QCD sum rules for the Δ isobar in neutron matter
Jesuel Marques L., Su Houng Lee, Aaron Park, R. D. Matheus, Kie Sang Jeong
aa r X i v : . [ nu c l - t h ] S e p QCD sum rules for the ∆ isobar in neutron matter Jesuel Marques L.,
1, 2, ∗ Su Houng Lee, † Aaron Park, ‡ R. D. Matheus, § and Kie Sang Jeong ¶ Instituto de Fsica Terica, Universidade Estadual Paulista, So Paulo 01140-070, Brazil Department of Physics and Institute of Physics and Applied Physics,Yonsei University, Seoul 03722, Republic of Korea Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Republic of Korea (Dated: September 5, 2018)We study the properties of the ∆ isobar in the symmetric and asymmetric nuclear matter using theQCD sum rules approach based on the energy dispersion relation. Allowing for different continuumthresholds for the polarization tensors with different dimensions, we find stable masses for the ∆in both the vacuum and the medium. Compared to the nucleon case, we find that the vectorrepulsion is smaller for the ∆ while the scalar attraction is similar (75 MeV vector repulsion and200 MeV scalar attraction in the symmetric matter). The smaller vector repulsion can be understoodusing the Pauli principle and a constituent quark model. Also, the isospin dependence of thequasiparticle energy, which mainly comes from the vector self-energy, is quite weak. We also allowfor an explicit π − N continuum contribution to the polarization function but find its effect to beminimal. Phenomenological consequences of our results are discussed. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] I. INTRODUCTION
The ∆ isobar is the lowest excitation of the nucleon that has been identified experimentally, which was done morethan 70 years ago [1]. Since then, it has been a subject of interest in relation to nuclear physics phenomena and inparticular to pion dynamics in nuclear matter [2]. Recently, there has been a renewed interest in the changes of ∆properties in asymmetric nuclear matter, as large changes will soften the neutron star equation of state and henceinfluence the maximum allowed mass for the neutron star [3]. Although the mass shift of the ∆ is expected to besmall in nuclear matter [4–6], there are indications from nuclear target experiments that there are non trivial changesin the ∆ properties in nuclear matter [7, 8]. Furthermore, heavy ion data points to larger mass shifts at higherdensity [9–13].The properties of the ∆ in symmetric nuclear matter were studied in a phenomenological model with nucleonresonances [4] and with self-consistent calculations considering the covariant particle-hole vertex [5, 6], in a chiralquark-meson model [14], in a relativistic chiral effective field theory [15], and in the QCD sum rule approaches [16–18].In Refs. [5, 6] in particular, a fully relativistic and self-consistent many body approach was developed that consistentlyincludes Migdals short range correlations effects. With a realistic parameters, the work predicted a downward shiftof about 50 MeV for the ∆ resonance at nuclear saturation density. Here, we will use the QCD sum rule approachto express the in-medium modification of ∆ state in terms of chiral symmetry restoration and subsequent change ofcondensate values [19–23, 30] and improve on the previous works by allowing for different continuum thresholds forthe polarization tensors with different dimensions, and treating the 4-quark condensates in greater detail. We thenapply this approach to study the ∆ in the asymmetric nuclear matter, where we allow for the difference in the densityof the protons and neutrons. Additionally, we will investigate the effect of explicitly allowing for the π − N continuumcontribution to the phenomenological side of the polarization function as the self-energy of the ∆ can be significantlyaffected by the π − N state in both the vacuum and medium [5, 6]. We find that while the strength of the scalarattraction is similar to that obtained in the nucleon case, the vector repulsion is smaller, which can be understoodusing the Pauli principle and a constituent quark model. Our result shows that the mass of ∆ ++ is reduced by 125MeV (200 MeV scalar attraction and 75 MeV vector repulsion) in symmetric nuclear matter and by 145 MeV (190MeV scalar attraction and 45 MeV vector repulsion) in neutron matter at nuclear matter density.The paper is organized as follows. In Sec. II, we introduce the currents for the ∆ and discuss the QCD sum ruleformalism for this particle. In Sec. III, the results of our sum rule analysis are discussed. In Sec. IV, we discusswhy the vector repulsion for the ∆ should be smaller than that for the nucleon within a constituent quark model.Discussions and conclusions are given in Sec. V. II. QCD SUM RULES FOR THE ∆ ISOBAR STATEA. Correlation of spin- state In this study, we have calculated in-medium properties of the ∆ isobar state. As discussed in the literature [16, 18–23], we first had to choose the proper interpolating current that has the same quantum number as the hadron ofinterest. If the coupling strength between the ∆ isobar state and the interpolating field is strong enough, informationon the physical properties of the ∆ can be obtained from the correlation function:Π µν ( q ) ≡ i Z d xe iqx h Ψ | T[ η µ ( x )¯ η ν (0)] | Ψ i , (1)where | Ψ i is the parity and time-reversal symmetric ground state and η µ ( x ) is an interpolating field for the ∆ isobarstate: h Ψ | η µ (0) | ∆( q, s ) i = λψ ( s ) µ ( q ) , (2)where λ is the coupling to the ∆ state and ψ ( s ) µ is the ∆ isobar wave function with spin index s and momentum q .In the phenomeological sense, the wave function can be regarded as a Rarita-Schwinger (RS) field [25]. In the RSformalism, the Lorentz indices represent the bosonic nature and the Dirac indices represent the fermionic nature of aspin n +12 system, where n is an integer. To account for the correct degrees of freedom of the spin- system, the RSfield should satisfy the following gauge constraints: q µ ψ ( s ) µ ( q ) = 0 , γ µ ψ ( s ) µ ( q ) = 0 , (3)where ‘ q ’ represents the momentum of the on-shell ∆ isobar state.In a vacuum, the propagator of the relativistic spin- field can be obtained as [27, 28] S / µν ( q ) = 1 /q − m ∆ (cid:18) g µν − γ µ γ ν − q µ q ν q − q ( /qγ µ q ν + q µ γ ν /q ) (cid:19) . (4)Making use of the Dirac equation ( /q − m ∆ ) ψ ( s ) µ ( q ) = 0 and the normalization condition ψ ( s ) µ ψ ( s ) µ = − m ∆ of the wavefunction, the ∆ isobar state in the correlator can be described as X s h | η µ (0) | ∆( q, s ) ih ∆( q, s ) | ¯ η ν (0) | i = − λ ( /q + m ∆ ) (cid:18) g µν − γ µ γ ν − q µ q ν q − q ( /qγ µ q ν + q µ γ ν /q ) (cid:19) . (5)On the other hand, as the interpolating field is not the properly gauged RS field, the correlator (1) has followingstructure:Π µν ( q ) = [Π ( q ) + Π q ( q ) /q ] P / ( q ) + (cid:2) Π ( q ) + Π ( q ) /q (cid:3) P / ( q ) + (cid:2) Π ( q ) + Π ( q ) /q (cid:3) P / ( q ) + · · · , (6)where the spin projections, satisfying the constraint P / ( q ) + P / ( q ) + P / ( q ) = g µν , can be summarized as P / ( q ) = g µν − γ µ γ ν − q ( /qγ µ q ν + q µ γ ν /q ) , (7) P / ( q ) = 13 γ µ γ ν − q µ q ν q + 13 q ( /qγ µ q ν + q µ γ ν /q ) , (8) P / ( q ) = q µ q ν q . (9)Thus, to extract the ∆ properties, one can concentrate on the g µν tensor structure and extract the [Π ( q ) + Π q ( q ) /q ]part of the polarization function that has information only about the spin system:Π µν ( q ) = [Π s ( q ) + Π q ( q ) /q ] g µν + · · · ⇒ λ /q − m ∆ g µν + · · · . (10)
1. Phenomenological side
Now, consider taking the medium expectation value of the correlation function characterized by the nuclear density ρ = ρ n + ρ p , the matter velocity u µ , and the iso-spin asymmetry factor I = ( ρ n − ρ p ) / ( ρ n + ρ p ). In the mean-fielddescription of the quasi-∆ isobar state in the medium, the quasi-particle wave function ˜ ψ ( s ) µ will satisfy the equationof motion [ /q − m ∆ − Σ( q, u )] ˜ ψ ( s ) µ ( q ) = 0, where Σ( q, u ) = Σ v ( q , qu ) /u + Σ s ( q , qu ). The functions Σ s , Σ v are the thescalar and vector self-energies, respectively. Then, the in-medium correlation can be written as X s h Ψ | η µ (0) | ∆( q, s ) ih ∆( q, s ) | ¯ η ν (0) | Ψ i = − λ ∗ ( /q − Σ v /u + m ∆ + Σ s )( g µν + · · · ) , (11)and the corresponding phenomenological structure of the in-medium correlator is [16]Π µν ( q ) = (cid:0) Π s ( q , qu ) + Π q ( q , qu ) /q + Π v ( q , qu ) /u (cid:1) g µν + · · · , ⇒ λ ∗ /q − Σ v /u − m ∆ − Σ s g µν + · · · . (12)
2. Operator product expansion:
Each invariant can be expressed in terms of the QCD degrees of freedom via the OPE in q → ∞ , | ~q | → fixed limit:Π i ( q , q ) = X n C in ( q , q ) h ˆ O n i ρ,I , (13)where C in ( q , q ) represent the Wilson coefficients and h ˆ O n i ρ,I represents the in-medium condensate.
