Quantum Field Theory of Correlated Bose-Einstein condensates: II. Ward-Takahashi Identities and Correlation Functions
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Quantum Field Theory of Correlated Bose-Einstein condensates:II. Ward-Takahashi Identities and Correlation Functions
Takafumi K ita
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the ex-pressions of the first-order variations ( δ Ψ , δG ) due to perturbations obtained in the preceding paper[T. Kita, J. Phys. Soc. Jpn. , 024001 (2021)] for the condensate wave function Ψ and Green’sfunction G . They enable us to obtain several exact results on the density and current correlationfunctions K νν ′ , and also express K νν ′ in terms of low-energy Green’s functions and vertices. Thelatter expressions open up the possibility of constructing theory of superfluid Bose liquids in thesame way as that for fermions at low temperatures. The vertices are found to have different limitsdepending on which of frequency ω and wavenumber q is set equal to zero first. I. INTRODUCTION
The Ward identities [1] relates derivatives of Green’sfunction to vertices. They were extended by Takahashi[2] from differential relations to finite-difference relations.The Ward identities have played a crucial role in pro-viding the Landau theory of Fermi liquids [3–5] with amicroscopic foundation [6–8]. Specifically, they enableus to describe low-temperature properties of a system ofhomogeneous identical fermions in terms of low-energyGreen’s functions and vertices, with no reference to thehigh-energy part of Green’s function at all. They werealso used by Leggett [9, 10] in extending the Landau the-ory to superfluid phases, and by Serene and Rainer [11]in formulating the quasiclassical theory of superfluid Hein terms of a few phenomenological Landau parameters.In contrast, no derivations of Ward or Ward-Takahashiidentities have been known for Bose-Einstein conden-sates, except that by Gavoret and Nozi`eres at zero tem-perature based on a perturbation expansion [12]. Specif-ically, their two Ward identities include neither the con-densate wave function Ψ nor three-point vertices Γ (3) characteristic of Bose-Einstein condensation. Moreover,their derivation is based on the fictitious gap method in-troduced to avoid the infrared divergence in the simpleperturbation expansion, with which the four-point ver-tices Γ (4) are concluded to be identical between the q -and ω -limits, i.e., independent of which of the frequency ω or wavenumber q is set equal to zero first. Whetherthe statements are valid or not should be clarified defi-nitely. Establishing the identities at finite temperatures,we may also be able to construct a correspondent of theLandau theory of Fermi liquids also for Bose-Einsteincondensates.This paper presents a non-perturbative derivation ofthe Ward-Takahashi identities for Bose-Einstein conden-sates at finite temperatures. The basic strategy is out-lined as follows. We apply the local-gauge and truncated-Galilean transformations following Serene and Rainer[11]. On the one hand, we know exactly how the con-densate wave function Ψ and Green’s function G changeunder these transformations. On the other hand, thetransformations generate apparent perturbations in the system to yield variations ( δ Ψ , δG ), whose exact expres-sions with vertices we have already derived in the preced-ing paper (which is referred to as I hereafter) [13]. Equat-ing the two results for each perturbation at the first orderyields a specific Ward-Takahashi identity. The resultingWard-Takahashi identities will be shown to have addi-tional contributions given in terms of Ψ and the three-point vertex Γ (3) besides those of G . Moreover, they ap-propriately reduce in the normal-state limit of Ψ → δ Ψ and δG obtained in I intothe energy-momentum space. Section III derives Ward-Takahashi identities. Section IV introduces the densityand current correlation functions, derives several exactresults on them based on the Ward-Takahashi identities,and expresses them in terms of low-energy Green’s func-tions and vertices. Section V presents a brief summary.We set ~ = 1 throughout. II. FOURIER TRANSFORM OF δ Ψ AND δG We focus on homogeneous systems throughout. Wehere transform the expressions of first-order variations( δ Ψ , δG ) obtained in I into the energy-momentum space.The final results are given in Eq. (10) below.It was shown in I that first-order variations of (Ψ , G )under perturbation S ext ≡ Z dx Z dx ′ ~ψ T ( x ) δ ˆ I ( x, x ′ ) ~ψ ( x ′ ) (1)are given exactly by δ~ Ψ = ˆ G (cid:18) δ ˆ I (s) ~ Ψ −
12 Γ (3)
GG δ~I (s) (cid:19) , (2a) δ ~G = GG (cid:20)(cid:18) −
12 Γ (4) GG (cid:19) δ~I (s) + Γ (3)T δ~ Ψ (cid:21) , (2b)with δ~I (s) ( x, x ′ ) ≡ δ~I ( x, x ′ ) + δ~I T ( x ′ , x ) . (3)See Eqs. (29), (41), and (42) of I.Let us expand the 2 × G ( x, x ′ ) in the Fourier series,ˆ G ( x, x ′ ) = 1 βV X p ˆ G ( p ) e ip ( x − x ′ ) , (4)where V is the volume of the system, x and p are definedby x ≡ ( r , τ ) and p ≡ ( p , ε n ) , (5)with ε n ≡ πn/β the boson Matsubara frequency, and px denotes a scalar product with the Minkowskii metricgiven explicitly by px ≡ p · r − ε n τ. (6)Accordingly, the matrix element of GG is expressiblethrough a change of variables as h ξ , ξ ′ | GG | ξ , ξ ′ i ≡ G j j ( x , x ) G j ′ j ′ ( x ′ , x ′ )= 1( βV ) X p,q G j j ( p + ) G j ′ j ′ ( − p − ) × e ip + ( x − x ) − ip − ( x ′ − x ′ ) , (7)where ξ ≡ ( j, x ) with j = 1 ,
