Effective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei
aa r X i v : . [ nu c l - t h ] M a y Effective Field Theory of Emergent SymmetryBreaking in Deformed Atomic Nuclei
T. Papenbrock , and H. A. Weidenm¨uller Department of Physics and Astronomy, University of Tennessee, Knoxville, TN37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA Max-Planck Institut f¨ur Kernphysik, D-69029 Heidelberg, GermanyE-mail: [email protected] , [email protected] Abstract.
Spontaneous symmetry breaking in non-relativistic quantum systemshas previously been addressed in the framework of effective field theory. Low-lyingexcitations are constructed from Nambu-Goldstone modes using symmetry argumentsonly. We extend that approach to finite systems. The approach is very general. Tobe specific, however, we consider atomic nuclei with intrinsically deformed groundstates. The emergent symmetry breaking in such systems requires the introductionof additional degrees of freedom on top of the Nambu-Goldstone modes. Symmetryarguments suffice to construct the low-lying states of the system. In deformed nucleithese are vibrational modes each of which serves as band head of a rotational band. ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei
1. Introduction
In this paper we present a detailed approach towards emergent symmetry breakingin finite non-relativistic quantum systems that uses concepts of effective field theory(EFT) [1, 2, 3]. A short summary of this approach was previously presented inRef. [4], but without any details. Spontaneous symmetry breaking and the use ofEFT in infinitely extended non-relativistic quantum systems such as ferromagnets iswell established [5, 6]. Our work is specifically tailored for finite systems. Examplesare finite Bose-Einstein condensates or BCS superconductors with a broken U (1) phasesymmetry and molecules and nuclei that possess deformed ground states and, thus, breakrotational invariance. In finite systems there is no spontaneous symmetry breakingin the strict sense, and we speak instead of emergent symmetry breaking [7]. Thedescription of such systems within an EFT requires, aside from the standard Nambu-Goldstone modes, additional degrees of freedom. To be specific we address here the caseof deformed nuclei. The additional degrees of freedom then describe rotations of theentire nucleus. The treatment of emergent symmetry breaking in other systems requiresonly minor modifications, however.To set the stage we recall some aspects of the theory of nuclei with deformed groundstates such as occur in rare-earth nuclei and in the actinides. Low-lying excitationsare traditionally described phenomenologically as rotations and vibrations, both in thecollective geometric model [8] and in the algebraic model [9]. Both models describecertain “leading order” aspects (or gross features) very well: Low-lying excitations arevibrational states that serve as band heads of rotational bands. Electromagnetic intra-band transitions are very strong, inter-band transitions are much weaker. These modelstypically fail to account quantitatively for finer details regarding “next-to-leading order”effects such as the change of the moment of inertia with the vibrational band head orthe magnitude of weak E abinitio calculations of rotational bands in light nuclei [12, 13]. Similarly, nuclear mean-field calculations yield microscopic evidence for ground-state deformations [14]. For suchmicroscopic approaches the description of emergent phenomena is a challenge, however,because they present multi-scale problems. Therefore, the traditional approach withinthe Wigner-Weyl (linear) realization of the underlying symmetry requires very largemodel spaces to describe the emergent symmetry breaking.The approach using an EFT is based entirely upon symmetry arguments. Theemergent broken symmetry is treated using the nonlinear Nambu-Goldstone realizationof the symmetry (as opposed to the linear Wigner-Weyl realization) plus those additionaldegrees of freedom that account for rotations. The combined treatment of these degreesof freedom is the novel technical aspect of our work. The approach has the advantageof being model independent. A controlled expansion in terms of well-defined small ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei
2. General Approach
We consider a deformed liquid drop with a space–fixed center of mass and with axialsymmetry about the body–fixed z ′ –axis. In the present Section we introduce the Nambu-Goldstone modes and in addition the time-dependent modes that describe rotationalmotion. We warn the reader ahead of time that in Section 3 we switch to anotherparameterization of these modes. The parameterization used in the present Sectionis physically transparent and easy to justify. It has the drawback that it leads toanalytically cumbersome expressions. The parameterization used in Section 3 is easy tohandle but based on a different physical picture. In order to exhibit the general schemewe follow the approach of the present Section up to the construction of the classical fieldtheory and its symmetries. The developments of Section 3 then run in parallel to thesedevelopments. We denote the Cartesian coordinates in the space–fixed system S (the body–fixed system S ′ ) by { x, y, z } (by { x ′ , y ′ , z ′ } , respectively). The nuclear ground state is invariant under SO (2) rotations about the body–fixed z ′ –axis while SO (3) symmetry is broken by thedeformation. The Nambu-Goldstone modes parameterize the two-dimensional cosetspace SO (3) /SO (2) and depend on two complex fields π ± . In the body–fixed system, amass element of the liquid drop has spherical coordinates ( r, θ, φ ). The classical fields π ± depend on these dynamical variables and on time t . We neglect the r -dependence of π ± because radial vibrations of the liquid drop are expected to have higher excitationenergies than surface vibrations. We denote the remaining coordinates θ, φ and time t jointly by ρ µ with µ = 1 , ,
3. The fields π ± generate the local rotation U ( ρ ) = exp {− iπ − ( ρ ) P + − iπ + ( ρ ) P − } . (1)We follow Ref. [16] and denote the total angular momentum operator in the body-fixedsystem by ~P and in the space-fixed system by ~J . Then P x ′ , P y ′ , P z ′ are the componentsof ~P in the body-fixed system, and P ± = ∓ √ P x ′ ± iP y ′ ) . (2)Although we use the expression “angular momentum operator” for ~P and ~J , the threecomponents of these operators represent in the present context only the three generators ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei π + = π ∗− ,the transformation U is unitary. For a physical interpretation of the fields π ± we write π ± = 12 ω exp {± iζ } (3)with ω and ζ as the new real dynamical fields so that Eq. (1) becomes U = exp {− iω [cos ζ P x ′ + sin ζ P y ′ ] } . (4)We interpret Eq. (4) geometrically. In the body–fixed system, rotations are around anaxis perpendicular to the symmetry axis. The azimuthal angle of the rotation axis isdenoted as ζ . The angle ω is the angle of rotation about that axis. Replacing ζ by ζ + π we change the orientation of the axis of rotation or, equivalently, the direction of rotation(clockwise rotation → counter–clockwise rotation). Clockwise rotation by the angle ω is equivalent to counter–clockwise rotation by the angle 2 π − ω . Therefore, we mightconfine the angle ω to 0 ≤ ω ≤ π . For reasons that will become apparent later we do notadopt that choice here so that the ranges of the angles ζ and ω are 0 ≤ ω, ζ ≤ π . Weexpect the effective Lagrangian to depend on trigonometric functions of ω and ζ only.The fields ω and ζ are functions of the dynamical variables ρ µ , µ = 1 , ,
3. Inspectionshows that a pure time dependence of ω and ζ describes the rotation of the deformednucleus as a whole while a genuine dependence of ω and ζ on the angles θ and φ accountsfor surface vibrations, i.e., local dislocations of the constituents of the deformed drop(fluid elements or nucleons, as the case may be). Therefore, we write ω = ω ( t ) + ω ( θ, φ, t ) ,ζ = ζ ( t ) + ζ ( θ, φ, t ) , (5)with the understanding that ω and ζ possess a non–trivial dependence on at leastone of the angles θ, φ (so that not all partial derivatives with respect to these anglesvanish identically). We expect (and verify later) that the functions ω and ζ define thetrue Nambu–Goldstone modes of the deformed nucleus that describe surface vibrations.For these we will use a small–amplitude approximation. The purely time–dependentfunctions ω ( t ) and ζ ( t ), on the other hand, are alien to the standard approach tospontaneously broken symmetry. They represent the novel element that accounts forthe finite size of the system and describe rotational motion. For these functions wecannot use the small-amplitude approximation. This situation is technically similar tofinite-volume EFTs of quantum chromodynamics [17, 18] and magnets [19]. Here – andin contrast to the genuine finite systems we consider – the finite volume stems fromlimitations of computational resources and obscures the physics of the infinite systemone is interested in. In both cases, the purely time-dpendent “zero mode” exhibits largefluctuations and has to be treated separately.We follow Ref. [5] and define the Nambu–Goldstone modes a ± µ and a z ′ µ of our problemby writing U † ( ρ ) i∂ µ U ( ρ ) = a − µ P + + a + µ P − + a z ′ µ P z ′ . (6) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei ∂ µ = ∂/∂ρ µ for µ = 1 , ,