3. Dispersion relation
The correlation function satisfies the following dispersion relation on the complex ω plane:Π i ( q , | ~q | ) = 12 πi Z ∞−∞ dω ∆Π i ( ω, | ~q | ) ω − q + F n ( q , | ~q | ) , (14)where F n ( q , | ~q | ) ≡ F en ( q , | ~q | ) + q F on ( q , | ~q | ) is a finite-order polynomial. The discontinuity ∆Π i ( ω, | ~q | ) is defined asfollows: ∆Π i ( ω, | ~q | ) ≡ lim ǫ → + [Π i ( ω + iǫ, | ~q | ) − Π i ( ω − iǫ, | ~q | )] = 2 i Im[Π i ( ω + iǫ, | ~q | )]= ∆Π ei ( ω , | ~q | ) + ω ∆Π oi ( ω , | ~q | ) . (15)All the possible resonances including the ground state are contained in the discontinuity (15). By using these relations,the invariants can be decomposed into an even and an odd part in powers of q , each part having the followingdispersion relation at fixed | ~q | :Π i ( q , | ~q | ) = Π ei ( q , | ~q | ) + q Π oi ( q , | ~q | ) , (16)Π ei ( q , | ~q | ) = 12 πi Z ∞−∞ dω ω ω − q ∆Π oi ( ω , | ~q | ) + F en ( q , | ~q | ) , (17)Π oi ( q , | ~q | ) = 12 πi Z ∞−∞ dω ω − q ∆Π ei ( ω , | ~q | ) + F on ( q , | ~q | ) , (18)where ∆Π ei ( q , | ~q | ) and ∆Π oi ( q , | ~q | ) are even functions of q . In the vacuum limit ( u µ → ρ → ei ( q , | ~q | )reduces to Π ei ( q , | ~q | ) ⇒ Π i ( s ) = 1 πi Z ∞ ds ∆Π i ( s ) s − q + F n ( s ) , (19)and Π oi ( q , | ~q | ) vanishes. B. Currents for the ∆ isobar state In the SU(2) limit, one can represent low-lying baryon states by a direct product of the light quarks [26]: (cid:20)(cid:18) , (cid:19) ⊕ (cid:18) , (cid:19)(cid:21) = (cid:20)(cid:18) , (cid:19) ⊕ (cid:18) , (cid:19)(cid:21) + 3 (cid:20)(cid:18) , (cid:19) ⊕ (cid:18) , (cid:19)(cid:21) + (cid:20)(cid:18) I = 12 (cid:19) representations (cid:21) , (20)where we have used the notation for chiral multiplets, with the first and second numbers in the parenthesis referingto SU(2) L and SU(2) R representations, respectively.As the ∆ isobar has the quantum numbers of spin- and isospin- , it can be described in either the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) or the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representations.For the ∆ ++ state, one takes all the quarks to be the u quarks. Then, in the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation, apossible interpolating current can be chosen as η ∆ µν ≡ ( u T Cσ αβ u ) σ αβ σ µν u, (21)where the antisymmetrized color indices are implied. The ∆ ++ state can be obtained in the parity-even mode in thecorrelation function. In the OPE of the correlation function of Eq. (21), the leading quark condensate contributionappears as a four quark condensate with α s correction. Subsequent sum rule analysis may contain non-negligibleambiguity as higher order quark condensates are not well understood at present.In the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation, one possible interpolating field is η (1) µ ≡ ( u T Cσ αβ u ) σ αβ γ µ u = 4 η (2) µ = 4( u T Cγ µ u ) u. (22)The η (1) µ current is renormalization covariant and can be shown to be equivalent up to a numerical factor to η (2) µ , introduced by Ioffe [20]. In this representation, the chiral condensate appears as the leading quark condensatecontribution in the correlation function. Also, as discussed in Ref. [20], the spin state of individual quark in η (2) µ =( u T Cγ µ u ) u can be understood in a consistent way within the constituent quark model picture. In this work, weconcentrate on this (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation and use the η (2) µ = ǫ abc ( u Ta ( x ) Cγ µ u b ( x )) u c ( x ) current that possessthe aforementioned benefits. C. Operator product expansion and Borel sum rules
We will follow the in-medium light quark and gluon condensates as described in Refs. [30–32]. Here we summarizethe density dependences of the condensates. The condensates in linear density approximation can be expressed asfollows: h ˆ O i ρ,I = h ˆ O i vac + h n | ˆ O | n i ρ n + h p | ˆ O | p i ρ p = h ˆ O i vac + 12 ( h n | ˆ O | n i + h p | ˆ O | p i ) ρ + 12 ( h n | ˆ O | n i − h p | ˆ O | p i ) Iρ, (23)where ρ = ρ n + ρ p and Iρ = ρ n − ρ p . Also, ρ = ρ = 0 .