2, and q and p ± are definedby q ≡ ( q , ω ℓ ) , (8a) p + ≡ ( p + q / , ε n + ω ℓ ) , (8b) p − ≡ ( p − q / , ε n ) . (8c)We also expand the other quantities in Eq. (2) as h ξ | δ~ Ψ = X q δ Ψ j ( q ) e iqx , (9a) h ξ, ξ ′ | δ ~G = 1 βV X pq δG jj ′ ( p ; q ) e ip + x − ip − x ′ , (9b) h ξ, ξ ′ | δ~I (s) = 1 βV X pq δI (s) jj ′ ( p ; q ) e ip + x − ip − x ′ , (9c) h ξ , ξ ′ | Γ (4) | ξ , ξ ′ i = 1( βV ) X pp ′ q Γ (4) j j ′ ; j j ′ ( p, p ′ ; q ) × e ip + x − ip − x ′ − ip ′ + x + ip ′− x ′ , (9d) h ξ , ξ ′ | | ξ , ξ ′ i = δ j j δ j ′ j ′ ( βV ) X pq e ip + ( x − x ) − ip − ( x ′ − x ′ ) , (9e) h ξ | Γ (3) | ξ , ξ ′ i = 1( βV ) X pq Γ (3) j ; j j ′ ( p ; q ) × e iqx − ip + x + ip − x ′ , (9f) h ξ , ξ ′ | Γ (3)T | ξ i = 1( βV ) X pq Γ (3)T j j ′ ; j ( p ; q ) × e ip + x − ip − x ′ − iqx . (9g)Substituting Eqs. (4), (7), and (9), we obtain theFourier transform of Eq. (2) as δ~ Ψ( q ) = ˆ G ( q ) (cid:20) δ ˆ I (s) ( q m / q ) ~ Ψ − βV X p Γ (3) ( p ; q ) GG ( p ; q ) δ~I (s) ( p ; q ) (cid:21) , (10a) δ ~G ( p ; q ) = GG ( p ; q ) " δ~I (s) ( p ; q ) − βV X p ′ Γ (4) ( p, p ′ ; q ) GG ( p ′ ; q ) δ~I (s) ( p ′ ; q )+ Γ (3)T ( p ; q ) δ~ Ψ( q ) , (10b)where q m ≡ ( q ,
0) with m denoting momentum , and δ ~G ,Γ (4) , etc., are vectors and matrices in the Nambu (i.e.,particle-hole) space defined by h j | ~ Ψ ≡ Ψ j , (11a) h j | δ~ Ψ( q ) ≡ δ Ψ j ( q ) , (11b) h j | ˆ G ( q ) | j ′ i ≡ G jj ′ ( q ) , (11c) h jj ′ | δ ~G ( p ; q ) ≡ δG jj ′ ( p ; q ) , (11d) h jj ′ | δ~I (s) ( p ; q ) ≡ δI (s) jj ′ ( p ; q ) , (11e) h j | Γ (3) ( p ; q ) | j j ′ i ≡ Γ (3) j ; j j ′ ( p ; q ) , (11f) h j j ′ | Γ (3)T ( p ; q ) | j i ≡ Γ (3)T j j ′ ; j ( p ; q ) , (11g) h j j ′ | GG ( p ; q ) | j j ′ i ≡ G j j ( p + ) G j ′ j ′ ( − p − ) , (11h) h j j ′ | Γ (4) ( p, p ′ ; q ) | j j ′ i ≡ Γ (4) j j ′ ; j j ′ ( p, p ′ ; q ) . (11i)The symmetries of these vectors and matrices are sum-marized in AppendixA. III. WARD-TAKAHASHI IDENTITIES
We derive Ward-Takahashi Identities of Bose-Einsteincondensates based on the local-gauge transformation,truncated-Galilean transformation [11], and a spatiallyinhomogeneous perturbation [7].
A. Local gauge transformation
First, we perform the local gauge transformation ~ψ ′ ( x ) = e i ˆ σ χ ( x ) ~ψ ( x ) (12)on τ ∈ [0 , β ] with periodicity χ ( r , τ + β ) = χ ( r , τ ). Thetransformation yields variations δ~ Ψ ≡ ~ Ψ ′ − ~ Ψ and δ ˆ G ≡ ˆ G ′ − ˆ G , which are given in the first order in χ ( x ) by δ~ Ψ( x ) = i ˆ σ χ ( x ) ~ Ψ , (13a) δ ˆ G ( x, x ′ ) = i h χ ( x )ˆ σ ˆ G ( x, x ′ ) + ˆ G ( x, x ′ )ˆ σ χ ( x ′ ) i . (13b)Let us expand χ ( x ) in the Fourier series χ ( x ) = X q χ ( q ) e iqx . (14)Substituting Eqs. (4), (9a), (9b), and (14) into Eq. (13),we obtain δ~ Ψ( q ) = iχ ( q )ˆ σ ~ Ψ , (15a) δ ˆ G ( p ; q ) = iχ ( q ) h ˆ σ ˆ G ( p − ) + ˆ G ( p + )ˆ σ i , (15b)where p ± are given in Eq. (8).On the other hand, transformation (12) yields an ap-parent perturbation in the system. Specifically, let ussubstitute ~ψ ( x ) = e − i ˆ σ χ ( x ) ~ψ ′ ( x ) into the kinetic part ofthe action given by S = 12 Z dx ~ψ T ( x ) (cid:20) − i ˆ σ ∂∂τ + ˆ σ (cid:18) ˆ p m − µ (cid:19)(cid:21) ~ψ ( x ) , (16)extract terms first order in χ ( x ) and set ~ψ ′ ( x ) → ~ψ ( x ) inthem as appropriate in the first order, and expand χ ( x )and ~ψ ( x ) as Eq. (14) and ~ψ ( x ) = 1 V / X p ~ψ ( p ) e ipx . (17)We thereby find that the perturbation is expressible as S ext = β X pq ~ψ T ( − p + ) δ ˆ I ( p ; q ) ~ψ ( p − ) , (18)with δ ˆ I ( p ; q ) = i χ ( q ) (cid:18) iω ℓ ˆ σ + p · q m i ˆ σ (cid:19) . (19)The corresponding Fourier coefficient of Eq. (3) is givenby δ ˆ I (s) ( p ; q ) = δ ˆ I ( p ; q ) + δ ˆ I T ( − p ; q ) = 2 δ ˆ I ( p ; q ) . (20) Let us substitute Eqs. (15) and (19) into Eq. (10). Not-ing that χ ( q ) is arbitrary, we obtainˆ σ ~ Ψ = ˆ G ( q ) "(cid:18) iω ℓ ˆ σ + q m i ˆ σ (cid:19) ~ Ψ − βV X p Γ (3) ( p ; q ) × GG ( p ; q ) (cid:18) iω ℓ ~σ + p · q m i~σ (cid:19) , (21a)ˆ σ ˆ G ( p − ) + ˆ G ( p + )ˆ σ = GG ( p ; q ) (X p ′ " δ pp ′ − βV Γ (4) ( p, p ′ ; q ) GG ( p ′ ; q ) × (cid:18) iω ℓ ~σ + p ′ · q m i~σ (cid:19) + Γ (3)T ( p ; q )ˆ σ ~ Ψ ) , (21b)where 1 denotes the 4 × A = ~B in Eq.