3. The coefficients a ± µ and a z ′ µ may alternatively be writtenas functions of the fields π ± or of the fields ω, ζ . The effective Lagrangian is obtainedby forming combinations of these coefficients that are invariant both with respect to thegroup operations and with respect to space rotations.We express the Nambu–Goldstone modes in terms of ω and ζ . The calculation islengthy but straightforward. We use the left-hand side of Eq. (6), the form (4) for U ,and a Taylor expansion for U . We find a − µ = (1 /
2) exp {− iζ } [ ∂ µ ω − i ( ∂ µ ζ ) sin ω ] ,a + µ = (1 /
2) exp { + iζ } [ ∂ µ ω + i ( ∂ µ ζ ) sin ω ] ,a z ′ µ = ( ∂ µ ζ )(1 − cos ω ) . (7) To determine the behavior of a ± µ and a z ′ µ under group operations we consider the actionof a fixed element g (independent of the ρ µ ) of the coset space on U . Equation (1)implies that under the action of g , U changes nonlinearly, U → [ gU ] h † ( g, U ) . (8)The product [ gU ] lies within the coset space SO (3) /SO (2) while h ( g, U ) = exp { i Ψ( g, U ) P z ′ } (9)is an element of SO (2). The function Ψ depends on the parameters characterizing both g and the transformation U . Thus, U † ( ρ ) i∂ µ U ( ρ ) → hU † ( ρ ) i [ ∂ µ U ( ρ )] h † + ∂ µ Ψ( g, U ) P z ′ (10)and a − µ P + + a + µ P − + a z ′ µ P z ′ → h ( a − µ P + + a + µ P − + a z ′ µ P z ′ ) h † + ∂ µ Ψ( g, U ) P z ′ . (11)We recall the commutation relations[ J x , J y ] = iJ z (cyclic) , [ P x ′ , P y ′ ] = − iP z ′ (cyclic) , [ J k , P l ′ ] = 0 (all k, l = x, y, z ) . (12)which imply [ J z , J + ] = J + , [ J z , J − ] = − J − , [ J + , J − ] = − J z , [ P z ′ , P + ] = − P + , [ P z ′ , P − ] = P − , [ P + , P − ] = P z ′ (13)and P mz ′ P ± = P ± ( P z ′ ∓ m . To calculate the right-hand side of expression (11) we usethese for ~P and the expansion of h in a Taylor series in P z ′ . We obtain a − µ → exp {− i Ψ } a − µ ,a + µ → exp { i Ψ } a + µ ,a z ′ µ → a z ′ µ + ∂ µ Ψ( g, U ) . (14) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei a ± µ , a z ′ µ that are invariant under groupoperations are [5] a + µ a − ν and ∂ µ a z ′ ν − ∂ ν a z ′ µ . (15)Here µ, ν = 1 , ,
3. The expressions (15) are obtained by treating the derivatives ∂ µ bothwith respect to the angles θ, φ and with respect to time as though they were derivativeswith respect to external variables. While that is appropriate for time t ( µ = 3), it isnot for the angles θ, φ ( µ = 1 ,
2) because these change under rotations. The effectiveLagrangian must be invariant with respect to such rotations. In actually constructingthe invariants we demand axial symmetry (see Section 3) and are guided by the analogyto coordinate transformations in Euclidean space in three dimensions. Here we wouldinterpret ∂ k with k = x, y, z as one component of the vector ~ ∇ , use the invariance ofthe scalar product of two vectors, and obtain the invariants ~ ∇ a + ~ ∇ a − . On the unitsphere we analogously replace ~ ∇ by the vector ~L of orbital angular momentum withcomponents L x ′ = + i sin φ ∂∂θ + i cos φ cot θ ∂∂φ ,L y ′ = − i cos φ ∂∂θ + i sin φ cot θ ∂∂φ ,L z ′ = − i ∂∂φ . (16)The lowest–order axially symmetric invariants formed from a ± are then L x ′ a + L x ′ a − + L y ′ a + L y ′ a − ,L z ′ a + L z ′ a − ,∂ t a + ∂ t a − . (17)Expressions like L x ′ a + are here understood as linear combinations of terms a + µ (read aspartial derivatives with respect to ρ µ and explicitly given in Eqs. (7)) with coefficientsgiven in the first of Eqs. (16). For simplicity, we employ in the present Section only thespherical invariant L x ′ a + L x ′ a − + L y ′ a + L y ′ a − + L z ′ a + L z ′ a − , (18)and use the more general axially symmetric invariants in Section 3.The substitution rules (14) apply in both the body-fixed and the space-fixedcoordinate system, albeit with different definitions of the function Ψ. Using this factand the well-known behavior of the components of ~L under rotations we easily confirmthat the form (18) is indeed rotationally invariant.The eigenstates of the nuclear Hamiltonian are (almost exactly) eigenstates of theparity operator. But that operator cannot be written in terms of rotations. Indeed,in three dimensions the determinant of the matrix describing parity inversion equalsminus one while the determinants of all rotation matrices are equal to plus one becausethese connect continuously to the unit matrix. Therefore, we cannot incorporate parity ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei R -parity ofRef. [8]) plays a role for the quantized version of the theory and is treated in Section 4.2,too. The explicit form of the invariants is obtained by using Eqs. (7) in expressions (17)and (18), L = 2 a − t a + t = 12 (cid:16) ˙ ω + ˙ ζ sin ω (cid:17) , L = 2 X k =1 ( L k a + )( L k a − ) = 12 (cid:16) ( ~Lω ) + sin ω ( ~Lζ ) (cid:17) . (19)The dot indicates the time derivative. The classical field theory is obtained by writing the effective Lagrangian density L as alinear combination of the two invariants (19), L = C L + D L . (20)The coefficients C and D are the parameters of the theory and are determined by a fitto the data. The total Lagrangian L is obtained by integrating L over the dynamicalangles θ, φ , L = Z d E L ≡ π Z π d θ sin θ Z π d φ L . (21)The normalization is chosen such that for L = 1 the integral in Eq. (21) yields unity.The factor sin θ in the measure (21) takes account of the fact that θ, φ define pointson the surface of a sphere so that we deal with curvilinear coordinates. We show inAppendix 1 that in such coordinates the equations of motion are X µ ∂ µ (cid:16) ∂ (sin θ L ) ∂ ( ∂ µ ω ) (cid:17) = ∂ (sin θ L ) ∂ω , X µ ∂ µ (cid:16) ∂ (sin θ L ) ∂ ( ∂ µ ζ ) (cid:17) = ∂ (sin θ L ) ∂ζ . (22)We recall that µ = 1 , θ, φ and µ = 3 for the time t , and thatboth ω and ζ are functions of θ, φ, t . Eqs. (22), (20), and (19) constitute the non–linearequations of motion for the two classical fields ω and ζ .In constructing L we have imposed rotational invariance. Therefore, we expect thatthe total angular momentum of the system is conserved. Using the Noether theorem asin Ref. [15], we now construct three constants of the motion. These correspond to the ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei k = x ′ , y ′ , z ′ we consider the rotation r = exp {− i X k δχ k P k } (23)by infinitesimally small angles δχ k about the three body–fixed axes. The ensuinginfinitesimal changes of ω and of ζ are denoted by δω and δζ , those of L by δ L . Astraightforward calculation yields δ L = X k =1 δχ k n X µ =1 ∂∂µ (cid:16) ∂ L ∂ ( ∂ µ ω ) M k + ∂ L ∂ ( ∂ µ ζ ) M k (cid:17)o . (24)With δω = δq and δζ = δq the matrix M is defined by δq ν = P k M νk δχ k andexplicitly given in Appendix 2. Rotational invariance implies δ L = 0 for every choice of δχ k with k = x ′ , y ′ , z ′ . That implies the vanishing of each of the three Noether currents k = x ′ , y ′ , z ′ in big curly brackets in Eq. (24).To obtain the constants of the motion we consider an infinitesimal change of thetotal Lagrangian given by δL = Z d θ d φ π δ (sin θ L ) , (25)with δ L defined in Eq. (24). The effective Lagrangian density L is periodic in the angle φ . With respect to θ , the same statement holds for L sin θ . Partial integration thenshows that the derivative terms with respect to θ and φ in Eq. (24) do not contributeto δL so that δL is given by a pure time derivative. The vanishing of δL for any choiceof the angles δχ k then implies the existence of three constants of the motion. These arethe three components Q k of the total angular momentum of the system. With M givenin Eq. (118) of Appendix 2 and the integration measure defined in Eq. (21) these are Q x ′ = Z d E (cid:16) ∂ L ∂ ˙ ω cos ζ − ∂ L ∂ ˙ ζ sin ζ cot ω (cid:17) ,Q y ′ = Z d E (cid:16) ∂ L ∂ ˙ ω sin ζ + ∂ L ∂ ˙ ζ cos ζ cot ω (cid:17) ,Q z ′ = − Z d E ∂ L ∂ ˙ ζ . (26) Without much justification given, we have constructed the effective Lagrangian inEqs. (20) and (19) from the lowest-order invariants in the Nambu-Goldstone modes.Further progress hinges on the identification of the energy scales that govern our problemand of the associated powers of higher-order terms in the Nambu-Goldstone modes.The relevant scales are the energy scale ξ of rotational motion, the energy scale Ω ofvibrational motion, and the cutoff parameter Λ beyond which other modes like fermionicexcitations play a role. In our approach it is assumed that ξ ≪ Ω ≪ Λ. ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei ω andof ζ in terms of spherical harmonics, ω = ∞ X L =2 X µ ω Lµ ( t ) Y Lµ ( θ, φ ) ,ζ = ∞ X L =2 X µ ζ Lµ ( t ) Y Lµ ( θ, φ ) . (27)The term with L = 1 describes center-of-mass motion and is suppressed. As mentionedearlier we take rotational motion fully into account by treating ω and ζ without anyapproximation. We use a small–amplitude approximation for the surface vibrationsdescribed by ω and ζ . A naive first approach would then consist in expanding L inpowers of ω and of ζ and in keeping only terms up to second order (but terms of allorders in ω and in ζ ). That would be in line with the treatment of ferromagnets andparamagnets in Ref. [5] where in the leading–order effective Lagrangian only terms upto second order in the fields are kept. However, because of the presence of rotationalmotion that procedure does not fully apply in our case. To see that we must analyzethe contributions to L in some detail. We use the symbol ∼ to define the relevant order,and we drop the magnetic quantum numbers on ω L and ζ L .The ratios ˙ ω /ω ∼ ξ and ˙ ζ /ζ ∼ ξ are governed by the energy scale of rotationalmotion. The range of the variables ω and ζ is of order unity. Therefore, ˙ ω ∼ ξ and˙ ζ ∼ ξ . Inserting Eqs. (5) into Eqs. (22) we obtain the terms ( C/