16 fm − = (110 MeV) is assigned for the normal nucleardensity. Consider an operator ˆ O u,d composed of either up or down quarks, respectively. Using the isospin symmetryrelation, h n | ˆ O u,d | n i = h p | ˆ O d,u | p i , (24)the neutron expectation value can be converted into the proton expectation value.The two-quark operators, Eq. (23), can be arranged as follows: h ˆ O u,d i ρ,I = h ˆ O u,d i vac + ( h p | ˆ O | p i ∓ h p | ˆ O | p i I ) ρ. (25)Here, ‘ ∓ ’ respectively stands for the u and d quark flavors and the isospin operators are defined asˆ O ≡
12 ( ˆ O u + ˆ O d ) , ˆ O ≡
12 ( ˆ O u − ˆ O d ) . (26)All the expectation values will be expressed in terms of the proton counterparts and be denoted as h p | ˆ O | p i → h ˆ O i p .The four quark condensates are denoted following the notation of Ref. [32] ǫ abc ǫ a ′ b ′ c h ¯ q a ′ Γ αm q a ¯ q b ′ Γ βm q b i ρ,I = g αβ h ¯ q Γ m q ¯ q Γ m q i tr. + (cid:18) u α u β − g αβ (cid:19) h ¯ q Γ m q ¯ q Γ m q i s.t. , (27) h ¯ q Γ m q ¯ q Γ m q i tr. = 23 h ¯ q Γ αm q ¯ q Γ mα q i vac − h ¯ q Γ αm t A q ¯ q Γ mα t A q i vac + X i = { n,p } (cid:18) h ¯ q Γ αm q ¯ q Γ mα q i i − h ¯ q Γ αm t A q ¯ q Γ mα t A q i i (cid:19) ρ i , (28) h ¯ q Γ m q ¯ q Γ m q i s.t. = X i = { n,p } (cid:18) h ¯ q Γ m q ¯ q Γ m q i i, s.t. − h ¯ q Γ m t A q ¯ q Γ n t A q i i, s.t. (cid:19) ρ i , (29)where Γ m = { I, γ , γ, γ γ, σ } , and the subscripts vac, i , and s.t. represent the vacuum expectation value, nucleonexpectation value, and symmetric traceless matrix elements, respectively. Twist-4 matrix elements are assigned byfollowing estimates as in Refs. [29, 31, 32]. The spin-0 and spin-1 condensates values are estimated by using thefactorization hypothesis: h q aα ¯ q bβ q cγ ¯ q dδ i ρ,I ≃ h q aα ¯ q bβ i ρ,I h q cγ ¯ q dδ i ρ,I − h q aα ¯ q dδ i ρ,I h q cγ ¯ q bβ i ρ,I , (30) h [¯ uu ] i ρ,I ⇒ k h ¯ qq i + 2 f ( h [¯ uu ] i p − h ¯ uu i p I ) h ¯ qq i vac ρ, (31) h u † u i ρ,I h ¯ uu i ρ,I ⇒ k (cid:0) h [ u † u ] i p − h [ u † u ] i p I (cid:1) h ¯ uu i vac ρ, (32)where the parameters k , k determine the factorization strength for the vacuum piece and f determine the mediumdependence of the scalar four-quark condensate. Both k and k are set to 0.7 to avoid an overly strong contributionof four quark condensates which may amplify ambiguity of the condensates. Furthermore, this choice leads a goodBorel curve that correctly reproduces the mass of the ∆ in the vacuum. f = 0 . f should be weak ( | f | ≪
1) [30–32].The correlation function (1) contains all possible states that overlap with the quantum numbers of the interpolatingfield as discussed before. Because our interest lies in the self-energies on the quasiparticle pole, the other excitationsshould be suppressed. Borel sum rules can be used for this purpose: The weight function W ( ω ) = ( ω − ¯ E q ) e − ω /M has been applied to the discontinuity in the dispersion relation and the corresponding differential operator ¯ B has beenapplied to the OPE side. Each transformed part will be denoted as W M [Π( q , | ~q | )] and ¯ B [Π( q , | ~q | )] respectively.Details for the weighting scheme and the corresponding differential operation in the Borel sum rules are brieflysummarized in Appendix B.Borel transformed invariants contain the quasi-antipole ¯ E q as an input parameter. As we are following relativisticmean field type phenomenology, the antipole ¯ E q is defined regardless of the clear pole like structure in the medium.The exact value can be determined by solving the self-consistent dispersion relation:¯ E q = Σ v ( ¯ E q ) − q ~q + m ∗ ∆ ( ¯ E q ) , (33)where m ∗ ∆ = m ∆ + Σ s and | ~q | = 0 MeV will be used, as the quasi-∆ state is not expected to have its own Fermi levelat the normal nuclear density ρ .The OPE of each invariant has been calculated as follows:Π e ∆ ,s ( q , | ~q | ) = − π q ln( − q ) h ¯ uu i ρ,I (34)Π o ∆ ,s ( q , | ~q | ) = 2 q h ¯ uγ u ¯ uu i ρ,I , (35)Π e ∆ ,q ( q , | ~q | ) = 1160 π ( q ) ln( − q ) + 19 π ln( − q ) h u † iD u i ρ,I + 49 π q q h u † iD u i ρ,I − π ln( − q ) D α s π G E ρ,I + 136 π ln( − q ) D α s π [( u · G ) + ( u · ˜ G ) ] E ρ,I + 34 q h ¯ uu ¯ uu i − q h ¯ uγ u ¯ uγ u i + 54 q h ¯ uγu ¯ uγu i tr. − q h ¯ uγ γu ¯ uγ γu i tr. + 18 q h ¯ uγu ¯ uγu i s.t. − q h ¯ uγ γu ¯ uγ γu i s.t. − q h ¯ uσu ¯ uσu i s.t. (36)Π o ∆ ,q ( q , | ~q | ) = − π ln( − q ) h u † u i ρ,I , (37)Π e ∆ ,u ( q , | ~q | ) = − π q ln( − q ) h u † u i ρ,I (38)Π o ∆ ,u ( q , | ~q | ) = 89 π ln( − q ) h u † iD u i ρ,I − π ln( − q ) D α s π [( u · G ) + ( u · ˜ G ) ] E ρ,I + 1 q h ¯ uγu ¯ uγu i s.t. + 1 q h ¯ uγ γu ¯ uγ γu i s.t. , (39)where the covariant derivative expansion is truncated at first order. Weighted invariants can be summarized as W subt. M [Π ∆ ,s ( q , | ~q | )] = ¯ B [Π e ∆ ,s ( q , | ~q | )] subt. − ¯ E ∆ ¯ B [Π o ∆ ,s ( q , | ~q | )] subt. = − π ( M ) h ¯ uu i ρ,I ˜ E L + 2 ¯ E ∆ ,q h u † u i ρ,I h ¯ uu i vac L , (40) W subt. M [Π ∆ ,q ( q , | ~q | )] = ¯ B [Π e ∆ ,q ( q , | ~q | )] subt. − ¯ E ∆ ¯ B [Π o ∆ ,q ( q , | ~q | )] subt. = − π ( M ) ˜ E L + 19 π M ˜ E h u † iD u i ρ,I L − π ~q h u † iD u i ρ,I L + 5288 π M ˜ E D α s π G E ρ,I L − π M ˜ E D α s π [( u · G ) + ( u · ˜ G ) ] E ρ,I L − h ¯ uu ¯ uu i L + 34 h ¯ uγ u ¯ uγ u i L − h ¯ uγu ¯ uγu i tr. L + 54 h ¯ uγ γu ¯ uγ γu i tr. L − h ¯ uγu ¯ uγu i s.t. L + 98 h ¯ uγ γu ¯ uγ γu i s.t. L + 34 h ¯ uσu ¯ uσu i s.t. L + ¯ E ∆ π M ˜ E h u † u i ρ,I L , (41) W subt. M [Π ∆ ,u ( q , | ~q | )] = ¯ B [Π e ∆ ,u ( q , | ~q | )] subt. − ¯ E ∆ ¯ B [Π o ∆ ,u ( q , | ~q | )] subt. = 14 π ( M ) ˜ E h u † u i ρ,I L + ¯ E ∆ (cid:20) π M ˜ E h u † iD u i ρ,I L − π M ˜ E D α s π [( u · G ) + ( u · ˜ G ) ] E ρ,I L − h ¯ uγu ¯ uγu i s.t. L − h ¯ uγ γu ¯ uγ γu i s.t. L (cid:21) , (42)where M is the Borel mass. Our OPE differs slightly from the one obtained in a previous study [16]. For thescalar invariant, the numeric coefficient of the h ¯ uγ u ¯ uu i ρ,I condensate we obtain is 2, whereas Jin obtains 4 /