(21b) signifies the equality A jj ′ = B jj ′ between their el-ements defined by A jj ′ ≡ h j | ˆ A | j ′ i and B jj ′ ≡ h jj ′ | ~B .Equation (21) is the most general form of the Ward-Takahashi identities concerning the local gauge transfor-mation. Let us operate ˆ G − ( q ) on Eq. (21a) from the leftand write GG ( p ; q ) ~B = ˆ G ( p + ) ˆ B ˆ G ( p − ) , (22)in Eq. (21b) based on Eqs. (11d), (11h), and (A1). Wecan thereby express Eq. (21) alternatively asˆ G − ( q )ˆ σ ~ Ψ = (cid:18) iω ℓ ˆ σ + q m i ˆ σ (cid:19) ~ Ψ − βV X p Γ (3) ( p ; q ) × GG ( p ; q ) (cid:18) iω ℓ ~σ + p · q m i~σ (cid:19) , (23a)ˆ G − ( p + )ˆ σ + ˆ σ ˆ G − ( p − )= X p ′ " δ pp ′ − βV Γ (4) ( p, p ′ ; q ) GG ( p ′ ; q ) × (cid:18) iω ℓ ~σ + p ′ · q m i~σ (cid:19) + Γ (3)T ( p ; q )ˆ σ ~ Ψ . (23b)Note that Eqs. (23a) and (23b) include the non-interacting limit with µ = 0 as trivial equalities, as seenby using ˆ G − ( p ) ≡ ε n ˆ σ − ( p / m )ˆ σ .Setting q = in Eq. (23) yieldsˆ G − ( q e ) iω ℓ ˆ σ ~ Ψ = ˆ σ ~ Ψ − βV X p Γ (3) ( p ; q e ) GG ( p ; q e ) ~σ , (24a)ˆ G − ( p +e )ˆ σ + ˆ σ ˆ G − ( p ) − Γ (3)T ( p ; q e )ˆ σ ~ Ψ iω ℓ = ˆ σ − βV X p ′ Γ (4) ( p, p ′ ; q e ) GG ( p ′ ; q e ) ~σ , (24b)where q e and p +e are defined by q e ≡ (0 , ω ℓ ) , (25a) p +e ≡ ( p , ε n + ω ℓ ) , (25b)with subscript e denoting energy . On the other hand,setting ω ℓ = 0 in Eq. (23) yieldsˆ G − ( q m )ˆ σ ~ Ψ = q m i ˆ σ ~ Ψ − βV X p Γ (3) ( p ; q m ) × GG ( p ; q m ) p · q m i~σ , (26a)ˆ G − ( p +m )ˆ σ + ˆ σ ˆ G − ( p − m )= p · q m i ˆ σ − βV X p ′ Γ (4) ( p, p ′ ; q m ) GG ( p ′ ; q m ) p ′ · q m i~σ + Γ (3)T ( p ; q m )ˆ σ ~ Ψ , (26b)where q m and p +m are defined by q m ≡ ( q , , (27a) p +m ≡ ( p ± q / , ε n ) , (27b)with subscript m denoting momentum . Taking the limit ω ℓ → q , we obtain the Ward identities from the localgauge transformation, which for ~ Ψ → ~ B. Truncated Galilean transformation
Next, we consider the truncated Galilean transforma-tion [11] ~ψ ′ ( x ) = ~ψ ( x + X ( τ )) , (28)where X ( τ ) ≡ ( R ( τ ) ,
0) with R ( τ + β ) = R ( τ ). Thecondensate wave function remains invariant through thetransformation as it does not depend on x for homoge-neous systems, i.e., δ~ Ψ = 0 . (29)In contrast, variation δ ˆ G ≡ ˆ G ′ − ˆ G in Green’s function isfinite, which in the first order is given by δ ˆ G ( x, x ′ ) = [ R ( τ ) · ∇ + R ( τ ′ ) · ∇ ′ ] ˆ G ( x, x ′ ) . (30)Let us expand R ( τ ) in the Fourier series R ( τ ) = X ℓ R ( q e ) e iq e x , (31) with q e defined by Eq. (25a). Substituting Eqs. (4), (9b),and (31) into Eq. (30), we obtain δ ˆ G ( p ; q ) = i R ( ω ℓ ) · p h ˆ G ( p ) − ˆ G ( p +e ) i , (32)where p +e is defined by Eq. (25b).On the other hand, transformation (28) yields an ap-parent perturbation in the system. Specifically, let ussubstitute ~ψ ( x ) = ~ψ ′ ( x − X ( τ )) into Eq. (16), extractterms first-order in R ( τ ) and set ~ψ ′ ( x ) → ~ψ ( x ) in themas appropriate in the first order, and expand R ( τ ) and ~ψ ( x ) as Eqs. (31) and (17), respectively. We thereby findthat the perturbation is expressible as Eq. (18) with δ ˆ I ( p ; q ) = δ q0 ω ℓ R ( ω ℓ ) · p i ˆ σ , (33)which satisfies Eq. (20).Let us substitute Eqs. (29), (32), and (33) into Eq.(10). Noting that R ( ω ℓ ) is arbitrary, we obtainˆ G ( q e ) 12 βV X p Γ (3) ( p ; q e ) GG ( p ; q e ) p i~σ = 0 , (34a) p ˆ G ( p +e ) − ˆ G ( p ) iω ℓ = GG ( p ; q e ) " p i~σ − βV X p ′ Γ (4) ( p, p ′ ; q e ) GG ( p ′ ; q e ) × p ′ i~σ . (34b)These identities are expressible alternatively by operatingˆ G − ( q e ) on Eq. (34a) from the left and using Eq. (22) inEq. (34b) as12 βV X p Γ (3) ( p ; q e ) GG ( p ; q e ) p i~σ = 0 , (35a) p m ˆ G − ( p ) − ˆ G − ( p +e ) iω ℓ = p m i ˆ σ − βV X p ′ Γ (4) ( p, p ′ ; q e ) GG ( p ′ ; q e ) p ′ m i~σ . (35b)Equation (35b) in the limit of ω ℓ → C. Spatially inhomogeneous perturbation
Finally, we apply a static inhomogeneous potential tothe system U ext ( r ). Let us expand U ext ( r ) = X q U ext ( q ) e iq m x , (36)with q m defined by Eq. (27a). The perturbation is ex-pressible as Eq. (18) with δ ˆ I ( p ; q ) = 12 δ ω ℓ U ext ( q )ˆ σ , (37)which also satisfies Eq. (20).In the limit of q → , the perturbation can be absorbedinto the shift of the chemical potential given by δµ = − U ext ( ). Let us substitute Eq. (37) into Eq. (10), takethe limit q → , and replace U ext ( ) by − δµ . We therebyobtain ∂ ~ Ψ ∂µ = − lim q m → ˆ G ( q m ) (cid:20) ˆ σ ~ Ψ − βV X p Γ (3) ( p ; q m ) GG ( p ; q m ) ~σ (cid:21) , (38a) ∂ ~G ( p ) ∂µ = − GG q ( p ) " ~σ − βV X p ′ Γ (4 q ) ( p, p ′ ) GG q ( p ′ ) ~σ − Γ (3 q )T ( p ) ∂ ~ Ψ ∂µ , (38b)where superscript q denotes the q -limit of setting ω ℓ = 0first and taking the limit q → subsequently [5, 7, 11],e.g., Γ (4 q ) ( p, p ′ ) ≡ lim q m → Γ (4) ( p, p ′ ; q m ) . (39)Equation (38b) is expressible alternatively by using Eq.(22) and δG − = − G − δG G − as ∂ ~G − ( p ) ∂µ = ~σ − βV X p ′ Γ (4 q ) ( p, p ′ ) GG q ( p ′ ) ~σ − Γ (3 q )T ( p ) ∂ ~ Ψ ∂µ . (40)Equation (40) is the superfluid-Bose-liquid version of Eq.(19.4) in Ref. 7 for normal Fermi liquids.Derivative ∂ ~ Ψ /∂µ should be finite, which implies thatthe quantity in the square brackets of Eq. (38a) shouldvanish in the q -limit, i.e.,ˆ σ ~ Ψ − βV X p Γ (3 q ) ( p ) GG q ( p ) ~σ = 0 . (41)It is shown in AppendixB that Eq. (41) can also be de-rived directly from Eq. (40). IV. DENSITY AND CURRENT CORRELATIONFUNCTIONS
In this section, we derive formally exact expressionsof the density and current correlation functions, i.e., Eq.(52) below, based on the linear-response treatment underexternal perturbations. We then clarify their propertiesand also derive low-energy expressions using the Ward-Takahashi identities.