2) ˙ ω and ( C/
2) ˙ ζ .We show below that these describe rotational motion. Therefore, ( C/
2) ˙ ω ∼ ξ and( C/
2) ˙ ζ ∼ ξ . Together with ˙ ω ∼ ξ and ˙ ζ ∼ ξ that implies C ∼ /ξ which is consistentwith the interpretation of C as moment of inertia. The ratios ˙ ω L /ω L ∼ Ω and ˙ ζ L /ζ L ∼ Ωwith L ≥ L ≥ C/
2) ˙ ω L and ( C/
2) ˙ ζ L . Thesedescribe vibrational motion. Therefore ( C/
2) ˙ ω L ∼ Ω and ( C/
2) ˙ ζ L ∼ Ω. Together with C ∼ /ξ these relations imply ˙ ω L , ˙ ζ L ∼ √ ξ Ω and, together with ˙ ω L /ω L ∼ Ω, ˙ ζ L /ζ L ∼ Ωalso ω L ∼ p ξ/ Ω, ζ L ∼ p ξ/ Ω for L ≥
2. These relations are used below when weexpand L in powers of ω L and ζ L . The terms in L (see Eqs. (19)) describe vibrationalmotion so we have ( D/
2) ( ~Lω L ) ∼ Ω and ( D/ ~Lζ L ) ∼ Ω.In our approach the operator ~L is dimensionless (see Eqs. (16)). Formally, however, ~L plays the same role as the momentum in the theory of Ref. [5] where only smallmomenta are kept for small energies. Physically we analogously expect that onlythe small eigenvalues of the operator ~L are relevant for the low–energy part of thespectrum in our case. Therefore we formally attach to ~L the scale Ω. Together with ω L , ζ L ∼ p ξ/ Ω that gives D ∼ /ξ . We summarize these assumptions and results bywriting ξ ≪ Ω ,ω , ζ ∼ , ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei ω L , ζ L ∼ p ξ/ Ω ≪ L ≥ , ˙ ω , ˙ ζ ∼ ξ , ¨ ω , ¨ ζ ∼ ξ , ˙ ω L , ˙ ζ L ∼ p ξ Ω for L ≥ , ¨ ω L , ¨ ζ L ∼ Ω p ξ Ω for L ≥ ,C ∼ ξ ,D ∼ ξ ,~L ∼ Ω . (28)These considerations imply the following rule for an approximate treatment of theproblem. The effective Lagrangian L in Eqs. (20, 19) contains terms of order Ω thatdescribe nuclear surface vibrations and terms of order ξ that describe rotational motion.In expanding L in powers of ω L and ζ L we must, therefore, keep terms of orders Ω, √ Ω ξ and ξ . We omit terms of order p ξ/ Ω or less. The resulting approximate expressionsare C L ≈ C (cid:16) ( ˙ ω + ˙ ω ) + ˙ ζ sin ω + 2 ˙ ζ ˙ ζ h sin ω + ω sin 2 ω i + ˙ ζ h sin ω + ω sin 2 ω + ω cos 2 ω i(cid:17) ,D L ≈ D (cid:16) ( ~Lω ) + ( ~Lζ ) h sin ω + ω sin 2 ω + ω cos 2 ω i(cid:17) . (29)These developments show which terms to keep in the effective Lagrangian. To ascertain consistency of our arguments we consider the case of nuclear rotationwithout surface vibrations. We accordingly assume that ω and ζ depend only upontime, so that in Eqs. (5) we have ω = 0 = ζ . For purely time–dependent fieldsclassical field theory changes into classical mechanics. The treatment becomes verysimilar to that of Ref. [15]. The effective Lagrangian is L = C (cid:16) ˙ ω + ˙ ζ sin ω (cid:17) , (30)and the canonical momenta are π ω = ∂ L ∂ ˙ ω = C ˙ ω ,π ζ = ∂ L ∂ ˙ ζ = C ˙ ζ sin ω . (31) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei H is given by H = 12 C (cid:16) π ω + 1sin ω π ζ (cid:17) . (32)The three components Q k of angular momentum are given by Eqs. (26) which now read Q k = π ω M k + π ζ M k . (33)We use Eq. (118) of Appendix 2 for M and find X k Q k = π ω + 1sin ω π ζ . (34)That shows that the Hamiltonian (32) is proportional to the square of the total angularmomentum, H = 12 C X k Q k . (35)In other words, we obtain the classical theory of the rotating top. The constant C isthe moment of inertia.
3. Another Parameterization
We have actually carried the approach of Section 2 further, deriving the Hamiltonianand quantizing it. The resulting equations are difficult to interpret, however. Theydo not display in an obvious fashion what is expected on physical grounds: Harmonicvibrational motion of the variables ω L and ζ L . As shown in Appendix 3, these difficultieshave to do with the non–Cartesian form of the measure d E in Eq. (21). That is why wenow introduce another parameterization of the matrix U defined in Eq. (1). We proceedin close analogy to Section 2. We use the space-fixed system, and we parameterize the matrix U in product form, U = g ( ζ , ω ) u ( x, y ) ,g ( ζ , ω ) = exp n − iζ ( t ) ˆ J z o exp n − iω ( t ) ˆ J y o ,u ( x, y ) = exp n − ix ˆ J x − iy ˆ J y o . (36)The purely time–dependent variables ω and ζ describe rotations of the finite system,similarly to the variables ω , ζ introduced in Eqs. (5). As in Section 2 we choose theranges as 0 ≤ ω, ζ ≤ π . This is convenient for Section 4.1. With θ and φ as definedin Section 2, the fields x = x ( θ, φ, t ) and y = y ( θ, φ, t ) play the role of the fields ω and ζ defined in Eqs. (5). They describe the small-amplitude vibrations of the liquid drop.To exclude the possibility that x and y induce a global rotation of the entire drop werequest Z d E x ( θ, φ, t ) = 0 = Z d E y ( θ, φ, t ) . (37) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei U = gu acts onto objects to the right. Thus, the localvibrations induced by the field u are followed by a global rotation g of the entire drop.We show in Appendix 3 that the parameterization of U in terms of the variables ω and ζ in Eqs. (4) and (5) and the one introduced in Eq. (36) are completely equivalent.Why then did we not start from the outset with the parameterization (36)? As shownin Section 2, the parameterization used in Eqs. (4) and (5) can be justified physically ina convincing manner. Moreover, it is tailored after the standard approach to symmetrybreaking in non-relativistic systems. The advantage of the new parameterization isthat it treats the rotational degrees of freedom separately while the parameterizationin Eqs. (4) and (5) treats the rotational mode and the vibrational modes on an equalfooting. It has an alternative physical interpretation. When acting from right to left, u induces a small-amplitude dislocation of a nucleon (or volume element) at ( θ, φ ) in theaxially-symmetric nucleus (whose symmetry axis is the z axis), while g then rotates theentire nucleus.Power counting as in Section 2.4 shows that ω, ζ ∼ O (1) , ˙ ω, ˙ ζ ∼ ξ , | x | , | y | ∼ ε / ≪ , ˙ x, ˙ y ∼ Ω ε / . (38)The parameter ε helps to identify (and omit) higher powers of x and y , consistent witha focus on small-amplitude harmonic surface vibrations. A physical interpretation ofthis parameter is given at the end of Section 3.3 below. From here on the developmentis similar to that of Section 2.In analogy to Eq. (6) we define U − i∂ µ U = a xµ J x + a yµ J y + a zµ J z . (39)As in Section 2 the symbol ∂ µ with µ = 1 , , θ, φ and time t while in the present Section ∂ ν with ν = 1 , g − ∂ t g = − i (cid:16) − ˙ ζ sin ωJ x + ˙ ωJ y + ˙ ζ cos ωJ z (cid:17) ,u − ∂ t u = − i (cid:16) ˙ x + y x ˙ y − y ˙ x ) (cid:17) J x − i (cid:16) ˙ y − x x ˙ y − y ˙ x ) (cid:17) J y − i (cid:18)
12 ( y ˙ x − x ˙ y ) (cid:19) J z , (40)and, from the Baker–Campbell–Haussdorf expansion, U − ∂ ν U = u − ∂ ν u ,U − ∂ t U = u − ∂ t u + u − (cid:0) g − ∂ t g (cid:1) u ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei − i (cid:16) ˙ x − ˙ ζ sin ω + y x ˙ y − y ˙ x ) − y ˙ ζ cos ω (cid:17) J x − i (cid:16) ˙ y + ˙ ω − x x ˙ y − y ˙ x ) + x ˙ ζ cos ω (cid:17) J y − i (cid:18)
12 ( y ˙ x − x ˙ y ) + ˙ ζ cos ω − x ˙ ω − y ˙ ζ sin ω (cid:19) J z + . . . . (41)Here and in what follows, the dots indicate terms of higher order in ε . From Eqs. (40)and (41) we obtain a xt = ˙ x + y x ˙ y − y ˙ x ) − ˙ ζ sin ω − y ˙ ζ cos ω + . . . ,a yt = ˙ y − x x ˙ y − y ˙ x ) + ˙ ω + x ˙ ζ cos ω + . . . ,a zt = −
12 ( x ˙ y − y ˙ x ) + ˙ ζ cos ω − y ˙ ζ sin ω − x ˙ ω + . . . , (42)and a xν = ∂ ν x + y x∂ ν y − y∂ ν x ) + . . . ,a yν = ∂ ν y − x x∂ ν y − y∂ ν x ) + . . . ,a zν = −