3. Theanomalous dimensional running corrections are included as L − η +Γ On ≡ (cid:20) ln( M/ Λ QCD )ln( µ/ Λ QCD ) (cid:21) − η +Γ On , (43)where Γ η (Γ O n ) is the anomalous dimension of the interpolating current η ( ˆ O n ), and µ is the separation scale of theOPE taken to be µ ≃ η = − came from symmetric quark configuration andcan be found in Ref. [33]. The OPE continuum effect above ground state has been subtracted by multiplying thecorresponding ˜ E n to all ( M ) n +1 terms in W M [Π ∆ ,i ( q , | ~q | )] [30]:˜ E ≡ − e − s ∗ /M , (44)˜ E ≡ − e − s ∗ /M (cid:0) s ∗ /M + 1 (cid:1) , (45)˜ E ≡ − e − s ∗ /M (cid:0) s ∗ / M + s ∗ /M + 1 (cid:1) , (46)where s ∗ ≡ ω − ~q and ω is the energy at the continuum threshold, as briefly explained in Appendix B. Thecontinuum subtracted invariants have been denoted as W subt. M [Π ∆ ,i ( q , | ~q | )]. III. SUM RULE ANALYSISA. Sum rule analysis for the quasi- ∆ isobar state If one could calculate the correlation function exactly, the sum rules should not depend on the Borel mass. However,as the OPE is truncated at a finite order, one should find the proper range of Borel mass. The upper bound canbe constrained by requiring the OPE continuum not to exceed 50% of the total OPE contribution while the lowerbound can be constrained by the condition that the contribution of four-quark condensates does not exceed 50% ofthe OPE contribution. We constrain the Borel window through the weighted invariant W subt. M [Π ∆ ,q ( q , | ~q | )] because M ( GeV ) (cid:1) q / Total . Conti. / Total = ( a ) M ( GeV ) Π q / Total . Conti. / Total ρ = ρ ( b ) FIG. 1. Borel window estimated from W subt. M [Π ∆ ,q ( q , | ~q | )] (a) in vacuum and (b) in medium. Black dotted line represents 50%. M ( GeV ) λ q λ s ρ = ( a ) M ( GeV ) λ q * λ s * λ u * ρ = ρ ( b ) FIG. 2. Borel curves for (a) the ∆ isobar pole residues in vacuum and (b) the quasi-∆ isobar pole residues in symmetricmedium at normal nuclear density. Units of vertical axis are GeV . this weighted invariant contains all the contribution of OPE diagrams. In Fig. 1, one can find that the proper Borelwindow is very narrow. As the criteria are satisfied near M ≃ . , the sum rules will be analyzed in the rangeof 1 . ≤ M ≤ . .We first examine the sum rules in vacuum. Considering the phenomenological structure (10), the ground state sumrules can be written as1 π Z s dse − s/M Im [Π µν ( s )] ≃ g µν (cid:16) W subt. M [Π ∆ ,s ( q , | ~q | )] + W subt. M [Π ∆ ,q ( q , | ~q | )] /q (cid:17) + · · ·≃ g µν (cid:16) − λ m ∆ e − m /M − λ e − m /M /q (cid:17) + · · · , (47)where the irrelevant part, the structures proportional to γ µ γ ν , p µ γ ν and so on, which contain the other spin projections,is omitted. Residue sum rules for each invariant can be independently expressed as − λ s m ∆ e − m /M = W subt. M [Π ∆ ,s ( q , | ~q | )] , (48) − λ q e − m /M = W subt. M [Π ∆ ,q ( q , | ~q | )] , (49)where the vacuum limit is taken for the weighted invariant.If the ∆ isobar couples strongly to the interpolating current, the pole residues λ s,q should reflect a good Borelbehavior and the two values extracted from their respective plateau should be similar. The two invariants have differentdimensions and hence the spectral density in the Borel transformed dispersion relations have different weightingfunctions. As we approximate the continuum part of the spectral density with a sharp step function at the threshold s , it is natural to expect that a different threshold would correctly reflect the contributions from the physicalcontinuum contribution in the respective sum rules. Therefore, the continuum threshold is differently assigned as s s = (1 . for W subt. M [Π ∆ ,s ( q , | ~q | )] and s q = (2 . for W subt. M [Π ∆ ,q ( q , | ~q | )], respectively.The large difference in the continuum threshold for the two invariants can also be understood by considering theexcited ∆ states below 1.9 GeV with I ( J ) = ( ). These are the two states ∆(1600) and ∆(1700) with positive andnegative parity respectively. Due to the difference in the parities, the two states have the following phenomenologicalside:Π µν ( q ) = (cid:18) λ /q + m + q − m + λ − /q − m − q − m − (cid:19) g µν = /q (cid:18) λ q − m + λ − q − m − (cid:19) g µν + (cid:18) λ m + q − m − λ − m − q − m − (cid:19) g µν , (50)where the subscript ± denotes the parity of the ∆’s. One notes that the two states contribute differently in the twopolarization tensors and hence will inevitably lead to different effective thresholds. Because of the finite truncationof the OPE, the scalar channel (40) does not reflect the cancellation tendency one can anticipate via Eq. (50). Thuslimitation of the threshold at s s = (1 . is a plausible choice. Furthermore, as the sum rule with differentthreshold leads to a stable sum rule for the ∆ with experimental masses, the different threshold seems to guaranteethe isolation of the ground state ∆ particle in the pole structure.As can be seen in Fig. 2(a) both Borel curves show stable behavior in the proper Borel window with values close toeach other within acceptable uncertainty. The corresponding mass curve is plotted in Fig. 3(a) where one can readthe ∆ isobar mass as m ∆ ≃ .
15 GeV.We did a rough estimate of our errors by the following procedure: We varied the values of s s and s q by (100 MeV) above and below the value we used in the final analysis, as well as the value of k by 0 . . π Z ω − ω dω ( ω − ¯ E q ) e − ω /M Im [Π µν ( ω )] ≃ g µν (cid:16) W subt. M [Π ∆ ,s ( q , | ~q | )] + W subt. M [Π ∆ ,u ( q , | ~q | )] /u + W subt. M [Π ∆ ,q ( q , | ~q | )] /q (cid:17) + · · ·≃ g µν (cid:16) − λ ∗ m ∗ ∆ e − ( m ∗ ∆ +Σ v ) /M + λ ∗ Σ v e − ( m ∗ ∆ +Σ v ) /M /u − λ ∗ e − ( m ∗ ∆ +Σ v ) /M /q (cid:17) + · · · , (51)where m ∗ ∆ = m ∆ + Σ s . The analogous independent expression can be written as − λ ∗ s m ∗ ∆ e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,s ( q , | ~q | )] , (52) λ ∗ u Σ v e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,u ( q , | ~q | )] , (53) − λ ∗ q e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,q ( q , | ~q | )] . (54)As in the vacuum case, the quasipole residues should be the same as they are defined from the same quasiparticlestate. Again, one can assign ω s = ω u = 1 . W subt. M [Π ∆ ,s ( q , | ~q | )] and W subt. M [Π ∆ ,u ( q , | ~q | )], and ω q = 2 . W subt. M [Π ∆ ,q ( q , | ~q | )]. The residue sum rules are plotted in Fig. 2(b), where the stability is less than in the vacuumreflecting the less distinct quasipole structure compared to the vacuum case. Still, one finds moderate behavior of theresidues for the scalar and medium vector invariant. The quasiparticle self-energies in the isospin symmetric matterand the neutron matter are plotted in Figs. 3(b) and 3(c). In the symmetric condition, the strong attraction leads to m ∗ ∆ = m ∆ + Σ s ≃ .