A. Derivation
Consider the perturbation S ext = β X q ( =0) 4 X ν =1 j ν ( − q ) A ν ( q ) , (42)where j ν ( − q ) is defined by j ν ( − q ) = 12 X p ~ψ T ( − p + )ˆ λ ν ( p ) ~ψ ( p − ) , (43)in terms of verticesˆ λ ν ( p ) ≡ p ν m i ˆ σ : ν = 1 , , σ : ν = 4 (44)satisfying ˆ λ ν ( p ) = ˆ λ T ν ( − p ). Equation (42) representsa perturbation linear in A ν ( q ) obtained from Eq. (16)by (i) adding the scalar potential A ( x ) in the squarebrackets, (ii) replacing ˆ p → ˆ p + A , and (iii) expanding A ν ( x ) = P q A ν ( q ) e iqx .The thermodynamic average of j ν ( q ), which vanisheswithout the perturbation, can be written up to the linearorder in A ν as h j ν ( q ) i A = 12 X p h ~ψ T ( − p − )ˆ λ ν ( p ) ~ψ ( p + ) i A = V (cid:20) ~ Ψ T ˆ λ ν (cid:18) q (cid:19) δ~ Ψ( q ) + δ~ Ψ T ( q )ˆ λ ν (cid:18) − q (cid:19) ~ Ψ (cid:21) − β X p ′ ~λ T ν ( − p ) δ ~G ( p ; q ) , (45)where subscript A denotes the thermodynamic averageunder the perturbation, and the primed sum over p sig-nifies excluding the states p ± = 0. In deriving the secondexpression, we have (i) used that h ~ψ ( p ) i ≈ V / (cid:2) δ p ~ Ψ + δ pq δ~ Ψ( q ) (cid:3) holds up to the first order, (ii) performed the transfor-mation for p ± = 0: h ~ψ T ( − p − )ˆ λ ν ( p ) ~ψ ( p + ) i A = Tr ˆ λ ν ( p ) h ~ψ ( p + ) ~ψ T ( − p − ) i A = − β − Tr ˆ λ ν ( p ) δ ˆ G ( p ; q ) = − β − Tr ˆ λ T ν ( − p ) δ ˆ G ( p ; q ) , and (iii) expressed the trace of the product of 2 × C T and ˆ D asTr ˆ C T ˆ D = X jj ′ C j ′ j D j ′ j = ~C T ~D, (46)where ~C T and ~D are defined by ~C T | jj ′ i = C jj ′ , h jj ′ | ~D = D jj ′ . (47)Note that the two terms in the square brackets of Eq. (45)yield the same contribution, as can be checked directlyby using Eq. (44).Our correlation functions are defined in terms of Eq.(45) by K νν ′ ( q ) ≡ h j ν ( q ) i A δA ν ′ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) A =0 = − β h j ν ( q ) j ν ′ ( − q ) i , (48)where the second expression results directly from thedefinition of the thermodynamic average in the pres-ence of the perturbation H A ≡ S ext /β . Comparing Eq.(42) with Eq. (18), we see the correspondence ˆ I ( p ; q ) = P ν ˆ λ ν ( p ) A ν ( q ) so that Eq. (20) is now given byˆ I (s) ( p ; q ) = X ν ˆ λ ν ( p ) A ν ( q ) . (49)Let us substitute Eq. (10) with Eq. (49) into Eq. (45) andperform the differentiation of Eq. (48) to obtain K νν ′ ( q ).The result is expressible symbolically as K νν ′ ( q ) = − β ~λ T ν GG (cid:18) − βV Γ (4) GG (cid:19) ~λ ν ′ + V (cid:20) ~ Ψ T ˆ λ ν ( q / − βV ~λ T ν GG Γ (3)T (cid:21) ˆ G ( q ) × (cid:20) ˆ λ ν ′ ( q / ~ Ψ − βV Γ (3) GG ~λ ν ′ (cid:21) . (50)To express it more concisely and specifically, we introducerenormalized vertices ˆΛ (2) ν ( q ) and ~ Λ (4) ν ( p ; q ) byˆΛ (2) ν ( q ) ~ Ψ ≡ ˆ λ ν (cid:18) q (cid:19) ~ Ψ − βV X p Γ (3) ( p ; q ) GG ( p ; q ) ~λ ν ( p ) , (51a) ~ Λ (4) ν ( p ; q ) ≡ ~λ ν ( p ) − βV X p ′ Γ (4) ( p, p ′ ; q ) GG ( p ′ ; q ) ~λ ν ( p ′ ) . (51b)Equation (50) can be written in terms of them as K νν ′ ( q ) = − β X p ~λ T ν ( − p ) GG ( p ; q ) ~ Λ (4) ν ′ ( p ; q )+ V (cid:2) ˆΛ (2) ν ( − q ) ~ Ψ (cid:3) T ˆ G ( q ) ˆΛ (2) ν ′ ( q ) ~ Ψ , (52a)or alternatively, K νν ′ ( q ) = − β X p ~ Λ (4)T ν ( − p ; − q ) GG ( p ; q ) ~λ ν ′ ( p )+ V (cid:2) ˆΛ (2) ν ( − q ) ~ Ψ (cid:3) T ˆ G ( q ) ˆΛ (2) ν ′ ( q ) ~ Ψ , (52b)which is the key result here. Here we have used the iden-tity on Eq. (51), ~ Ψ T ˆ λ ν ( q / − βV X p ~λ T ν ( − p ) GG ( p ; q )Γ (3)T ( p ; q )= (cid:2) ˆΛ (2) ν ( − q ) ~ Ψ (cid:3) T , (53a) X p ′ ~λ T ν ( − p ′ ) (cid:20) δ pp ′ − βV GG ( p ′ ; q )Γ (4) ( p ′ , p ; q ) (cid:21) | jj ′ i = h jj ′ | ~ Λ (4) ν ( − p ; − q ) ≡ ~ Λ (4)T ν ( − p ; − q ) | jj ′ i , (53b)the latter of which is given in the notation of Eq. (47);they can be shown to hold by using the symmetries givenin AppendixA.Two comments are in order concerning the correlationfunctions. First, Eq. (52) has the structure obtained byGavoret and Nozi`eres at T = 0, where its first and sec-ond terms on the right-hand side are called the regular and singular parts, respectively. Thus, the structure per-sists also at finite temperatures. Second, Eq. (52) can beregarded as the second derivative K νν ′ ( q ) = 12 δ ∆Ω δA ν ( − q ) δA ν ′ ( q )of the variation ∆Ω in the grand potential due to theperturbation. Noting Eqs. (10a), (49), and (50), we canidentify ∆Ω as∆Ω = − β X νν ′ X pq A ν ( − q ) ~λ T ν ( − p ) GG ( p ; q ) ~ Λ (4) ν ′ ( p ; q ) × A ν ′ ( q ) + X q δ~ Ψ T ( − q ) ˆ G − ( q ) δ~ Ψ( q ) . (54)Thus, ∆Ω cannot be expressible in terms of the change δ~ Ψ( q ) of the condensate wave function alone, contraryto the considerations by Baym [14] and Holtzmann andBaym [15], but there is additional one from the excitedstates given by the first term on the right-hand side,which Gavoret and Nozi´eres call the regular part [12].Moreover, it follows from the isotropy of the system that δ Ψ( q ) for q = q m ≡ ( q ,
