⇒ ω ′ = ω − E + E ′ = ω − ǫ − ǫ F + ǫ ′ + ǫ F = ǫ ′ − Q , (C.24) ~k − ~q = ~k ′ − ~q ′ ⇒ ~q ′ = Ω (cid:0) k F + l ′ (cid:1) − Ω ( k F + l ) , (C.25)where Q ≡ ǫ − ω and Ω ( K F + l ) ≡ ~k − ~q . Then the loop energy-momentum is ω ′ = ǫ ′ − Q , (C.26) ~q ′ = Ω (cid:0) k F + l ′ (cid:1) − Ω ( k F + l ) , (C.27)where Q is the external energy difference between the fermion and meson, satisfying thedispersion relations, ǫ = ~k m ∗ − ǫ F , (C.28) ω = ~q + m φ . (C.29)From the above energy-momentum conservation, we have ~q ′ ≈ (cid:0) l ′ − l (cid:1) + 2 k F (1 − cos θ ) + O (1 /k F ) . (C.30)22he contribution from the diagram (a) is given by δS ( a ) ≡ Z (cid:0) dωd q (cid:1) (cid:0) dǫd k (cid:1) (2 π ) h φ ∗ < φ < ¯ ψ < ψ < δ (cid:16) ǫ, ω, ~k, ~q (cid:17) × Z dǫ ′ d k ′ (2 π ) − iǫ ′ − ~v F · ~l ′ ǫ ′ − Q ) + ( l ′ − l ) + 2 k F (1 − cos θ ) + m φ , (C.31) ≈ Z (cid:0) dωd q (cid:1) (cid:0) dǫd k (cid:1) (2 π ) h φ ∗ < φ < ¯ ψ < ψ < δ (cid:16) ǫ, ω, ~k, ~q (cid:17) × Z dǫ ′ k F dl ′ d Ω(2 π ) − iǫ ′ − ~v F · ~l ′ ǫ ′ − Q ) + ( l ′ − l ) + m ′ φ , (C.32)where we used k ′ = (cid:16) ~k F + ~l ′ (cid:17) ≈ k F with ¯Λ ≪ k F and defined m ′ φ ≡ m φ + 2 k F (1 − cos θ ).Let us evaluate the integral I ≡ Z dǫ ′ k F dl ′ d Ω(2 π ) ǫ ′ − Q ) + ( l ′ − l ) + m ′ φ − iǫ ′ − ~v F · ~l ′ (C.33)= k F (2 π ) Z ∞−∞ dǫ ′ Z s ¯Λ < | l ′ | < ¯Λ d Ω dl ′ ( ǫ ′ − Q ) + ( l ′ − l ) + m ′ φ − iǫ ′ − ~v F · ~l ′ . (C.34)Since the RGE of h should be independent of Q and l , we set Q = l = 0 in Eq. (C.34).Integrating over ǫ ′ , we get I ± = k F (2 π ) Z d Ω Z s ¯Λ < | l ′ | < ¯Λ dl ′ × ( − ± θ ( ∓ v F l ′ ) ∓ (cid:0) − v F (cid:1) l ′ + m ′ φ + v F l ′ (cid:0) − v F (cid:1) l ′ + m ′ φ q l ′ + m ′ φ , (C.35)where I ± is for the upper/lower hemisphere in integrating ǫ ′ in the ǫ ′ complex plane.23n doing the integration over l ′ , we use the relations , Z s ¯Λ < | l ′ | < ¯Λ dl ′ f ( l ′ ) = Z ¯Λ¯Λ(1 − δt ) dl ′ f ( l ′ ) + Z − ¯Λ(1 − δt ) − ¯Λ dl ′ f ( l ′ ) , (C.39) Z ¯Λ¯Λ(1 − δt ) dl ′ f ( l ′ ) = f ( ¯Λ) ¯Λ δt , (C.40) Z − ¯Λ(1 − δt ) − ¯Λ dl ′ f ( l ′ ) = f ( − ¯Λ) ¯Λ δt . (C.41)After integrating over l ′ , we get I ± = (cid:0) − ¯Λ δt (cid:1) k F Z d Ω(2 π ) × ± θ (cid:0) ∓ v F ¯Λ (cid:1) ∓ (cid:0) − v F (cid:1) ¯Λ + m ′ φ + v F ¯Λ (cid:0) − v F (cid:1) ¯Λ + m ′ φ q ¯Λ + m ′ φ + ± θ (cid:0) ± v F ¯Λ (cid:1) ∓ (cid:0) − v F (cid:1) ¯Λ + m ′ φ − v F ¯Λ (cid:0) − v F (cid:1) ¯Λ + m ′ φ q ¯Λ + m ′ φ (C.42)= 0 . (C.43) References [1] D. B. Kaplan and A. E. 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