12 ( x∂ ν y − y∂ ν x ) + . . . . (43)Eqs. (42) and (43) give the lowest-order contributions to the Nambu-Goldstone modesfor the parameterization (36). As in Section 2.2 we build the effective Lagrangian upon invariants constructed fromthe Nambu-Goldstone modes. To this end we need to determine the behavior of thesemodes under transformations. We consider a rotation r about infinitesimal angles δχ k around the space-fixed k = x, y, z axes. We use Eq. (36) for U . With rg ( ζ , ω ) = g ( ζ ′ , ω ′ ) h ( γ ′ ) , h ( γ ′ ) = exp { iγ ′ J z } , (44)and ζ ′ , ω ′ , γ ′ given below we have rU = rg ( ζ , ω ) u ( x, y ) = g ( ζ ′ , ω ′ ) h ( γ ′ ) u = g ( ζ ′ , ω ′ ) [ h ( γ ′ ) uh † ( γ ′ )] h ( γ ′ ) = U ′ h . (45)The last of equations (45) defines U ′ ≡ U ( ζ ′ , ω ′ , x ′ , y ′ ) as an element of the coset space SO (3) /SO (2). We accordingly write U ′− i∂ µ U ′ = ( a xµ ) ′ J x + ( a yµ ) ′ J y + ( a zµ ) ′ J z . (46)Proceeding as in Section 2.2 we have U − ∂ µ U → ( rU ) − ∂ µ ( rU )= ( U ′ h ) − ∂ µ ( U ′ h )= h − ( U ′ ) − ( ∂ µ U ′ ) h + h − ( ∂ µ h ) . (47) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei ( a xµ ) ′ ( a yµ ) ′ ! = cos γ ′ − sin γ ′ sin γ ′ cos γ ′ ! a xµ a yµ ! (48)and ( a zµ ) ′ = a zµ + δ µt ˙ γ ′ . (49)We have used that γ ′ as given in Eq. (52) below is a function of t only.It remains to work out the relation between the variables ζ ′ , ω ′ , x ′ , y ′ and ζ , ω, x, y .A calculation similar to that of Appendix 2 shows that g ( ζ , ω ) transforms into g ( ζ ′ , ω ′ )with ζ ′ = ζ + δζ and ω ′ = ω + δω where δζδω ! = − cot ω cos ζ − cot ω sin ζ − sin ζ cos ζ ! δχ x δχ y δχ z . (50)According to Eq. (45) the matrix u transforms under the action of r into h ( γ ′ ) uh † ( γ ′ ).That transformation differs from that of Eq. (8) because of the prefactor g in thedefinition of U in Eqs. (36). Therefore, x and y transform into x ′ and y ′ accordingto x ′ y ′ ! = cos γ ′ − sin γ ′ sin γ ′ cos γ ′ ! xy ! (51)where γ ′ = cos ζ sin ω δχ x + sin ζ sin ω δχ y . (52)Eq. (51) shows that x + y is invariant under rotations. Moreover, since γ ′ dependson time, under rotations the four quantities x, y, ˙ x, ˙ y are transformed into linearcombinations of x ′ , y ′ , ˙ x ′ , ˙ y ′ .We are now ready to construct the invariants. We begin with the time derivativesin Eqs. (42). Eq. (48) shows that ( a xt ) + ( a yt ) is invariant. We use the power countingof Eqs. (38) (see also Section 3.4 below) and drop terms of order ξ ε and ξ Ω ε k with k ≥ /
2. We also omit terms linear in x, y, ˙ x , or ˙ y as these vanish upon integration over θ and φ , see Eqs. (37). That gives( a xt ) +( a yt ) ≈ ˙ ω + ˙ ζ sin ω + ˙ x + ˙ y +2( x ˙ y − y ˙ x ) ˙ ζ cos ω −
13 ( x ˙ y − y ˙ x ) . (53)The invariant form (53) is the sum of three homogeneous polynomials of orders zero,two, and four, respectively, in the variables x , y , and their time derivatives. Underrotations, each of these is transformed into another homogeneous polynomial of thesame order, see the text below Eq. (52)). Invariance of the form (53) implies that eachof the said polynomials is invariant by itself. Hence the invariants are L a = ˙ ω + ˙ ζ sin ω , L b = ˙ x + ˙ y + 2( x ˙ y − y ˙ x ) ˙ ζ cos ω , L c = ( x ˙ y − y ˙ x ) . (54) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei L d = ( x + y )[ ˙ x + ˙ y + 2( x ˙ y − y ˙ x ) ˙ ζ cos ω ] (55)is obtained by multiplying L b with the invariant ( x + y ). The invariant L d is of thesame order as L c .We turn to the invariants constructed from the derivatives with respect to the angles θ, φ in Eqs. (43). We confine ourselves to terms of up to fourth order in x and y andtheir derivatives. Eq. (48) shows that for all ν = θ, φ the form ( a xν ) + ( a yν ) is invariant,and so are a zν and a zν a zν ′ . Forming suitable linear combinations of these and multiplyingwith the additional invariant ( x + y ) we find the invariants L a = ( ~Lx ) + ( ~Ly ) , L a ′ = ( L z x ) + ( L z y ) , L b = ( x~Ly − y~Lx ) , L c = ( x + y ) (cid:16) ( ~Lx ) + ( ~Ly ) (cid:17) . (56)The construction of the invariants in Eqs. (54) to (56) is based on the fact that underrotations, u transforms into huh † . This feature does not apply in the absence of anyrotational motion, i.e., for g = 1 or ω = 0 = ζ . It is easy to see that in that casewe would have rU = ru = u ( x ′ , y ′ ) h where x ′ , y ′ are nonlinear functions of x, y . Theargument shows why the invariants in Eqs. (54) to (56) occur specifically in the case ofrotational motion but differ in cases like ferromagnetism or paramagnetism where allmodes considered are true Nambu-Goldstone modes. It also shows that care is neededwhen considering the limit of an infinitely large moment of inertia: In the framework ofthe present formalism, that limit differs from the one where rotational motion is ruledout from the outset.In the construction of the invariants, time-reversal invariance has been taken intoaccount in the same manner as in Section 2.2. When we apply our formalism to atomicnuclei, an additional symmetry (the R -symmetry) comes into play. That symmetrymatters for the quantized version of the theory and is, therefore, deferred to Section 4.2. As in Section 2.3 the effective Lagrangian L is given in terms of an arbitrary linearcombination of the invariants constructed in Section 3.2 and involves eight constants C i , i = a, b, c, d and D i , i = a, a ′ , b, c that must be determined by a fit to data. Inobvious notation we have L = L + L = Z d E L = Z d E (cid:16) X i = a,b,c,d C i L i − X i = a,a ′ ,b,c D i L i (cid:17) . (57)The integration over angles is defined in Eq. (21). ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei x in two ways, eitherin spherical harmonics Y Lµ = ( − ) µ Y ∗ L − µ or in terms of the real orthonormal functions Z Lµ ≡ √ (cid:0) Y Lµ + Y ∗ Lµ (cid:1) , µ > ,Y L , µ = 0 , i √ (cid:0) Y Lµ − Y ∗ Lµ (cid:1) , µ < . (58)We note that the functions Z Lµ do not form the components of a spherical tensor. Wewrite x = ∞ X L =2 L X µ = − L x Lµ Z Lµ = ∞ X L =2 L X µ = − L ˜ x Lµ Y Lµ , (59)and correspondingly for the real variables y , ˙ x , ˙ y . As in Eqs. (27), contributions with L = 0 and L = 1 are excluded. The coefficients x lµ are real. For the complex coefficients˜ x Lµ we have ˜ x ∗ Lµ = ( − ) µ ˜ x L − µ . For every value of L the coefficients ˜ x Lµ and x Lµ ′ arelinearly related in an obvious way. The first of Eqs. (59) is useful for quantization. Thesecond is more useful when the calculation requires angular-momentum algebra.Using the first Eq. (59) and carrying out the integration over angles, we obtain forthe total Lagrangian in Eq. (57) L = C a (cid:16) ˙ ω + ˙ ζ sin ω (cid:17) + C b X L X µ (cid:0) ˙ x Lµ + ˙ y Lµ (cid:1) + C b ˙ ζ cos ω X L X µ ( x Lµ ˙ y Lµ − y Lµ ˙ x Lµ ) − X Lµ (cid:0) D a L ( L + 1) + D a ′ µ (cid:1) (cid:0) x Lµ + y Lµ (cid:1) . (60)We have restricted ourselves to terms up to and including the orders O (Ω), O ( ε Ω), and O ( ξ ). The canonical momenta are p ω = ∂ L ∂ ˙ ω , p ζ = ∂ L ∂ ˙ ζ , p xLµ = ∂ L ∂ ˙ x Lµ , p yLµ = ∂ L ∂ ˙ y Lµ . (61)In analogy to the first Eq. (59) we define the real function p x ( θ, φ ) = ∞ X L =2 L X µ = − L p xLµ Z Lµ ( θ, φ ) , (62)and correspondingly for p y ( θ, φ ).We use the Noether theorem as in Section 2.3 and find the conserved quantities( k = x, y, z ) Q k = Z d E (cid:26) δωδχ k p ω + δζδχ k p ζ + δxδχ k p x + δyδχ k p y (cid:27) . (63) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei Q k with the three components of angular momentum. Explicitly weobtain from Eq. (50) and from the differential form of Eq. (51) Q x = − p ω sin ζ − p ζ cot ω cos ζ + cos ζ sin ω K ,Q y = p ω cos ζ − p ζ cot ω sin ζ + sin ζ sin ω K ,Q z = p ζ . (64)Here K = Z d E ( xp y − yp x ) . (65)The terms in Eqs. (64) that do not involve the factor K correspond to the angularmomentum of the rigid rotor. That is shown below and is analogous to Section 2.5. Theintegral K over solid angle in Eq. (65) is the angular momentum of the two–dimensionaloscillators that describe the surface vibrations. That is easily shown by applying theNoether theorem to R d E ( x + y ). As remarked below Eq. (52), that expression isinvariant under SO (2) transformations of ( x, y ). The conserved quantity associatedwith this invariance is K . The square of the total angular momentum is Q = p ω + 1sin ω (cid:0) p ζ − Kp ζ cos ω + K (cid:1) . (66) Given the full Lagrangian in Eq. (60) we can now complete the arguments leading tothe relations (38). To identify the terms that are kept we use arguments similar to theones in Section 2.4. In addition to Eqs. (38) we assume D a , D a ′ ∼ Ω /ε ,C a ∼ ξ − ,C b , C c , C d ∼ ( ε Ω) − , (67)which implies C a L a ∼ ξ ,C b L b ∼ Ω ,C c L c , C d L d ∼ ε Ω , (68)and p ω , p ζ ∼ ,p x , p y ∼ ε − / . (69)If we were to scale C c ∼ ( ε Ω) − we would find C c L c ∼ Ω, and that would be unexpected(or “unnatural”) for a term with such a high power in the coordinates x, y . We cannotcompletely rule out this possibility, however, as effective field theories with unnaturallylarge scale do exist. As an example we mention the pion-less nuclear effective field theory ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei p x , p y implied bythe relations (69) do not seem natural but cannot be avoided if we insist on x, y ∼ ε / .To illuminate the role of the dimensionless parameter ε we consider the limit ofan infinite system (with infinite moment of inertia C a , or ξ → ζ and ω become static and have constant values that depend on the orientation of the rotor.Spontaneous symmetry breaking gives rise to the Nambu–Goldstone fields x and y thatdescribe the low–energy modes at the scale Ω. The effective field theory breaks downat the scale Λ ≫ Ω. The assumption ε ∼ Ω / Λ ≪ x or y that were suppressed in Eq. (53) would beof order Ω ε k with k ≥ / / Λ. A similar power countingfor the spatial derivatives results from the replacement ~L → Ω ~L as in Section 2.4.In the opposite case where Λ → ∞ but where ξ ≪ Ω is finite (i.e., differs fromzero), the terms of order Ω ε ∝ / Λ disappear. The Hamiltonian describes a rotorcoupled to a set of oscillators. A problem occurs once the excitation energy is so largethat the amplitudes x, y are of order unity and compete with the finite rotations of thetop. A distinction between the two types of modes is then no longer meaningful, andspontaneous symmetry breaking does not give an adequate description of the system.