0) is proportional to q · A ( q m )so that the second term can only contribute to the longi-tudinal part of the current correlation functions. Thus,any microscopic study on the superfluid density, whichis relevant to the transverse response, should be basedon the first term with GG in Eq. (54) instead of thesecond term with δ~ Ψ. In this connection, the “proofs”of the Josephson sum rule [14, 15] and the inequality − G ( p , ≥ mV Ψ /N p [14], which rely on the secondterm of Eq. (54) alone, cannot be justified [16]. Neithercan the claim by Watabe [17] based on these sum ruleand inequality against the emergence of the anomalousexponent in G predicted by our recent renormalizationgroup study [18, 19]. B. Properties of renormalized vertices
It follows from the Ward-Takahashi identities that therenormalized vertices in Eq. (51) satisfy several identities.First, we replace q by q e ≡ ( , ω ℓ ) in Eqs. (51a) and(51b), set ν = 4 and substitute ˆ λ = ˆ σ from Eq. (44),and compare the resulting expressions with Eqs. (24a)and (24b), respectively. We thereby obtainˆΛ (2)4 ( q e ) ~ Ψ = ˆ G − ( q e ) iω ℓ ˆ σ ~ Ψ , (55a) ~ Λ (4)4 ( p ; q e ) = ˆ G − ( p +e )ˆ σ + ˆ σ ˆ G − ( p ) − Γ (3)T ( p ; q e )ˆ σ ~ Ψ iω ℓ . (55b)It should be noted that the numerator on the right-handof Eq. (55b) certainly vanishes in the limit of ω ℓ → q by q m ≡ ( q ,
0) in Eqs. (51a) and(51b), multiply it by q ν , take the sum over ν = 1 , , λ ν = ( p ν /m ) i ˆ σ from Eq. (44), and com-pare the resulting expressions with Eqs. (26a) and (26b),respectively. We thereby obtain X ν =1 q ν ˆΛ (2) ν ( q m ) ~ Ψ = ˆ G − ( q m )ˆ σ ~ Ψ , (56a) X ν =1 q ν ~ Λ (4) ν ( p ; q m ) = ˆ G − ( p +m )ˆ σ + ˆ σ ˆ G − ( p − m ) − Γ (3)T ( p ; q m )ˆ σ ~ Ψ . (56b)Alternatively, we differentiate Eq. (26b) with respect to q ν for ν = 1 , ,
3, take the limit of q → , and comparethe resulting expression with Eq. (51b) in the q -limit de-fined generally by Eq. (39). We thereby obtain ~ Λ (4 q ) ν ( p ) = ∂∂q ν h ˆ G − ( p +m )ˆ σ + ˆ σ ˆ G − ( p − m ) − Γ (3)T ( p ; q m )ˆ σ ~ Ψ i q = , (56c)for ν = 1 , , q by q e in Eqs. (51a) and (51b), set ν = 1 , , λ ν ( p ) = ( p ν /m ) i ˆ σ from Eq.(44), and compare the resulting expressions with Eqs.(34a) and (35b). We thereby obtainˆΛ (2) ν ( q e ) ~ Ψ = ~ , (57a) ~ Λ (4) ν ( p ; q e ) = p ν m ~G − ( p ) − ~G − ( p +e ) iω ℓ , (57b)for ν = 1 , , ν = 4 in Eqs. (51a) and (51b) and taketheir q -limits. Comparing the resulting expressions withEqs. (41) and (40), respectively, we obtainˆΛ (2 q )4 ~ Ψ = ~ , (58a) ~ Λ (4 q )4 ( p ) = ∂ ~G − ( p ) ∂µ + Γ (3 q )T ( p ) ∂ ~ Ψ ∂µ . (58b) C. Properties of K νν ′ ( q ) Let us enumerate properties of Eq. (52), which aregiven by Eqs. (59)-(63) below.First, Eq. (48) satisfies K νν ′ ( q ) = K ν ′ ν ( − q ) , (59)as seen from Eq. (48). One can also confirm based onEq. (53a) and the symmetries given in AppendixA thatEq. (52) satisfies Eq. (59).Second, K νν ′ ( q e ) with q e = ( , ω ℓ ) vanishes identically,i.e., K νν ′ ( q e ) = 0 . (60)This is proved in the three steps of (i)-(iii) below: (i) Weset ν ′ = 1 , , q = q e in Eq. (52a), substitute Eq.(57), and transform the resulting expression as follows: K νν ′ ( q e )= − β X p ~λ T ν ( − p ) GG ( p ; q e ) ~G − ( p ) − ~G − ( p +e ) iω ℓ p ν ′ m = − β X p Tr ˆ λ T ν ( p ) ˆ G ( p +e ) − ˆ G ( p ) iω ℓ p ν ′ m , where we have used Eq. (22) and ~λ T ν ( − p ) ~B = Tr ˆ λ ν ( p ) ˆ B .The last expression can be shown to vanish for each of ν = 1 , , ν = 4 by making a change of variables p +e → p in the summation over ˆ G ( p +e ), which does notaffect the vertex of Eq. (44). Thus, we arrive at Eq. (60)for ν ′ = 1 , ,
