Nuclear Symmetry Energy from QCD sum rules Kie Sang Jeong1 and Su Houng Lee1 1Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea (Dated: September4, 2012) We calculate the nucleon self-energies in an isospin asymmetric nuclear matter using QCD sum rules. Takingthedifference of these for theneutron and proton enables usto express thepotential part of the nuclear symmetry energy in terms of local operators. We find that the scalar (vector) self energy part gives negative (positive) contribution to the nuclear symmetry energy which are consistentwiththeresultsfromrelativisticmeanfieldtheories. Moreover,wefindthatanimportant contribution to the negative contribution of the scalar self energy comes from the twist-4 matrix elements,whose leadingdensitydependencecan beextracted from deepinelastic scattering experi- ments. Thissuggeststhattwist-4contributionpartlymimicstheexchangeoftheδ mesonandthat 2 it constitutes an essential part in theorigin of thenuclearsymmetry energy from QCD.Ourresult 1 also extends an early success of QCD sum rule method in understanding the symmetric nuclear 0 matter in terms of QCD variables to the asymmetric nuclear matter case. 2 PACSnumbers: 21.65.Cd,21.65.Ef,12.38.Lg p e S I. INTRODUCTION the strong optical potentials were found to have a basis 1 from QCD. The first pioneering work of applying QCD ] With the recent constructions and plans for the next sum rule method to nucleons in medium were performed h generation low energy rare isotope accelerators world- by Drukarev and Levin [9, 10]. Later, the relation be- t came clearer through the work by Cohen, Furnstahl and - wide,thereisarenewedinterestinthenuclearsymmetry l Griegel [11], who used the energy dispersion relation to c energy, as the topic could be studied further with finer showthattheshowedthestrongscalar-vectorselfenergy u details in these experiments [1]. Understanding the de- n tailsofthenuclearsymmetryenergywillprovidevaluable appearinginthequasi-nucleonpoleinthesymmetricnu- [ insightsintoexoticnuclearmatterrangingfromrareiso- clear matter [11–14] can be traced back to the scalar- 1 topes toneutronrichnuclearmattersuchasthe neutron vectorquarkcondensateinthe nuclearmediumandthat thecontributionfromhigherdimensionalcondensatesdo v star [2, 3]. One of the main puzzle to be solved is the 0 behavior of the nuclear symmetry energy at high den- not change the general behavior. 8 sity [4, 5]. Motivated by these results, and to express and eluci- 0 From a phenomenological point of view, insights into datetheoriginofnuclearsymmetryenergydirectlyfrom 0 nuclear symmetry energy can be obtained from under- QCD, we have applied QCD sum rule to calculate the . 9 standing the nuclearbinding expressedthroughthe semi neutron and proton energy in the asymmetric nuclear 0 empiricalmassformula [6]. There,thesymmetryenergy matter. Identifying the difference with appropriate fac- 2 can be understood as originating from the energy differ- tors to the nuclear symmetry energy, we show that this 1 : ence between the proton and the neutron in an isospin energy canbe expressedin terms of quarkand gluonde- v asymmetric nuclear matter. Hence, within the approxi- greeof freedom. Attempts to calculate the nucleonmass i X mation,theproblemofcalculatingthenuclearsymmetry in an asymmetric matter using QCD sum rules were re- energy can be reduced to obtaining the optical potential ported before [15–17]. But here, we will follow the for- r a and thereby calculating the energy of the nucleon quasi- malism adopted in Ref. [11]. We have performed the particleneartheFermisurfacesinanasymmetricnuclear operator product expansion (OPE) up to dimension 6 matter. operatorsandhaveidentified allthe independent twist-4 In Dirac phenomenology of nucleon-nucleus scatter- operators, where for most of the operators, the leading ing [7, 8], the optical potential of the nucleon is com- density dependence can be extracted from the deep in- posed of a vector and scalar part, U ≃ S +Vγ0. It is elastic scattering data on higher twist effects. From the well known that in order to fit the spin