00 GeV, which corresponds to a scalar attraction of about 150 MeV and a weak vector repulsionΣ v ≃
75 MeV, which lead to negative pole shift on the order of ∼
75 MeV in comparison with the ∆ isobar mass invacuum. This reduction pattern is quite similar to the results of Refs. [4–6]. In neutron matter, the vector repulsionbecomes even weaker Σ v ≃ .
045 GeV and the quasi-∆ ++ state energy can be read as ≃ .
07 GeV corresponding toscalar attraction of 140 MeV, which is quite similar in magnitude to the quasi-neutron energy in neutron matter [32]. Finally, our result shows that the quasipole position, which corresponds to the mass of the ∆ ++ in the medium, isreduced by 75 MeV (150 MeV scalar attraction and 75 MeV vector repulsion) in symmetric nuclear matter and by95 MeV (140 MeV scalar attraction and 45 MeV vector repulsion) in neutron matter at nuclear matter density.0 M ( GeV ) m Δ ρ = ( a ) (cid:11)(cid:12)(cid:13) (cid:14)(cid:15)(cid:16) (cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22) M ( GeV ) (cid:23)(cid:24)(cid:25)(cid:26) GeV E Δ m Δ * Σ Δ ρ = ρ , I = ( b ) (cid:27)(cid:28)(cid:29) (cid:30)(cid:31) !" M ( GeV ) ’()* GeV E Δ m Δ * Σ Δ ρ = ρ , I = ( c ) FIG. 3. Borel curves for (a) the ∆ isobar mass in vacuum, and (b) the quasi-∆ isobar self energies in symmetric matter, and(c) in neutron matter. (Orange) long-dashed lines represent reference values. Units of vertical axis are GeV.
B. Consideration of π − N continuum state as ground state The interpolating field η (2) µ can couple to any state or continuum states with the same quantum numbers as the∆ isobar. As the energy threshold of π − N continuum state is expected to be lower than the mass of the ∆ isobar,one may explicitly consider the π − N continuum like contribution. It should be noted that this contribution ispresent in the QCD sum rule approach, because one introduces an interpolating field that couples not only to the ∆but also to the π − N directly. Once this contribution is subtracted out, one can study the ∆ property in mediumwithin the sum rule approach, which can now be compared to any phenomenological calculation that includes the π − N type of contribution with short-range correlation effects such as introduced by the Migdal vertices [34, 35]. Asmentioned before, our sum rule for the ∆ indeed seems to give a result similar to the self-consistent phenomenologicalapproach [6]. Therefore, to identify and isolated the π − N contribution that directly couples to the current, weintroduce a hybrid state to discriminate from the OPE continuum where the other excited states are expected toreside.As briefly discussed in Sec. II, η (2) µ interpolates the isospin- state where the corresponding spin- ground state isexpected. A phenomenological current for the hybrid state can be inferred from the following vertex [24, 36]: L int = gǫ µναβ (cid:0) ∂ µ ¯ ψ ν γ γ α Ψ p ∂ β π + ¯Ψ p γ γ α ∂ µ ψ ν ∂ β π (cid:1) , (55)where Ψ p and π represents proton and pion, respectively. This vertex satisfies the transverse gauge condition q µ ψ µ ( q ) = 0 and the spin- part contained in RS field ψ µ is excluded as shown in Ref. [24]. Identifying the wavefunction of spin- state interpolated via η (2) µ to ψ µ , one can identify the π − N current as J µπN ( x ) = ǫ µανβ γ γ α ∂ ν Ψ p ( x ) ∂ β π ( x ) . (56)Now one can calculate the hybrid ground state from the current (56) correlator:Π µνπN ( q ) ≡ i Z d xe iqx h Ψ | T[ J µπN ( x ) ¯ J νπN (0)] | Ψ i = i/q (cid:16) P / ( q ) − P / ( q ) (cid:17) µρ Z d k (2 π ) S ( i ) p ( k ) D π ( q − k ) k ρ k σ (cid:16) P / ( q ) − P / ( q ) (cid:17) σν /q, (57)1where the hadron propagators are given as S p ( k ) = /k + m p k − m p , (58) S ∗ p ( k ) = /k − Σ pv /u + m ∗ p ( k − Σ pv u ) − m ∗ p , (59) D π ( q − k ) = 1( q − k ) − m π . (60)The quasinucleon propagator S ∗ p ( k ) is used with the self energies calculated in Refs. [31, 32]: m ∗ p = (0 . . I ) GeV,Σ pv = (0 . − . I ) GeV. D π ( q − k ) used in both the vacuum and the in-medium calculation as the in-mediumcorrection is expected to be small in the chiral limit ( m π → W subt. M [Π µνπN ( q )] ⇒ (cid:16) P / ( q ) + 4 P / ( q ) (cid:17) µν π Z ω m ∗ p +Σ pv dωe − ω /M ( − ImΠ q ( ω ) /q − ImΠ u ( ω ) /u − ImΠ s ( ω )) + · · ·≡ (cid:16) P / ( q ) + 4 P / ( q ) (cid:17) µν (cid:0) − ρ q ( M ) /q − ρ u ( M ) /u − ρ s ( M ) (cid:1) + · · · , (61)where only P / ( q ) + 4 P / ( q ) proportional invariants are kept and the ImΠ i ( ω ) are calculated to beImΠ q ( ω ) = ( ω )
24 + ( m ∗ p ) ω − m ∗ p ω − ( m ∗ p ) ω ) − (Σ pv ) ω
24 + ( m ∗ p ) ω ) − m ∗ p − ( m ∗ p ) ω ) ! , (62)ImΠ u ( ω ) = Σ pv ( ω + Σ pv ) ω
24 + ( m ∗ p ) ω ) − m ∗ p − ( m ∗ p ) ω ) ! , (63)ImΠ s ( ω ) = m ∗ p ( ω + Σ pv ) ω
12 + ( m ∗ p ) ω − m ∗ p − ( m ∗ p ) ω ) ! . (64)The weighted invariants reduce to the vacuum structure in the u µ → ρ → η (2) µ -hybridcoupling strength as c πN , one can rewrite the sum rules for the ground state (51) as12 π Z ω ∗ − ω ∗ dω ( ω − ¯ E q ) e − ω /M Im [Π µν ( ω )] ≃ g µν (cid:18) − λ ∗ m ∗ ∆ e − ( m ∗ ∆ +Σ v ) /M + λ ∗ Σ v e − ( m ∗ ∆ +Σ v ) /M /u − λ ∗ e − ( m ∗ ∆ +Σ v ) /M /q − c πN ρ s ( M ) − c πN ρ u ( M ) /u − c πN ρ q ( M ) /q (cid:19) , (65)which leads to following sum rules: − λ ∗ s m ∗ ∆ e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,s ( q , | ~q | )] + c πN ρ s ( M ) , (66) λ ∗ u Σ v e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,u ( q , | ~q | )] + c πN ρ u ( M ) , (67) − λ ∗ q e − ( m ∗ ∆ +Σ v ) /M = W subt. M [Π ∆ ,q ( q , | ~q | )] + c πN ρ q ( M ) . (68)As plotted in Fig. 4(a), the Borel curves for the pole residues do not change drastically but provide a betterbehavior in the vacuum sum rules. Although the hybrid contribution is minimal in the range of 0 . ≤ c πN ≤ .