The effective Hamiltonian H is obtained from the effective Lagrangian L via a Legendretransformation. To perform that transformation we write the kinetic part L of L in a form that displays its bilinear dependence on the velocities. We define theinfinite-dimensional velocity vector V T = { ˙ ζ, ˙ ω, ˙ x Lµ , ˙ y Lµ } where L = 2 , , . . . and µ = L, L − , . . . , − L and write L = (1 / V T ˆ GV . The matrix ˆ G is easily foundfrom Eq. (60). We analogously define the vector of momenta P T = { p ζ , p ω , p xLµ , p yLµ } .Then the effective classical Hamiltonian is H = 12 P T ˆ G − P + 12 X Lµ (cid:0) D a L ( L + 1) + D a ′ µ (cid:1) (cid:0) x Lµ + y Lµ (cid:1) . (70)To calculate the inverse ˆ G − we write ˆ G = ˆ G + ˆ G , where ˆ G is the diagonal part of ˆ G ,and use perturbation theory in ˆ G so that ˆ G − = ˆ G − − ˆ G − ˆ G ˆ G − + ˆ G − ˆ G ˆ G − ˆ G ˆ G − ± . . . . Keeping only terms up to and including the orders O (Ω), O ( ε Ω), and O ( ξ ) we obtain H = 12 X Lµ (cid:20) C b (cid:0) ( p xLµ ) + ( p yLµ ) (cid:1) + (cid:0) D a L ( L + 1) + D a ′ µ (cid:1) (cid:0) x Lµ + y Lµ (cid:1)(cid:21) + 12 C a (cid:20) p ω + 1sin ω (cid:0) p ζ + 2 Kp ζ cos ω + K cos ω (cid:1)(cid:21) = 12 X Lµ (cid:20) C b (cid:0) ( p xLµ ) + ( p yLµ ) (cid:1) + (cid:0) D a L ( L + 1) + D a ′ µ (cid:1) (cid:0) x Lµ + y Lµ (cid:1)(cid:21) + 12 C a (cid:0) Q − K (cid:1) . (71) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei K = X Lµ (cid:0) x Lµ p yLµ − y Lµ p xLµ (cid:1) . (72)In the Hamiltonian (71), the Nambu-Goldstone modes undergo harmonic vibrations.These are coupled via K to a rigid rotor. The vibrations are of order O (Ω), while therotations are of order O ( ξ ).We mention in passing that an alternative (and simpler) form of the Legendretransformation seems to exist. Instead of defining the momenta as in Eqs. (61) and usingEq. (70), one might define the canonical momenta p x ( θ, φ ) and p y ( θ, φ ) as functionalderivatives of P i C i L i with respect to x ( θ, φ ) and y ( θ, φ ), use ( p , p , p ) = ( p ζ , p x , p y )and ( q , q , q ) = ( ζ , x, y ), and define H as H = p ω ˙ ω + Z d E X i =1 p i ˙ q i − L . (73)That procedure yields the same result as the one used above only to first order in ˆ G .The terms of second order differ, and the Hamiltonian resulting from Eq. (73) is notrotationally invariant. More precisely: Upon quantization H does not commute withthe three components Q k of angular momentum in Eq. (64). Therefore, we have notused Eq. (73).
4. Quantized Hamiltonian
In the present Section we complete the program of the paper. We quantize the effectiveHamiltonian (71), we discuss two important discrete symmetries, and we investigate theresulting spectra.
In quantizing H and the three components Q k of angular momentum, we encounterthe problem that ω and ζ are curvilinear coordinates, and that quantization in suchcoordinates is ambiguous, see Ref. [22] and references therein. The quantization dependson the physical constraints that limit the dynamics to the curved manifold, and therebyon the physical situation. We focus attention on the relevant parts of H and of the Q k .These are given by the Hamiltonian H rot of the rigid rotor, H rot = 12 C a (cid:16) p ω + 1sin ω p ζ (cid:17) , (74)and by the associated rigid-rotor parts of Eqs. (64), Q rot x = − p ω sin ζ − p ζ cot ω cos ζ ,Q rot y = p ω cos ζ − p ζ cot ω sin ζ ,Q rot z = p ζ . (75) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei H rot . In Appendix 4 we describe adifferent approach to quantization which avoids the ambiguity encountered below andyields the same result. With G − = (cid:18) / sin ω (cid:19) (76)we write H rot in matrix form, H rot = 12 C a ( p ω , p ζ ) G − p ω p ζ ! . (77)Quantization is achieved upon puttingˆ H rot = 12 C a √ det G ( − i∂ ω , − i∂ ζ ) G − √ det G − i∂ ω − i∂ ζ ! = − C a (cid:16) ∂ ω + cot ω∂ ω + 1sin ω ∂ ζ (cid:17) . (78)The transition from Eq. (74) to Eq. (78) is tantamount to puttingˆ p ω = − i √ sin ω ∂ ω √ sin ω , ˆ p ζ = − i∂ ζ . (79)The expressions (75) must be symmetrized with respect to ζ , p ζ prior to using Eqs. (79).That gives ˆ Q rot x = i sin ζ ∂ ω + i cot ω cos ζ ∂ ζ , ˆ Q rot y = − i cos ζ ∂ ω + i cot ω sin ζ ∂ ζ , ˆ Q rot z = − i∂ ζ . (80)It is easy to check that the three components obey [ ˆ Q rot x , ˆ Q rot y ] = i ˆ Q rot z (cyclic). Therefore,ˆ Q rot = { ˆ Q rot x , ˆ Q rot y , ˆ Q rot z } is a bona fide angular-momentum operator.For the remaining variables, we impose the usual quantization conditions and choosea representation where all the x ’s and y ’s are ordinary variables so thatˆ p xLµ = − i ∂∂x Lµ , ˆ p yLµ = − i ∂∂y Lµ . (81)The components of the quantized angular momentum operator are given byˆ Q x = i sin ζ ∂ ω + i cot ω cos ζ ∂ ζ + cos ζ sin ω ˆ K , ˆ Q y = − i cos ζ ∂ ω + i cot ω sin ζ ∂ ζ + sin ζ sin ω ˆ K , ˆ Q z = − i∂ ζ , (82)with the operator ˆ K defined in Eq. (65) and the quantization condition (81). The ˆ Q k obey the commutation relations[ ˆ Q x , ˆ Q y ] = i ˆ Q z (cyclic) . (83) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei Q = − ∂ ω − cot ω∂ ω + 1sin ω ( − ∂ ζ + 2 i ˆ K cos ω∂ ζ + ˆ K ) . (84)It is obvious that [ ˆ Q k , ˆ K ] = 0 for k = x, y, z . A complete set of commuting angular-momentum operators is, thus, ˆ Q , ˆ Q z , ˆ K . That is expected: For the axially symmetricrotor the square ˆ Q of the total angular momentum and its projections ˆ Q z onto thespace–fixed and ˆ K onto the symmetry axes are constants of the motion, with quantumnumbers J ( J + 1), M , K , respectively.We expect that[ ˆ Q k , ˆ H ] = 0 for k = x, y, z . (85)To prove Eqs. (85) we observe that the first line of Eq. (71) does not involve ˆ p ω orˆ p ζ . Therefore, it suffices to show that the terms in this line all commute with ˆ K .That is straightforward. We conclude that ˆ H consists of two commuting parts: Thesquare of the total angular momentum with quantum numbers J ( J + 1), M , K , andthe Hamiltonian for the surface vibrations which carry the quantum number K . Asa consequence, a rotational band occurs upon every eigenstate of the vibrational partof ˆ H . All rotational bands have the same moment of inertia. Differences arise onlythrough terms not considered in the approximation leading to Eq. (60), see Section 4.4. Discrete symmetries may restrict the spectrum beyond the requirement imposed bytime-reversal invariance. We follow Bohr and Mottelson [8, 23] and Weinberg [2].We first consider R -symmetry. That symmetry is realized if an axially symmetricnucleus is, in addition, symmetric under a rotation about π around an axis perpendicularto the symmetry axis. For definiteness, we choose a rotation r around the y axis. Weconsider the product g ( φ, θ ) r (0 , π,
0) where (with operators acting to the right) therotation r is applied prior to g . We have g ( φ, θ ) r (0 , π,
0) = r ( φ, θ, g (0 , π )= g ( φ + π, π − θ ) h ( π ) . (86)We note that the naive evaluation g ( φ, θ ) r (0 , π,
0) = g ( φ, θ + π ) would carry θ outsideof its domain of definition 0 ≤ θ ≤ π . Eqs. (86) show that the rotational degrees offreedom θ and φ behave under R as a particle on the sphere under the usual parity, i.e.( θ, φ ) → ( π − θ, φ + π ). The operation R acts also on the intrinsic variables x, y , andwe have u ( x, y ) r (0 , π,