3. (ii) Next, we set ν ′ = 4 and q = q e in Eq.(52a), substitute Eq. (55), and transform the resultingexpression as follows: K ν ( q e )= − β X p ~λ T ν ( − p ) GG ( p ; q e ) " ˆ G − ( p +e )ˆ σ + ˆ σ ˆ G − ( p ) iω ℓ − Γ (3)T ( p ; q e )ˆ σ ~ Ψ iω ℓ + V (cid:2) ˆΛ (2) ν ( − q e ) ~ Ψ (cid:3) T ˆ σ ~ Ψ iω ℓ = 12 β X p Tr ˆ λ ν ( p ) ˆ σ ˆ G ( p ) + ˆ G ( p +e )ˆ σ iω ℓ + V ~ Ψ T ˆ λ T ν ( ) ˆ σ ~ Ψ iω ℓ , where we have used Eqs. (22) and (53a). The last expres-sion can be shown to vanish for each of ν = 1 , , ν = 4 by substituting Eq. (44) and making a change ofvariables p +e → p in the summation over ˆ G ( p +e ). Thus,we arrive at Eq. (60) for ν ′ = 4. (iii) Finally, the sym-metry of Eq. (59) completes the proof of Eq. (60).Third, K νν ′ ( q ) for q = q m ≡ ( q ,
0) and ν, ν ′ = 1 , , longitudinal sum rule or f-sum rule [20] givenby X ν ′ =1 K νν ′ ( q m ) q ν ′ = − Nm q ν , (61)where N is the number of particles in the system. Theidentity is proved as follows. We multiply Eq. (52a) for q = q m by q ν ′ and take the sum over ν ′ = 1 , ,
3, sub-stitute Eq. (56), and transform the resulting equation asfollows: X ν ′ =1 K νν ′ ( q m ) q ν ′ = − β X p ~λ T ν ( − p ) GG ( p ; q m ) h ˆ G − ( p +m )ˆ σ + ˆ σ ˆ G − ( p − m ) − Γ (3)T ( p ; q m )ˆ σ ~ Ψ i + V (cid:2) ˆΛ (2) ν ( − q m ) ~ Ψ (cid:3) T ˆ σ ~ Ψ= − β X p Tr ˆ λ T ν ( p ) h ˆ σ ˆ G ( p − m ) + ˆ G ( p +m )ˆ σ i + V ~ Ψ T ˆ λ ν ( q / σ ~ Ψ= − " − β X p Tr ˆ σ ˆ G ( p ) + V ~ Ψ T ˆ σ ~ Ψ q ν m = − Nm q ν , where we have used Eqs. (22) and (53a), substituted λ ν ( p ) = ( p ν /m ) i ˆ σ from Eq. (44), made a change of vari-ables p ± m → p for the sum over ˆ G ( p ± m ), noted that thesum of p ˆ G ( p ) over p vanishes, and replaced G ( p ) by G ( p ) e iε n + in the final sum over p following the standardprocedure for the equal-time Green’s function [21].Fourth, the current correlation functions ( ν, ν ′ =1 , ,
3) of the normal state ( ~ Ψ = ~
0) in the q -limit sat-isfy K n νν ′ ( q m →
0) = − Nm δ νν ′ , (62)which implies that the normal density is equal to the par-ticle density in the normal state, as it should. The proofproceeds as follows. We take the q -limit of Eq. (52a),substitute Eq. (56c), and set ~ Ψ = ~
0. The resulting ex-pression can be transformed by using Eq. (22) and noting ∂ ˆ G − ( p ± m ) /∂q ν (cid:12)(cid:12) q = = ± ∂ ˆ G − ( p ) /∂p ν as K n νν ′ ( q m → − β X p ~λ T ν ( − p ) GG q ( p ) ~ Λ (4 q ) ν ( p )= − β X p ˆ λ ν ( p ) ˆ G ( p ) " ∂ ˆ G − ( p ) ∂p ν ′ ˆ σ − ˆ σ ∂ ˆ G − ( p ) ∂p ν ′ ˆ G ( p )= − β X p Tr ˆ λ ν ( p ) " ∂ ˆ G ( p ) ∂p ν ′ ˆ σ − ˆ σ ∂ ˆ G ( p ) ∂p ν ′ = 12 β X p δ νν ′ m Tr ˆ σ ˆ G ( p )= − δ νν ′ Nm , where we have (i) also used ˆ G ˆ σ = − ˆ σ ˆ G that holds forthe normal state and ˆ G δ ˆ G − ˆ G = − δ ˆ G , (ii) substitutedˆ λ ν ( p ) = ( p ν /m ) i ˆ σ from Eq. (44), and (iii) performedintegration by parts with respect to p . Thus, we havederived Eq. (62) for the normal state.Fifth, the density correlation function K ( q ) in the q -limit satisfies the compressibility sum rule [20], K q = − ∂N∂µ . (63)To prove it, we set ν = ν ′ = 4 in Eq. (52a), take the q -limit and express the second term on the right-hand sidein terms of Eq. (38a), substitute Eqs. (44) and (58), andtransform the resulting expression as follows K q = − β X p ~σ GG q ( p ) ~ Λ (4 q )4 ( p ) − V (cid:0) ˆΛ (2 q )4 ~ Ψ) T ∂ ~ Ψ ∂µ = − β X p ~σ GG q ( p ) " ∂ ~G − ( p ) ∂µ + Γ (3 q )T ( p ) ∂ ~ Ψ ∂µ = 12 β X p Tr ˆ σ ∂ ˆ G ( p ) ∂µ − V ~ Ψ T ˆ σ ∂ ~ Ψ ∂µ = − ∂N∂µ , where we have also used Eq. (22), ˆ G δ ˆ G − ˆ G = − δ ˆ G , andEq. (41). D. Correlation functions in terms of Γ (4 ω ) We finally express the correlation functions in terms oflow-energy vertices Γ (4 ω ) in the ω -limit. The four-pointvertex Γ (4) ( p, p ′ ; q ) satisfiesΓ (4) ( p, p ′ ; q ) = Γ (4i) ( p, p ′ ; q ) − βV X p ′′ Γ (4i) ( p, p ′′ ; q ) × GG ( p ′′ ; q )Γ (4) ( p ′′ , p ′ ; q ) , (64)where Γ (4i) ( p, p ′ ; q ) is the irreducible four-point vertex;see Eq. (58a) of I. Its solution can be written symbolicallywith omitting the factor ( βV ) − for simplicity asΓ (4) = (cid:18) (4i) GG (cid:19) − Γ (4i) . (65)Let us express GG as a sum of the two contributions, GG = GG ω + GG L , (66a)where GG ω ≡ GG ω ( p ) denotes the ω -limit of GG definedby GG ω ( p ) ≡ lim q e → GG ( p ; q e ) (66b)with q e ≡ ( , ω ℓ ), and superscript L signifies low energy .Then Γ (4) can be written alternatively asΓ (4) = (cid:18) (4 ω ) GG L (cid:19) − Γ (4 ω ) , (67)with Γ (4 ω ) ≡ (cid:0) Γ (4i) GG ω (cid:1) − Γ (4i) ; see AppendixC forthe derivation.It is also shown in AppendixC that Eq. (52) is express-ible in terms of GG L as K νν ′ ( q ) = K r ωνν ′ − β X p ~ Λ (4 ω )T ν ( − p ) GG L ( p ; q ) ~ Λ (4) ν ′ ( p ; q )+ V (cid:2) ˆΛ (2) ν ( − q ) ~ Ψ (cid:3) T ˆ G ( q ) ˆΛ (2) ν ′ ( q ) ~ Ψ , (68)where K r ωνν ′ denotes K r ωνν ′ = δ ν δ ν ′ (cid:26) − ∂N∂µ + 12 β X p ~ Λ (4 ω )T4 ( − p ) × (cid:2) GG q ( p ) − GG ω ( p ) (cid:3) ~ Λ (4 q )4 ( p ) (cid:27) , (69)and ˆΛ (2) ν ( q ) ~ Ψ and ~ Λ (4) ν ( p ; q ) are given byˆΛ (2) ν ( q ) ~ Ψ = ˆΛ (2 ω ) ν ~ Ψ − βV X p Γ (3) ( p ; q ) GG L ( p ; q ) × ~ Λ (4 ω ) ν ( p ) , (70a) ~ Λ (4) ν ( p ; q ) = ~ Λ (4 ω ) ν ( p ) − βV X p ′ Γ (4) ( p, p ′ ; q ) GG L ( p ′ ; q ) × ~ Λ (4 ω ) ν ( p ′ ) . (70b)Comparing Eq. (70) with Eq. (51), we observe that thebare vertices ˆ λ ν ~ Ψ and ~λ ν have been replaced by renor-malized vertices ˆΛ (2 ω ) ν ~ Ψ and ~ Λ (4 ω ) ν ( p ) in the ω -limit,which are expressible in terms of ˆ G − by Eq. (55) for ν = 4 and by Eq. (57) for ν = 1 , , q e → V. SUMMARY
We have derived four kinds of Ward-Takahashi identi-ties for correlated Bose-Einstein condensates, which aregiven by Eqs. (24), (26), (35), and (38). Each of themconsists of the condensate part and the quasiparticle part,and taking the limit of Ψ → T = 0 based on the fictitious gap method[12], i.e., their Eqs. (5.22) and (5.24), our identities (i)have the condensate part, (ii) are distinct between the ω -and q -limits, and (iii) contain additional terms with thethree-point vertices. We have also obtained expressions of the density andcurrent correlation functions as Eq. (52), which are com-posed of the regular part and singular part in agree-ment with the result by Gavoret and Nozi`eres. Using theWard-Takahashi identities, we have derived exact prop-erties of the correlation functions, which are given byEqs. (59)-(63). They include the well-known longitudinaland compressibility sum rules. It follows from Eqs. (61)and (62) that the finite superfluid density emerges dueto ~ Ψ = ~ q . Thefact that the vertices are different between the ω -limitand q -limit strongly suggests that the regular part of thecorrelation functions, i.e., the first term of Eq. (52), cansustain collective oscillations in the same way as in Fermiliquids [5, 7] and superfluid Fermi liquids [10, 11].Finally, the correlations functions have been shown tobe expressible in terms of low-energy Green’s functionsand vertices as Eq. (68). They will form a basis forconstructing the low-energy effective theory of correlatedBose-Einstein condensates.Although the present consideration is restricted toequilibrium ones, the extension to nonequilibrium sys-tems can be performed straightforwardly based on theformulation on the Keldysh contour of I by choosing p ± symmetrically as p ± ≡ p ± q/ ACKNOWLEDGMENT
This work was supported by JSPS KAKENHI GrantNumber JP20K03848.
Appendix A: Symmetry Properties of Eq. (11)
It follows from Eq. (6b) of I that Green’s func-tions in the coordinate space obey G jj ′ ( x, x ′ ) = G j ′ j ( x ′ , x ). Moreover, they satisfy [18] G jj ′ ( x, x ′ ) = (cid:2) G − j ′ , − j ( r ′ τ, r τ ′ ) (cid:3) ∗ . The relations translate throughEq. (4) into G jj ′ ( p ) = G j ′ j ( − p ) = (cid:2) G − j ′ , − j ( p , − ε n ) (cid:3) ∗ = G − j ′ , − j ( p ) . (A1)The last equality holds in the gauge where the condensatewave function is real. Using Eq. (A1), one can show easilythat Eq. (11h) satisfies h j j ′ | GG ( p ; q ) | j j ′ i = h j j ′ | GG ( − p ; − q ) | j j ′ i . (A2)It also follows from Eqs. (28a) and (35a) of I that thefour-point vertices obeyΓ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) = Γ (4) ( ξ ′ , ξ ; ξ , ξ ′ )= Γ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) . (4) j j ′ ; j j ′ ( p, p ′ ; q ) = Γ (4) j ′ j ; j j ′ ( − p +e , p ′ ; q )= Γ (4) j j ′ ; j j ′ ( − p ′ , − p ; − q ) , (A3)with p +e ≡ ( p , ε n + ω ℓ ).Regarding Γ (3) ( p ; q ), one can show based on Eq. (35b)of I that it is connected with Γ (4) ( p, p ′ ; q ) byΓ (3) j ; j j ′ ( p ; q ) = X j ′ ( − j + j ′ − Ψ j ′ Γ (4) j j ′ ; j j ′ ( q m / , p ; q ) , (A4)with q m ≡ ( q , (3) j ; j j ′ ( p ; q ) = Γ (3) j ; j ′ j ( − p +e ; q ) = Γ (3)T j j ′ ; j ( − p ; − q ) , (A5)as shown by using Eqs. (35b) and (35c) of I and Eq. (A3)above. Appendix B: Derivation of Eq. (41)