2, thesubtraction of this hybrid like structure makes the pole contribution even more clear with m ∆ = 1 .
21 GeV [Fig. 5(a)].However, the same subtraction scheme does not change the in-medium sum rules very much. However, it should bepointed out that if the softening of the pion spectrum is taken into account, the situation might change. This is sobecause although the direct π − N continuum is a contribution appearing in the sum rule approach, it is neverthelesscorrelated to the continuum appearing in the ∆ self-energy as the total spectral density in the correlation function2 M ( GeV ) λ q λ s ρ = c π N = ( a ) M ( GeV ) λ q * λ s * λ u * ρ = ρ , c π + = ( b ) FIG. 4. π − N continuum subtracted Borel curves for (a) the ∆ isobar pole residues in vacuum and (b) the quasi-∆ isobar poleresidues in medium. Units of vertical axis is GeV and c πN = 0 . is identified to the changes in the operator product expansion. The quasipole residues plotted in Fig. 4(b) and theself-energies plotted in the curves in Figs. 5(b) and 5(c) are almost unchanged from the plots in Figs. 2(b), 3(b), and3(c), respectively. The negative pole shift becomes of order 100 MeV (200 MeV scalar attraction and 75 MeV vectorrepulsion in the symmetric matter), which agrees with the experimental observation [11].The other apparent feature is that the scalar self-energy is almost isospin independent [Figs. 3(c) and 5(c)]. Thistendency agrees with the phenomenology discussed in Ref. [3] where the quasibaryon vector self-energy has beenarranged as Σ v ≡ g ω ∆ ¯ ω + τ i g ρ ∆ ¯ ρ (3)0 , (69)Σ Nv ≡ g ωN ¯ ω + τ p/n g ρN ¯ ρ (3)0 , (70)where g i represents the meson exchange coupling and τ p/n = ±
1. Following this notation, the vector self-energy inthis work can be expressed asΣ v ( ρ, I ) = − W subt. M [Π ∆ ,u ( q , | ~q | )] W subt. M [Π ∆ ,q ( q , | ~q | )] ≃ Σ v ( ρ ,
0) + (Σ v ( ρ , − Σ v ( ρ , I ≃ (0 . − . τ I ) GeV , (71)where τ i is defined as τ ++ = 3, τ + = 1, τ = −
1, and τ − = −
3. If one considers the proton self-energiescalculated in Refs. [31, 32], where Σ pv = (0 . − . I ) GeV, the ratio x ρ ≡ g ρ ∆ /g ρN becomes very small ( x ρ ≃ . IV. WEAK VECTOR REPULSION IN A CONSTITUENT QUARK MODEL
Our result shows that the vector repulsion for the ∆ in medium is weaker than that for the nucleon. This resultcan be understood by making use of the Pauli principle and constituent quark model. In the quark cluster model, theshort-range vector repulsion can be shown to arise from combining Pauli principle and the quark two-body interactionsbetween quarks [37, 38]. The repulsion can be estimated by comparing the static energy of dibaryon configuration tothat of the two separated baryon states.If we represent the dibaryon state using coefficients of fractional parentage (cfp) [39], then we can estimate howmuch baryon-baryon state is included in dibaryon state with specific quantum number. If we only consider an s -wavedibaryon state so that the orbital wave function is totally symmetric, the corresponding dibaryon states are as follows.For ( I, S )=(1,0) or (0,1), Ψ = 13
N N + 23 √ √ CC, (72)where CC means hidden color states that cannot be represented in terms of free baryons.3 ,-. /23 M ( GeV ) m Δ ρ = c π : = ( a ) ;<= >?@ ABCDEF M ( GeV ) E Δ m Δ * Σ Δ ρ = ρ , c GH = = ( b ) IJK LMO
PQRSTU M ( GeV ) VWXY
GeV E Δ m Δ * Σ Δ ρ = ρ , c Z[ = = ( c ) FIG. 5. π − N continuum subtracted Borel curves for (a) the ∆ isobar mass in vacuum, (b) the quasi-∆ isobar self energiesin symmetric matter, and (c) in neutron matter. Orange long-dashed lines represent reference values. Units of vertical axis isGeV and and c πN = 0 . I, S ) (3,0) (2,1) (1,2) (1,0) (0,3) (0,1) V d
16 8 16 ∆ V
16 24 0
TABLE I. V d is the expectation value of − P i I, S )=(2,1) or (1,2), Ψ = 13 ∆∆ + 23 √ N ∆ + 2 √ CC, (73)where the relative orbital state for N and ∆ is symmetric. These compositions determine the fractional weight ofrepulsion in the dibaryon configuration that contributes to the respective repulsion in the two-baryon interaction atshort distance. It can be shown that the difference in the interaction energy between the dibaryon configurationsand two separated baryons are dominated by the color-spin interaction, while all other s -wave interaction strengthcancel [40, 41]. Assuming that the dibaryon and the two-baryon occupy the same spatial configurations, the colorspin interaction strength can be shown to be proportional to the following matrix element: V = − n X i 1) and (1 , N N states.Making use of the N N component in the respective dibaryon configuration given in Eq. (72), we find the relativerepulsion to be as follows:12 (cid:18) H NN , + H NN , (cid:19) = 12 (cid:18) (cid:19) 563 + 12 (cid:18) (cid:19) 24 = 6427 ≃ . . (75)42. Spin- and isospin-averaged N -∆ interaction:Similar to N N case, only two ( I, S ) states, which are (1 , 2) and (2 , N ∆ state. Making use of the N − ∆ component in the respective dibaryon configuration given in Eq. (73), we find the relative repulsion tobe as follows: 12 (cid:18) H N ∆1 , + H N ∆2 , (cid:19) = 12 (cid:18) √ (cid:19) 16 + 12 (cid:18) √ (cid:19) 803 = 256135 ≃ . . (76)Therefore, we can conclude that the repulsion in N ∆ is 20% smaller than that in N N . This trend of a weak vectorrepulsion from the ∆ isobar is consistent with the above QCD sum rule results. V. DISCUSSION AND CONCLUSIONS In this work, we calculated the quasi-∆ isobar energy in the isospin asymmetric matter. We allowed for differentcontinuum thresholds for the invariants with different dimensions and obtained an stable Borel curve for the ∆ isobarmass in the vacuum. The quasi-∆ ++ self energies in the medium are also obtained within the stable Borel curves. Wefind that the vector self energy in the medium, which can be understood as the repulsive vector potential in the meanfield approximation, is very weak in comparison to the nucleon case [22, 23, 26, 30–32]. As an order of magnitudeestimate, the attraction in the scalar channel is 200 MeV and the repulsion in the vector channel is less than 100MeV.As the interpolating field η (2) µ can couple to the π − N continuum state with energy threshold lower than the ∆ ++ mass, we also explicitly considered the phenomenological structure coming from this continuum. For the vacuumsum rules, the subtraction of π − N continuum makes the pole signal more prominent, whereas in the medium case,it does not make any significant changes with respect to the quasi-∆ ++ sum rules without the π − N continuum.The situation