0) = u ( − κ sin ψ, κ cos ψ ) r (0 , π, g ( ψ, κ ) h ( − ψ ) r (0 , π, g ( ψ, κ ) r (0 , π, h ( ψ )= g ( ψ + π, π − κ ) h ( ψ + π )= u ( − ( π − κ ) sin( ψ + π ) , ( π − κ ) cos( ψ + π )) h (2 ψ ) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei u ( − ( κ − π ) sin ψ, ( κ − π ) cos ψ ) h (2 ψ )= u ( − x, − y ) h (2 ψ ) . (87)Here, we used Eqs. (120) that relate g and u , and geometric considerations in goingfrom the second to the third line, and from the sixth to the last line. Eqs. (87) showthat the intrinsic variables transform under R as ( x, y ) → ( − x, − y ). According to Bohrand Mottelson [8, 23], eigenfunctions of the intrinsic and external variables must havethe same R parity, i.e., both must simultaneously be either positive or negative.We next consider ordinary parity P . Following Weinberg (Ref. [2], Sect. 19.2), thefields x and y have the same parity as the generators J x and J y . As components of anaxial vector, J x and J y have positive parity. Thus P ( x, y ) = ( x, y ). That implies thatall quantized modes (i.e., single excitations, double excitations ...) of the fields x and y have positive parity and are allowed. The quantized Hamiltonian is given by Eq. (71) with the understanding that themomenta are operators as defined in Eqs. (79) and (81). The term of leading order( O ( ω )) is given by the first line and written asˆ H ω = X Lµ ( p xLµ ) + ( p yLµ ) C b + C b ω Lµ (cid:0) x Lµ + y Lµ (cid:1)! . (88)It describes an infinite set of uncoupled harmonic oscillators with frequencies ω Lµ =[( L ( L + 1) D a + µ D b ) /C b ] / . In practice, the breakdown scale Λ serves as a cutoff forthis sum. It is useful to combine x Lµ and y Lµ into a two-dimensional SO (2)-symmetricharmonic oscillator with quantum numbers n Lµ = 0 , , , . . . , k Lµ = 0 , ± , ± , . . . , andenergies (2 n Lµ + | k Lµ | + 1) ω Lµ . The intrinsic angular momentum of the oscillators isgiven by the eigenvalues K = P L ≥ P Lµ = − L k Lµ of the operator ˆ K , see Eq. (65). Thedouble sum extends over occupied states only.For the ground state all quantum numbers vanish. For the excited states, we assume D b >
0. The lowest vibrational state corresponds to the single-quantum excitation of themode ( x , y ). As shown in Section 4.2 the fields x and y have the same positive parityas the corresponding generators J x and J y in Eqs. (36). Thus, the lowest vibrationalstate has | K | = 1 and negative R -parity. That is indeed observed in linear molecules [24].Nuclei, however, are different. Here, low-lying states are built from paired Fermions andhave positive R -parity. States with negative R -parity correspond to pair breaking andhave high excitation energies at or above the breakdown scale Λ of the EFT. Thus, stateswith odd K and positive parity are absent in the low-energy spectrum of nuclei. As anexample we mention the absence of low-lying magnetic dipole excitations [25, 26], i.e., K = 1 states with positive parity, in the spectra of deformed even-even nuclei [27, 28, 29].For nuclei, two quanta need to be excited in the lowest ( x , y ) mode, yieldinga degenerate pair of states with K = 0 and | K | = 2. Data [27, 29] indeed show thatthe low-lying vibrations in even-even deformed nuclei have K = 0 and | K | = 2. In the ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei ≈ − K = 0 or | K | = 2 vibrations, see Refs. [30, 31] for recent reviews. Theimpressive spectra of Er [27] and of
Dy [29] confirm this picture. The positive-parity states in those spectra must be viewed as anharmonically distorted quantizedvibrations corresponding to our Nambu-Goldstone modes. These spectra also shownegative-parity states. These states cannot be understood within the EFT discussed inthis paper; they can possibly be viewed as vibrations on top of the intrinsic odd-paritystate with lowest energy.The quantized version of the full Hamiltonian of Eq. (71) isˆ H ω,ξ = ˆ H ω + ˆ I − ˆ K C a . (89)The last term causes a rotational band to appear on top of each of the vibrationalstates (band heads). The eigenfunctions are Wigner D -functions D IM,K ( α, β,
0) withtotal integer spin I and projections − I ≤ M, K ≤ I [16, 15]. The eigenvalues of ˆ I are I ( I + 1) with I ≥ | K | . At this order, all rotational bands have the same moment ofinertia, and deviations from this picture are due to higher-order corrections, see Ref. [10]and Section 4.4.We conclude that the EFT predicts that in leading order, the Nambu-Goldstonemodes due to emergent breaking of rotational symmetry yield a large number ofharmonic vibrations. In next-to-leading order each of these serves as head of a rotationalband. All bands have identical moments of inertia. Corrections of higher orderconsidered in Section 4.4 lead to anharmonicities of the vibrational states and causethe moments of inertia to differ.Given the close proximity of the breakdown scale in nuclei to the vibrationalexcitation energy, it is reasonable to consider a simpler – but still model-independent –approach to deformed nuclei. That approach [15] uses an effective theory (as opposedto the effective field theory of the present paper). It combines the quantized rotationsas lowest-energy excitations with the lowest vibrational modes. The latter correspondto the K = 0 and K = 2 modes of the present paper. Thus, the effective theory replacesthe quantum fields x and y by their quantized modes of longest wave length. Our resultsshow that the effective theory is based upon a solid field-theoretical foundation. The extension of the effective field theory to higher-order terms is straightforwardbut tedious. The program is this. (i) Use the power counting of Section 3.4 toidentify all terms that contribute to the effective Lagrangian in a given order. Thatincludes terms with higher time derivatives. These can be treated by perturbative ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei x and y into their normal modes asin Eqs. (59), and compute the Lagrangian by integration of the Lagrangian densityas in Eqs. (57). The resulting expressions are (complicated) sums involving Clebsch-Gordan coefficients. (iii) Perform the Legendre transformation to the Hamiltonianwithin perturbation theory to the desired order of the power counting. (iv) Quantizethe Hamiltonian and compute the spectrum.Obviously, steps (i) to (iii) are quite laborious. Furthermore, the kinetic part ofthe resulting Hamiltonian will have low-energy constants that are complicated linearcombinations of the corresponding coefficients of the Lagrangian. The latter are notknown, and the former need to be determined from data. It is, therefore, desirableto understand the transformation properties of the coordinates x Lµ and y Lµ and of thecanonical momenta p xLµ and p yLµ , and to directly construct the most general Hamiltonianthat is invariant under rotations at a given order of the power counting.The construction of the invariants is guided by the following observations. Eqs. (51)show that under rotations, the fields x ( θ, φ ) and y ( θ, φ ) transform as the x and y components of a two-dimensional vector. These transformation properties hold forevery point ( θ, φ ) on the unit sphere. Using the expansion of the second of Eqs. (59)we conclude that the complex normal modes ˜ x Lµ and ˜ y Lµ themselves, too, transformas the x and y components of a two-dimensional vector. The same is true for thecorresponding canonical momenta (denoted by ˜ p xLµ and ˜ p yLµ , respectively) as these stemfrom time derivatives of the field modes ˜ x Lµ and ˜ y Lµ . The integration R dE over theLagrangian density in Eq. (57) singles out scalars. For instance, the invariant that isbilinear in the momenta is X L (˜ p xL · ˜ p xL + ˜ p yL · ˜ p yL ) = X Lµ ( − µ (cid:0) ˜ p xLµ ˜ p xL − µ + ˜ p yLµ ˜ p yL − µ (cid:1) = X Lµ (cid:0) ( p xLµ ) + ( p yLµ ) (cid:1) . (90)Here ˜ p xL and ˜ p yL denote spherical tensors of degree L with components ˜ p xLµ and ˜ p yLµ ,respectively.We apply these considerations first to the kinetic terms in Eqs. (54) and (55)and then to the potential terms in Eqs. (56). In calculating the kinetic part of theHamiltonian we encounter the need to invert the generalized form of the matrix ˆ G inEq. (70). So far we have taken into account only the terms L a and L b in Eqs. (54).As in Eqs. (57) we now consider the sum L of all four terms in Eqs. (54) and (55).The kinetic part L of the effective Lagrangian is given by integration over d E of L .We note that L is bilinear in the time derivatives of all dynamical variables. We omitthe term ˙ ω