We here derive Eq. (41) from Eq. (40). Let us multiplyEq. (40) at p = 0 by − ˆ σ Ψ (3 σ ) from the left, where Ψ (3 σ ) is defined by h j | Ψ (3 σ ) | j j ′ i ≡ δ j j ( − j ′ − Ψ j ′ . (B1)The left-hand side of the resulting equation yields − ˆ σ Ψ (3 σ ) ∂ ~G − (0) ∂µ = − ˆ σ ∂ ˆ G − (0) ∂µ ˆ σ ~ Ψ = ~ , (B2)where we have (i) made a transformation similar to Eq.(42) of I and also (ii) used the Hugenholtz-Pines relation[22] ˆ G − (0)ˆ σ ~ Ψ / Ψ = 0 in the gauge Ψ = Ψ ≡ Ψ. Onthe other hand, the three terms on the right-hand sideare transformed as − ˆ σ Ψ (3 σ ) ~σ = − ˆ σ ˆ σ ˆ σ ~ Ψ = ˆ σ ~ Ψ , (B3a) − βV X p ′ (cid:0) − ˆ σ Ψ (3 σ ) (cid:1) Γ (4 q ) (0 , p ′ ) GG q ( p ′ ) ~σ = − βV X p ′ Γ (3 q ) ( p ′ ) GG q ( p ′ ) ~σ , (B3b) − (cid:0) − ˆ σ Ψ (3 σ ) (cid:1) Γ (3 q )T (0) ∂ ~ Ψ ∂µ = − h ˆΣ(0) + ˆ σ ˆΣ(0)ˆ σ i ∂ ~ Ψ ∂µ , (B3c)where we have used the Fourier transforms of Eqs. (35b)and (19) in I; see also Eq. (40) of I for the transformationof Eq. (B3c). Let us collect Eqs. (B3a)-(B3c), equate itwith Eq. (B2), and use the fact that Eq. (B3c) vanishesdue to the Nepomnyashchi˘i identity [23] Σ jj (0) = 0. Wethereby obtain Eq. (41). Appendix C: Derivations of Eqs. (67)
It follows from Eq. (64) that Γ (4) , Γ (4i) , and g ≡ GG (C1)obey Γ (4) = Γ (4i) − Γ (4i) g Γ (4) ; we omit factors β − and V ± . Its formal solution is given byΓ (4) = (cid:16) (4i) g (cid:17) − Γ (4i) = Γ (4i) (cid:16) g Γ (4i) (cid:17) − . (C2)Let us decompose g into two parts g = g ω + g L , (C3)where g ω is the ω -limit of g defined by Eq. (66b). Wealso introduce Γ (4 ω ) byΓ (4 ω ) = (cid:16) (4i) g ω (cid:17) − Γ (4i) . (C4)Then Γ (4) is expressible in terms of Γ (4 ω ) and g L as [5, 9]Γ (4) = (cid:16) (4 ω ) g L (cid:17) − Γ (4 ω ) . (C5)This can be shown from Eq. (C2) by writing 1 + Γ (4i) g = A + B with A ≡ (4i) g ω and B ≡ Γ (4i) g L , and usingthe matrix identity ( A + B ) − = (1 + A − B ) − A − .Next, Γ (4) g , 1 − Γ (4) g , and g (1 − Γ (4) g ) can be trans-formed into [9]Γ (4) g = Γ (4 ω ) g ω + Γ (4) g L R, (C6a)1 − Γ (4) g = (cid:0) − Γ (4) g L (cid:1) R, (C6b) g (1 − Γ (4) g ) = g ω R + R T g L (1 − Γ (4) g L ) R, (C6c)with R ≡ − Γ (4 ω ) g ω , (C7a) R T ≡ − g ω Γ (4 ω ) . (C7b)Equation (C6a) can be proved by substituting Eq. (C3)into the left-hand side, expressing Γ (4) g ω = (cid:0) Γ (4 ω ) − Γ (4) g L Γ (4 ω ) (cid:1) g ω based on Eq. (C5), and collecting termswith Γ (4) g L . Equation (C6b) results directly from Eqs.(C6a) and (C7a). Finally, proof of Eq. (C6c) proceedsby writing g (1 − Γ (4) g ) = g R − g Γ (4) g L R based on Eq.(C6b), expressing g Γ (4) = g ω Γ (4 ω ) + R T g L Γ (4) similarlyas Eq. (C6a), and collecting terms with g L .We can also write Γ (3) g and g Γ (3)T asΓ (3) g = Γ (3 ω ) g ω + Γ (3) g L R, (C8a) g Γ (3)T = g ω Γ (3 ω )T + R T g L Γ (3)T , (C8b)1as shown by using Eqs. (A4), (A5), and (C6a).Substitution of Eqs. (C6) and (C8) into Eq. (52) yields K νν ′ = K r ωνν ′ − ~λ T ν R T g L ~ Λ (4) ν + (cid:0) ˆΛ (2) ν ~ Ψ (cid:1) T ˆ G ˆΛ (2) ν ~ Ψ , (C9)where K r ωνν ′ ( r denoting “regular”) is defined by K r ωνν ′ ≡ − ~λ T ν g ω R ~λ ν , (C10)and ˆΛ (2) ν ~ Ψ and ~ Λ (4) ν are now given in terms of g L and R by ˆΛ (2) ν ~ Ψ = ˆ λ ν ~ Ψ − Γ (3 ω ) g ω ~λ ν − Γ (3) g L R ~λ ν , (C11a) ~ Λ (4) ν = (cid:16) − Γ (4) g L (cid:17) R ~λ ν . (C11b)The renormalized vertex R ~λ ν in Eqs. (C10) and (C11)can be approximated as R ( p ) ~λ ν ( p ) ≈ ~ Λ (4 ω ) ν ( p ) . (C12a)Indeed, R~λ ν ≡ (1 − Γ ω g ω ) ~λ ν differs from Eq. (51b) inthe ω -limit in that the argument of Γ (4i) in Eq. (C4) is q instead of q e ( → q . Similarly, the first two terms on the right-hand side of Eq. (C11a)is expressible as the ω -limit of Eq. (51a), i.e.,ˆ λ ν ~ Ψ − Γ (3 ω ) g ω ~λ ν ≈ ˆΛ (2 ω ) ν ~ Ψ . (C12b)Note that Eq. (C12) results directly by taking the ω -limitof Eq. (C11) where g L ω vanishes, as seen from Eq. (C3).Similarly, taking the ω -limit of Eq. (C9) yields K r ωνν ′ ≈ K ωνν ′ − (cid:0) ˆΛ (2 ω ) ν ~ Ψ (cid:1) ˆ G ω ˆΛ (2 ω ) ν ′ ~ Ψ . within the approximation of omitting the q dependencein R . Noting Eqs. (57a) and (60), we can conclude that K r ωνν ′ = 0 (C13a)holds except for ν = ν ′ = 4; see also Eq. (34a). On theother hand, Eq. (C10) for ν = ν ′ = 4 can be transformedas K r ω = K q − ( K q − K r ω ) ≈ − ∂N∂µ + ~λ T4 R T g L q ~ Λ (4 q )4 − (cid:0) ˆΛ (2 q )4 ~ Ψ (cid:1) T ˆ G q ˆΛ (2 q )4 ~ Ψ= − ∂N∂µ + ~ Λ (4 ω )T4 ( g q − g ω ) ~ Λ (4 q )4 , (C13b)where we have successively used Eq. (63), the q -limit ofEq. (C9), Eqs. (38a), (58a), (C12a), and finally Eq. (C3)in the q -limit.Let us substitute Eqs. (C11)-(C13) into Eq. 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