might change if the softening of the pion spectrum is taken into account. This is so because althoughthe direct π − N continuum is a contribution appearing in the sum rule approach, it is nevertheless correlated tothe broadening appearing in the ∆ self-energy as the total spectral density in the correlation function is identified tothe changes in the operator product expansion. The broadening of the ∆ can be found in studies that consistentlytake into account the quasiparticle and the quasiparticle loop structure [4, 5] within the mean field self-energies. Inparticular, it would be extremely useful to find an improved interpolating current with which one can identify thepart of the OPE in the ∆ correlation function that corresponds to explicit structures of a quasiparticle such as theMigdal contact interaction vertex [6, 34, 35]. We will leave the subject for a future discussion and research for theQCD sum rule study in the spin- state.In the neutron matter, the quasiparticle energy is found to be similar to that in the isospin symmetric condition.This is because the scalar self-energy does not depend strongly on the isospin asymmetry and the vector self-energy isweak. If this attractive tendency is still valid in the dense regime, the stiff symmetry energy shown in Ref. [31] wouldallow nn ↔ p ∆ − in the equilibrium. This strong attraction would lead to a soft equation of state even if there is nohyperon condensation in the matter as discussed in Ref. [3].Although the attractive tendency of the quasi-particle energy agrees with the experimental observations [9–13], astrong scalar attraction with weak vector repulsion seems problematic in relation to the recently established maximumneutron star mass which is close to 2M ⊙ [42, 43]. In the mean field approximation, the potential can be understoodas being the average interaction between the quasi-∆ isobar and the surrounding nucleons, namely the averagedtwo-particle interaction. A strong three-body repulsion would provide a possible resolution to the problem [40]. Athree-body repulsion between the ∆ and two nucleons at short distance will only become important when the densitybecomes large. Then the linear density extrapolation of strong scalar attraction will be saturated by the higher densityrepulsive force which will prevent the ∆ condensation from occuring. ACKNOWLEDGMENTS This material is based upon work supported by the So Paulo Research Foundation (FAPESP) under Grants No.2017/15346-0 and No. 2016/02717-8 (J.M.L. and R.D.M) and Korea National Research Foundation Grants. No. NRF-2016R1D1A1B03930089 (S.H.L.), No. NRF-2018R1D1A1B07043234 (A.P.), and No. NRF-2017R1D1A1B03033685(K.S.J.). A part of the calculation of the OPE was checked using the package “FeynCalc 9.0” [44, 45].5 Appendix A: Interpolating currents for the ∆ isobar1. Possible interpolating currents and relations between them For the purpose of describing the ∆ within QCD sum rules, we initially considered the following set of currents inthe (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation: η (1) µ (1 , = ( q T Cσ αβ q ) σ αβ γ µ q , (A1) η (2) µ (1 , = ( q T Cγ µ q ) q , (A2) η (3) µ (1 , = ( q T Cγ ν q ) σ µν q , (A3) η (4) µ (1 , = ( q T Cσ µν q ) γ ν q , (A4)where q , q and q are one of the light quark fields, according to the hadron we want to describe. If q = q = q = u ,that is, when all quark fields are the ones for the up quark, the sum rule calculated with this current should describethe ∆ ++ resonance, and respectively when they are all down quark fields, this should describe the ∆ − . In this case,with all quarks being the same, it is possible to show through Fierz rearrangement that η (1) µ = 4 η (2) µ = 4 iη (3) µ = − iη (4) µ , (A5)so the current choice is unique in this representation up to an overall numerical factor. If one of the quark fields inthe current corresponds to a different flavor from the other two, the currents should couple to the ∆ + (two up quarksand one down) and the ∆ (two down quarks and one up). In this case, however, taking the example for the ∆ + current, there are two possible configurations, η µ ( u,u ) d and η µ ( u,d ) u . Again, through Fierz rearrangement, one proves η (1) µ ( u,d ) u = 3 η (2) µ ( u,u ) d + iη (3) µ ( u,u ) d , (A6) η (2) µ ( u,d ) u = 18 η (1) µ ( u,u ) d + 14 η (2) µ ( u,u ) d + i η (3) µ ( u,u ) d , (A7) η (3) µ ( u,d ) u = − i η (1) µ ( u,u ) d − i η (2) µ ( u,u ) d − η (3) µ ( u,u ) d , (A8) η (4) µ ( u,d ) u = i η (1) µ ( u,u ) d + 3 i η (2) µ ( u,u ) d − η (3) µ ( u,u ) d − η (4) µ ( u,u ) d . (A9)From these relations one can show that η (1) µ ( u,u ) d + 2 η (1) µ ( u,d ) u = 4 (cid:16) η (2) µ ( u,u ) d + 2 η (2) µ ( u,d ) u (cid:17) = 4 i (cid:16) η (3) µ ( u,u ) d + 2 η (3) µ ( u,d ) u (cid:17) = − i (cid:16) η (4) µ ( u,u ) d + 2 η (4) µ ( u,d ) u (cid:17) , (A10)which is the generalization of the relations between the currents when all flavors are the same, Eq. A5. As explainedin Appendix A 2, the current η (1) µ is renormalization covariant, irrespective of the quark flavors. Hence we find thatthe combination (2 η ( i ) µ ( u,d ) u + η ( i ) µ ( u,u ) d ), for i = 1 to 4, is renormalization covariant, since this is true for i = 1 and theyare all proportional to each other.If one considers explicit light quark flavors, the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation contains the isospin- configurationwhich corresponds to the nucleon state, so simply using the current η ( i ) µ ( u,d ) u or η ( i ) µ ( u,u ) d is not enough to ensure thatwe are describing the ∆ + state, because there can be mixing with the proton spectral density.We expected that the sum rule done with the current (2 η µ ( u,d ) u + η µ ( u,u ) d ) would be describing the ∆ + only, sincethis configuration seems to be the natural generalization of η µ ( u,u ) u . However, the requirement that in the vacuumand symmetric nuclear matter that the sum rule obtained using this current and the one using η µ ( u,u ) u be the same,from the isospin symmetry of these systems, was not fulfilled. The issued is that although the OPE obtained withthe (2 η µ ( u,d ) u + η µ ( u,u ) d ) current is proportional to the one obtained with the η ( i ) µ ( u,u ) u up to dimension 4 operators,for the dimension 6, the 4-quark operators, the situation seems to be more subtle. We intend to address this puzzlein a subsequent work and in this article we focused on the case for ∆ ++ and ∆ − only. 2. The renormalization of the interpolating currents As