which gives a trivial contribution. Defining ˙ x j as the totality of the timederivatives { ˙ ζ, ˙ x Lµ , ˙ y Lµ } we proceed as in Section 3.5 and write L = 12 X ij ˙ x i A ij ˙ x j . (91)In order not to overburden the notation we have chosen the letter A rather than G for ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei L . As in Section 3.5 we write ˆ A for the matrix and A ij for itselements. We have A ij = A ji , and we write ˆ A = ˆ A (0) + ˆ A (1) . Here ˆ A (0) ( ˆ A (1) ) is thesum of the contributions arising from L a and L b (from L c and L d , respectively). Thelatter are small of order ε with respect to the former. The momenta are defined by p i = P j A ij ˙ x j , and the kinetic part of the effective Hamiltonian is H = 12 X ij p i ( A − ) ij p j . (92)We calculate ˆ A − perturbatively and use the explicit form of L c and L d in Eqs. (54)and (55). Details are given in Appendix 5. The contributions of order Ω ε to the effectiveHamiltonian stemming from L c ( L d ) are denoted by H c ( H d , respectively). We find H c = − C c C b Z dΩ ( xp y − yp x ) ,H d = − C d C b Z dΩ ( x + y )( p x + p y ) . (93)A contribution of order Ω ε arises also from L b . It is given by H b = 12 C a cos ω sin ω Z dΩ ( xp y − yp x ) . (94)We turn to the potential terms. For the modes x Lµ and y Lµ we deal, in analogy toEqs. (90), with the invariant P Lµ ( x Lµ + y Lµ ). In evaluating the remaining invariantsin Eqs. (56) we have to deal with the angular-momentum operators ~L and L z acting onthe fields x ( θ, φ ) and y ( θ, φ ). Denoting by L ν the spherical components of ~L , we haveˆ L ν x ( θ, φ ) = X Lµ ˜ x Lµ (cid:16) ˆ L ν Y Lµ ( θ, φ ) (cid:17) = X Lµ ˜ x Lµ C Lµ + ν νLµ Y Lµ + ν ( θ, φ ) . (95)Upon multiplication with Y ∗ aα ( θ, φ ) (where a and α are arbitrary) and integration overd E we find Z d E Y ∗ aα ( θ, φ ) ˆ L ν x ( θ, φ ) = X Lµ ˜ x Lµ C Lµ + ν νLµ Z d E Y ∗ aα ( θ, φ ) Y Lµ + ν ( θ, φ )= C aα νLα − ν ˜ x aα − ν = ( − ν ˆ L − ν ˜ x aα . (96)In the last line the operator ˆ L − ν is understood to act on the spherical tensor of rank a with components ˜ x aα . Invariants built upon the normal modes ˜ x Lµ and ˜ y Lµ are, thus,obtained by viewing these modes as components of spherical tensors of rank L . Scalarsare constructed after acting with ~L onto these components. For example Z d E ( ~Lx ) = X ν ( − ν X aα X bβ ˜ x aα ˜ x bβ Z d E C aα − ν − νaα C bβ + ν νbβ Y aα − ν Y bβ + ν = X ν ( − ν X aα X bβ ˜ x aα ˜ x bβ ( − α − ν C aα − ν − νaα C bβ + ν νbβ δ ba δ β + ν − α + ν ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei − X aα ( − α ˜ x aα ˜ x a − α = − ˜ x L · ˜ x L . (97)The evaluation of the terms in Eqs. (56) is now straightforward. Upon applying thequantization rules, we obtain higher-order terms in the Hamiltonian ˆ H .The terms in Eqs. (93) and (56) do not depend on the rotational degrees of freedom ω and ζ . These terms lift the degeneracies of the vibrational modes of excitation butdo not affect the moment of inertia. The term in Eq. (94) depends parametrically upon ω and couples the rotational bands with the vibrational modes.
5. Summary and Conclusions
We have constructed the EFT for emergent symmetry breaking in deformed nuclei. Inaddition to the Nambu-Goldstone modes, the theory contains two additional degreesof freedom that describe rotations about the body-fixed symmetry axis. Starting froma physically intuitive and mathematically standard parameterization where rotationaland vibrational degrees of freedom are treated on an equal footing, we have switched toa much more practical parameterization where the rotational degrees of freedom receivea separate treatment. The theory is characterized by three small parameters. Theseare (i) the ratio ξ/ Ω of the energies of rotational motion ξ and of vibrational motionΩ, (ii) the ratio Ω / Λ where Λ is the breakdown scale of the EFT (typically given bythe pairing energy or the energy of single-particle excitation in the shell model), and(iii) the parameter ε which characterizes deviations from harmonicity of the vibrations.In lowest order, the spectrum consists of vibrations each of which serves as head of arotational band. The vibrations are due to the quantized Nambu-Goldstone modes thatdescribe the emergent breaking of SO(3) symmetry. In leading order, the vibrationalmodes are degenerate, and the rotational bands all have the same moment of inertia.Terms of next order remove both degeneracies. Acknowledgments
This material is based upon work supported in part by the U.S. Department of Energy,Office of Science, Office of Nuclear Physics, under Award Numbers DE-FG02-96ER40963(University of Tennessee), and under contract number DEAC05-00OR22725 (Oak RidgeNational Laboratory).
Appendix 1: Equations of Motion inCurvilinear Coordinates
In curvilinear coordinates the equations of motion are obtained by variation ofthe product of the Lagrangian density L and the integration measure. Wedemonstrate that fact for the simplest case. We consider a Lagrangian density ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei L ( ψ, ∂ψ/∂t, ∂ψ/∂x , . . . , ∂ψ/∂x N ) that depends on the field ψ (a function of time t and of N Cartesian variables x , x , . . . , x N ), and on the N + 1 derivatives of the field.Standard variation of the action integral yields Z d t Z N Y ν =1 d x ν (cid:16) ∂∂t ∂ L ∂ ( ∂ t ψ ) + N X µ =1 ∂∂x µ ∂ L ∂ ( ∂ x µ ψ ) + ∂ L ∂ψ (cid:17) δψ . (98)We introduce N curvilinear coordinates ζ k ( x , x , . . . , x N ) with k = 1 , . . . , N that arefunctions of the N Cartesian coordinates x µ . We define the N –dimensional matrix M µk = ∂x µ ∂ζ k with D = det M . (99)Then N Y ν =1 d x ν = D N Y k =1 d ζ k , ∂∂x µ = N X k =1 ( M − ) kµ ∂∂ζ k , ∂ L ∂ ( ∂ x µ ) = N X l =1 M µl ∂ L ∂ ( ∂ ζ l ) . (100)We insert all this into expression (98) and obtain Z d t Z N Y k =1 d ζ k D (cid:16) ∂∂t ∂ L ∂ ( ∂ t ψ )+ N X µ =1 N X l =1 ( M − ) lµ ∂∂ζ l N X n =1 M µn ∂ L ∂ ( ∂ ζ n ψ ) + ∂ L ∂ψ (cid:17) δψ . (101)The triple sum in expression (101) can be written as N X l =1 ∂∂ζ l ∂ L ∂ ( ∂ ζ l ψ ) + N X µln =1 ( M − ) lµ n ∂∂ζ l M µn o ∂ L ∂ ( ∂ ζ n ψ ) . (102)The identities det ln D = ln Trace D and ∂ ζ l M µn = ∂ ζ n M µl imply that expression (102)is equal to N X l =1 ∂∂ζ l ∂ L ∂ ( ∂ ζ l ψ ) + D − n N X l =1 ∂∂ζ l D o ∂ L ∂ ( ∂ ζ n ψ ) . (103)Using all that and the independence of D of t and ψ we rewrite expression (101) as Z d t Z N Y k =1 d ζ k (cid:16) ∂∂t ∂ ( D L ) ∂ ( ∂ t ψ ) + N X l =1 ∂∂ζ l ∂ ( D L ) ∂ ( ∂ ζ l ψ ) + ∂ ( D L ) ∂ψ (cid:17) δψ . (104)Comparing expression (104) with expression (98) we conclude that variation of L in curvilinear coordinates is tantamount to varying D L and otherwise treating thecurvilinear coordinates like Cartesian ones. That is what we use in Section 2.3, with D = sin θ . ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei Appendix 2: The Matrix M We calculate the matrix M defined by δq ν = P k M νk δχ k . We define H = ω [cos ζ P x ′ + sin ζ P y ′ ] ,G = ω [ − sin ζ P x ′ + cos ζ P y ′ ] . (105)Then [ H, G ] = − iω P z ′ , [ H, − iP z ′ ] = G . (106)With r given by Eq. (23) we write r exp {− iH } = exp {− i ˜ H } exp {− iδξ P z ′ } . (107)Here ˜ H = ˜ ω [cos ˜ ζ P x ′ + sin ˜ ζ P y ′ ] , ˜ ω = ω + δω , ˜ ζ = ζ + δζ . (108)We calculate δω, δζ and δξ to first order in δχ k . Keeping only terms of first order wehave with k = x ′ , y ′ , z ′ X k δχ k P k = δωω H + δξ exp {− iH } P z ′ exp { iH } + δζ X (109)where X = (cid:16) ∞ X k =0 ( − i ) k − k ! k − X l =0 H l GH k − l − (cid:17) exp { iH } . (110)Since X must be of order zero in H only terms with l = k − HH l G → Gω l for l even ,H l G → − iP z ′ ω l +1 for l odd . (111)Thus, X = ( − sin ζ P x ′ + cos ζ P y ′ ) sin ω + P z ′ (cos ω − . (112)Similarly, H l ( − iP z ′ ) → Gω l − for l odd ,H l ( − iP z ′ ) → − iP z ′ ω l for l even . (113)Therefore, exp {− iH } P z ′ exp { iH } = P z ′ cos ω + ( − sin ζ P x ′ + cos ζ P y ′ ) sin ω . (114)Inserting Eqs. (114) and (112) into Eq. (109) we find X k δχ k P k = δω (cos ζ P x ′ + sin ζ P y ′ ) − P z ′ δζ + ( δξ + δζ )[ P z ′ cos ω + ( − sin ζ P x ′ + cos ζ P y ′ ) sin ω ] . (115) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei P x ′ , P y ′ , and P z ′ on both sides of that equation.With δω = δq , δζ = δq , δξ + δζ = δq we obtain δχ k = X l =1 ( M − ) kl δq l (116)where M − = cos ζ − sin ζ sin ω sin ζ ζ sin ω − ω . (117)The inverse matrix is M = cos ζ sin ζ − sin ζ cot ω cos ζ cot ω − − sin ζ sin ω cos ζ sin ω . (118) Appendix 3: Equivalence of theParameterizations (36) and (4, 5) for U We start from Eqs. (36) and derive Eqs. (4, 5). We define κ ≥ x = − κ sin Ψ ,y = κ cos Ψ . (119)We use a power-series expansion in κ , valid for κ ≪