introduced in Sec. II, the interpolating current for ∆ ++ is uniquely determined in the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representa-tion. However, for the ∆ + and ∆ , the choice is non-trivial as the local operator itself contains isospin- configurationand, also, the anomalous running can be different by the choice of local composite operator.6 (a) (c) q q q (b) q q q q q q FIG. 6. Anomalous correction of the interpolating current in one-loop order. Shaded ellipse in the dotted circle represents thediquark in the interpolating current. Consider the first type of interpolating field with explicit light quark flavors. η (1) µ (12)3 ≡ ( q T Cσ αβ q ) σ αβ γ µ q , (A11)where the numbers in parentheses represent the light quark flavors in the diquark. Then, the one-loop corrections canbe calculated as presented in Fig. 6: δ ( a ) h η (1) µ (12)3 i + δ ( b ) h η (1) µ (12)3 i = α s π ( γ + ln 4 π + · · · ) ( q T Cσ αβ q ) σ αβ γ µ q , δ ( c ) h η (1) µ (12)3 i = 0 , (A12)where γ ≃ . η (1) µ (12)3 is renormalization covariant.The second type is defined as η (2) µ (12)3 ≡ ( q T Cγ µ q ) q . (A13)For this current, the one-loop correction can be calculated as δ ( a ) h η (2) µ (12)3 i + δ ( b ) h η (2) µ (12)3 i = α s π ( γ + ln 4 π + · · · ) (cid:0) q T Cγ µ q ) q − i ( q T Cγ ρ q ) σ µρ q (cid:1) ,δ ( c ) h η (2) µ (12)3 i = α s π ( γ + ln 4 π + · · · ) ( q T Cγ µ q ) q , (A14)where ( q T Cγ ρ q ) σ ρµ q = i ( q T Cγ µ q ) q in q = q = q = q limit. The second type is renormalization covariant onlywhen q = q = q condition is satisfied.A specific linear combination of the operators can have renormalization covariance. Using relations obtained throughFierz rearrangement, ( u T Cσ αβ u ) σ αβ γ µ d = 8( u T Cγ µ u ) d − u T Cγ ρ u ) γ ρ γ µ d, (A15)( u T Cσ αβ d ) σ αβ γ µ u = 2( u T Cγ µ u ) d + ( u T Cγ ρ u ) γ ρ γ µ d, (A16)one can obtain a renomalization covariant expression for the ∆ + : η ∆ + µ ≡ ( u T Cσ αβ u ) σ αβ γ µ d + 2( u T Cσ αβ d ) σ αβ γ µ u = 4( u T Cγ µ u ) d + 8( u T Cγ µ d ) u, (A17)where the running off-diagonal terms are canceled. Appendix B: Borel weighting scheme1. Borel sum rules in the vacuum Using analyticity of the correlation function, the invariants can be written in a dispersion relation:Π i ( q ) = 12 πi Z ∞ ds ∆Π i ( s ) s − q + P n ( q ) , (B1)7where P n ( q ) is a finite-order polynomial in q which comes from the integration on the circle of contour on complexplane and the discontinuity ∆Π i ( s ) ≡ lim ǫ → + [Π i ( s + iǫ ) − Π i ( s − iǫ )] = 2 i Im[Π i ( s + iǫ )] is defined on the positivereal axis. All the possible physical states are contained in this discontinuity. From phenomenological considerations,the invariant can be assumed to have a pole and continuum structure,∆Π i ( s ) = ∆Π pole i ( s ) + θ ( s − s )∆Π OPE i ( s ) , (B2)where s represents the continuum threshold. To suppress the continuum contribution, weight function W ( s ) = e − s/M can be used as W M [Π i ( q )] = 12 πi Z ∞ ds e − s/M ∆Π i ( s ) . (B3)The corresponding differential operator B can be defined as B [ f ( − q )] ≡ lim − q ,n →∞− q /n = M ( − q ) n +1 n ! (cid:18) ∂∂q (cid:19) n f ( − q ) . (B4)By using this operator to Eq. (B1), one obtains following relation: W M [Π i ( q )] = 12 πi Z ∞ ds e − s/M ∆Π i ( s ) = B [Π i ( q )] , (B5)where following relation has been used: B (cid:20) s − q (cid:21) = e − s/M . (B6)The residues located after the continuum threshold with finite s can be subtracted as W subt. M [Π i ( q )] = 12 πi Z s ds e − s/M ∆Π i ( s ) = B [Π i ( q )] subt. . (B7)Via simple integrations, Z ∞ s dse − s/M = M e − s /M , (B8) Z ∞ s dsse − s/M = ( M ) e − s /M (cid:0) s /M + 1 (cid:1) , (B9) Z ∞ s dss e − s/M = ( M ) e − s /M (cid:0) s / M + s /M + 1 (cid:1) , (B10)and the subtraction of continuum contribution for the OPE side can be summarized as E ≡ − e − s /M , (B11) E ≡ − e − s /M (cid:0) s /M + 1 (cid:1) , (B12) E ≡ − e − s /M (cid:0) s / M + s /M + 1 (cid:1) . (B13) E n is multiplied to all ( M ) n +1 terms in B [Π i ( q )]. This weighting scheme and subsequent spectral sum rules areknown as Borel transformation and Borel sum rules. 2. Borel sum rules in the medium Now one can extend this argument to the in-medium case. The OPE of the correlation function has been executedin the q → −∞ , | ~q | → fixed limit and the dispersion relation (14) has been written from Cauchy relation in thecomplex energy q space. As one can not expect the symmetric spectral density assumed in the vacuum sum rules,8the contour (b) has been set to be different from the vacuum case, the contour (a). Similarly, one can try followingweighting in the complex energy plane: I contour( b ) dω W ( ω )Π i ( ω ) = 0 , (B14) Z circle( b ) | ω | =˜ ω dω W ( ω )Π i ( ω ) = − Z ˜ ω − ˜ ω dω W ( ω )∆Π i ( ω ) . (B15)As the phenomenological structure (12) is given with the quasi-poles E q = Σ v + p ~q + m ∗ and ¯ E q = Σ v − p ~q + m ∗ ,the weight function should suppress not only the OPE continuum but also the quasi-anti-∆ pole ¯ E q . W ( ω ) =( ω − ¯ E q ) e − ω /M has been used for the in-medium weight function for this purpose. This choice emphasizes theOPE quasi-∆ pole and suppresses the quasi-anti-∆ pole and the continuum. Moreover, in the zero density limit, thein-medium sum rules obtained with W ( ω ) = ( ω − ¯ E q ) e − ω /M can be reduced to the vacuum sum rules with an overallfactor e − ~q /M as the discontinuity ∆Π i ( ω ) is odd in the vacuum case. With the dispersion relations (17) and (18),the weighted sum of the residues can be written as W M [Π i ( q , | ~q | )] = 12 πi Z ∞−∞ dω ( ω − ¯ E q ) e − ω /M ∆Π i ( ω , | ~q | )= 12 πi (cid:20)Z ∞−∞ dω ω e − ω /M ∆Π oi ( ω , | ~q | ) − ¯ E q Z ∞−∞ dω e − ω /M ∆Π ei ( ω , | ~q | ) (cid:21) = ¯ B [Π ei ( q , | ~q | )] − ¯ E q ¯ B [Π oi ( q , | ~q | )] , (B16)where the discontinuity ∆Π i ( ω , | ~q | ) has been defined in Eq. (15) and the differential operator ¯ B is defined analogouswith the the operator (B4): ¯ B [ f ( − q , | ~q | )] ≡ lim − q ,n →∞− q /n = M ( − q ) n +1 n ! (cid:18) ∂∂q (cid:19) n f ( − q , | ~q | ) . (B17)The subtraction of the residues after s ∗ = ω − ~q can be summarized as Eqs. (44)-(46), analogous with the subtractionscheme of the vacuum case.The main difference from the vacuum Borel sum rules is the appearance of the quasi-anti-pole ¯ E q : W subt. M [Π i ( q , | ~q | )] = (cid:2) ¯ B [Π ei ( q , | ~q | )] − ¯ E q ¯ B [Π oi ( q , | ~q | )] (cid:3) subt. . 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