1. We prove the equivalence onlyto leading order in κ . Terms of higher order can be treated analogously. In addition toEqs. (36) we use the following definitions and identities, valid for arbitrary values of κ ,of the Euler angles α, β, γ , and of Ψ, h ( γ ) ≡ e iγJ z ,r ( α, β, γ ) ≡ g ( α, β ) h ( γ ) ,u ( − κ sin Ψ , κ cos Ψ) = g (Ψ , κ ) h † (Ψ) . (120)The first of Eqs. (120) defines h ( γ ) for an arbitrary angle γ . The second of Eqs. (120)defines the rotation r ( α, β, γ ) and shows that in Eqs. (36), the factor g ( ζ , ω ) acts on u like a rotation r with third Euler angle γ = 0. The third of Eqs. (120) is an identityfor the function u ( x, y ) defined in the third of Eqs. (36). That identity can easily bederived with the help of the relation exp {− i Ψ ˆ J z } ˆ J y exp { i Ψ ˆ J z } = − sin Ψ ˆ J x + cos Ψ ˆ J y .We use the transformation law r ( ζ , ω, g (Ψ , κ ) = g ( ζ ′ , ω ′ ) h ( γ ′ ) . (121)Here ζ ′ , ω ′ , γ ′ are functions of the angles ζ and ω and of the variables Ψ and κ and aregiven in Ref. [15]. We havecot( ζ ′ − ζ ) = cos ω cot Ψ + cot κ sin ω sin Ψ , cos ω ′ = cos κ cos ω − sin κ sin ω cos Ψ , cot γ ′ = − cos κ cot Ψ − cot ω sin κ sin Ψ . (122) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei | κ | ≪ ω ′ ≈ ω + κ cos Ψ ≈ ω + y ,ζ ′ ≈ ζ + κ sin Ψsin ω ≈ ζ − x sin ω , − γ ′ ≈ Ψ − κ cot ω sin Ψ = Ψ + x cot ω . (123)Therefore, g ( ζ , ω ) g (Ψ , κ ) h † (Ψ) ≈ g ( ζ − x sin ω , ω + y ) h ( x cot ω ) . (124)We use the third of Eqs. (120) once again to write the result as g ( ζ , ω ) g (Ψ , κ ) h † ( ψ ) ≈ u (cid:16) − ( ω + y ) sin( ζ − x sin ω ) , ( ω + y ) cos( ζ − x sin ω ) (cid:17) × h (Ψ) h ( x cot ω ) . (125)We recall that U is defined in the coset space SO3/SO2. Hence U ≈ u (cid:16) − ( ω + y ) sin( ζ − x sin ω ) , ( ω + y ) cos( ζ − x sin ω ) (cid:17) . (126)Eq. (126) gives the connection with the parameterization of U in Eqs (4) and (5) tolowest order in x, y . Differences in sign are due to the fact that here we work in thespace-fixed system. The variables ω and ζ in Eqs. (5) are seen to correspond to ω and ζ , respectively. To lowest order in κ , the variable ω corresponds to y whereasthe variable ζ corresponds to − x/ sin ω . The occurrence of the factor 1 / sin ω in thelast relation causes the difficulties in the attempt to derive the equations of harmonicmotion directly from the parameterization (4) via an expansion in powers of ω and ζ and explains why we have introduced the parametrization (36). The calculation canobviously be carried to higher orders. That establishes the complete equivalence of theparameterizations of U in Eqs. (36) and (4, 5). Appendix 4: Quantization in CurvilinearCoordinates
It is instructive to use another approach to quantization which shows how theambiguities that are associated with the prescription (79) are avoided. We assumethat ω and ζ are independent so that[ ω, ζ ] = 0 , [ ω, p ζ ] = 0 , [ p ω , ζ ] = 0 , [ p ω , p ζ ] = 0 . (127)For simplicity we have suppressed the symbol ˆ on the operators. It remains todetermine the commutators [ p ω , ω ] and [ p ζ , ζ ]. We choose a representation where ω and ζ are ordinary real variables. Quantization is subject to three requirements. (i)The expressions for Q rot k and for H rot must be Hermitian. (ii) The Q rot k must obey thestandard commutation relations [ Q rot x , Q rot y ] = iQ rot z (cyclic). (iii) When expressed interms of the quantized components Q rot k of angular momentum, the Hamiltonian of thepure rotor must be given by Eq. (35), with C → C a . As for point (i), Hermitecity isdefined with respect to an integration measure for the variables ω and ζ . The matrix U ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei SO (3) /SO (2). That suggests usingfor the volume element the expressiond V = d ω sin ω d ζ . (128)An operator O is Hermitian if the equality Z d V Ξ ∗ O Ψ = Z d V ( O Ξ) ∗ Ψ (129)holds for any two integrable functions Ξ( ω, ζ ) and Ψ( ω, ζ ) that are periodic with period2 π with respect to both ω and ζ . (That is why we have chosen in Section 3 the rangesof integration as 0 ≤ ω , ζ ≤ π ). Requirement (ii) on the commutators of the Q rot k andthe explicit form of the Q rot k in Eqs. (75) imply[ p ω , ω ] = − i , [ p ζ , ζ ] = − i . (130)Eqs. (75) show that for Q rot k to be Hermitian, p ω and p ζ must be Hermitian, too. Asformulated in Eq. (129), that condition is consistent with Eqs. (130) if p ω and p ζ obeyEqs. (79). Calculating H rot from Eq. (35) with C → C a we obtain the second ofEqs. (78). Thus, H rot is the quantized Hamiltonian of a rotor with C a the moment ofinertia. Appendix 5: Kinetic Part of the Effective Hamiltonian
We use ˆ A − ≈ ( ˆ A (0) ) − − ( ˆ A (0) ) − ˆ A (1) ( ˆ A (0) ) − + . . . . With L b = X Lµ [( ˙ x Lµ ) + ( ˙ y Lµ ) ] + 2 ˙ ζ cos ω X Lµ [ x Lµ ˙ y Lµ − y Lµ ˙ x Lµ ] (131)we have ˆ A (0) = sin ω − y Lµ cos ω x Lµ cos ω − y Lµ cos ω x Lµ cos ω . (132)The non-diagonal terms ∝ x Lµ , y Lµ are of order ε / . Therefore, we invert ˆ A (0) byexpanding in powers of these terms and obtain to first order( ˆ A (0) ) − ≈ ω y Lµ cos ω sin ω − x Lµ cos ω sin ωy Lµ cos ω sin ω − x Lµ cos ω sin ω . (133)The matrix ˆ A (1) receives contributions from both L c and L d . The contribution from L c is proportional to X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ ˙ y L ′ µ ′ − ˙ x Lµ y L ′ µ ′ (cid:17) L × (cid:16) x L ′′ µ ′′ ˙ y L ′′′ µ ′′′ − ˙ x L ′′ µ ′′ y L ′′′ µ ′′′ (cid:17) L i . (134) ffective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei L d we have correspondingly X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ x L ′ µ ′ + y Lµ y L ′ µ ′ (cid:17) L (cid:16) ˙ x L ′′ µ ′′ ˙ x L ′′′ µ ′′′ + ˙ y L ′′ µ ′′ ˙ y L ′′′ µ ′′′ (cid:17) L i + 2 ˙ ζ cos ω X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ x L ′ µ ′ + y Lµ y L ′ µ ′ (cid:17) L × (cid:16) x L ′′ µ ′′ ˙ y L ′′′ µ ′′′ − ˙ x L ′′ µ ′′ y L ′′′ µ ′′′ (cid:17) L i . (135)Both L c and L d are small. Therefore, we calculate − ( ˆ A (0) ) − ˆ A (1) ( ˆ A (0) ) − to lowestorder, i.e., by taking for ( ˆ A (0) ) − only the diagonal part of the matrix on the right-handside of Eq. (133). The resulting terms in the effective Hamiltonian are then given by H c = − C c C b X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ p yL ′ µ ′ − p xLµ y L ′ µ ′ (cid:17) L × (cid:16) x L ′′ µ ′′ p yL ′′′ µ ′′′ − p xL ′′ µ ′′ y L ′′′ µ ′′′ (cid:17) L i ,H d = − C d C b X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ x L ′ µ ′ + y Lµ y L ′ µ ′ (cid:17) L × (cid:16) p xL ′′ µ ′′ p xL ′′′ µ ′′′ + p yL ′′ µ ′′ p yL ′′′ µ ′′′ (cid:17) L i − C d C a C b p ζ cos ω sin ω X LL ′ L ′′ L ′′′ L h(cid:16) x Lµ x L ′ µ ′ + y Lµ y L ′ µ ′ (cid:17) L × (cid:16) x L ′′ µ ′′ p yL ′′′ µ ′′′ − p xL ′′ µ ′′ y L ′′′ µ ′′′ (cid:17) L i . (136)The terms inversely proportional to C b are of order ε Ω, the term inversely proportionalto C a C b is of order ε Ω and, therefore, negligible. That result justifies a posteriori thediagonal approximation for ( ˆ A (0) ) − . With the help of the expansions (59) and (62), weobtain Eqs. (93). References [1] Leutwyler H 1994
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