observables, one QCD sum rule analysis, we find that the scalar (vector) needs a strong scalar attraction(Re S <0) and a strong self energy part gives negative (positive) contribution to vector repulsion (Re V > 0) both of several hundred the nuclear symmetry energy which are consistent with MeV,butsuchthatthe combinedsumgivesonlyasmall the results from relativistic mean field theories. More- attraction of few tens of MeV, consistent with the tradi- over,we find that an important contribution to the neg- tional low energy nuclear physics. The strong scalar and ative contribution of the scalar self energy comes from vector potential comes about naturally also in relativis- the twist-4 matrix elements, whose higher density be- tic mean field theories (RMFT), where meson exchange havior will determine the still controversial property of interaction between nucleons on the Fermi sea produce the symmetry energy at these densities. Our result also strong scalar and vector potentials for the nucleons. extends an early success of QCD sum rule method in ButitwasonlyaftertheworksinQCDsumrulesthat understanding the symmetric nuclear matter in terms of 2 QCD operators to the asymmetric nuclear matter case. by from the second term Eq. (2). That is, (E −E ) = n p The paper is organized as follows. In Sec. II, we start 1IA∆, hence, 4 withbriefreviewandsimple ideaforthenuclearsymme- tryenergy. QCDsumrulefornucleonsintheasymmetric 1 1 a = A∆= (E (A,I)−E (A,I)). (4) nuclear matter is presented in Sec. III. Finally, sum rule A 8 4I n p analysis and conclusion are given in Sec. IV. For infinite nuclear matter case, one can get E as: N 1 II. A SIMPLIFIED DESCRIPTION FOR THE E = d3k d3k E (ρ ,ρ ). (5) NUCLEAR SYMMETRY ENERGY N d3k d3k n p N n p n p Z AndthenucleaRrsymmetryenergycanbeobtainedbycol- We start from a finite nuclei with A nucleons. In the lectingcoefficientsofI2 in(2). This generalizedEsym(ρ) Bethe-Weizs¨aker formula for the nuclear binding energy in Eq. (3) can be decomposed into kinetic like part and given as potential like part by mean field type quasi-particle ap- m =Nm +Zm −E /c2, proximation, in which one can treat both parts sepa- tot n p B rately. The kinetic part of Esym can be obtained ac- EB =aVA−aSA23 −aC(Z(Z−1))A−13 cording to Ref. [18]; −a I2A+δ(A,Z), (1) A 1 k2 Esym = F , (6) where I = (N −Z)/A, the fourth term accounts for the K 6 k2 +E2 total shifted energy due to the neutron number excess. F q,V(I=0) Taking the infinite nuclear matter limit of this formula, q where k is the Fermi momentum and E is the one notes that a reduces to the nuclear symmetry en- F q,V(I=0) A potential part of the quasi-nucleon self energy in sym- ergy [6]. metric nuclear matter at saturation density. To derive the formula for a that can be generalized A to the infinite nuclear matter, we start from a simple formula for the total energy: A. Linear density approximation E =NE +ZE tot n p 1 1 InQCDsumrulecalculations,wewillbeusingthelin- = A(1+I)E + A(1−I)E n p ear density approximation, because in medium conden- 2 2 1 1 satesinQCDsumrulecanbe mostreliablyestimatedto = A(En+Ep)+ AI(En−Ep), (2) leading order in density. This means that for either the 2 2 proton or the neutron, the mass will be given as follows: whereE (E )istheaverageneutron(proton)quasipar- n p ticle energy in the asymmetric nuclear matter. Now, the EVn(ρn,ρp)=m0+aρp+bρn coreofthe modeliswhatapproximationgoesintocalcu- 1 1 =m + ρ(a+b)+ ρI(b−a), lating the average energy. 0 2 2 Thesymmetryenergyinanasymmetricnuclearmatter 1 1 Ep(ρ ,ρ )=m + ρ(a+b)− ρI(b−a), (7) is defined as V n p 0 2 2 Etot(ρ,I)=E0(ρ)A+Esym(ρ)I2A+O(I4), (3) where m0 is the vacuum mass and a,b are the constants to be determined later. We then have, where ρ is the nuclear medium density and I = (N − Z)/A→(ρn−ρp)/(ρn+ρp)andthe neutronandproton EN = 1 d3k d3k EN(ρ ,ρ ) densitiesareρn = 12ρ(1+I), ρp = 21ρ(1−I)respectively. V d3knd3kp Z n p V n p Therefore,inEq.(2),thesymmetryenergywillhavecon- 1 1 tributions from the term proportionalto I in (E −E ) =Rm + aρ + bρ n p 0 p n 2 2 and the term proportional to I2 in (En+Ep). (cid:18) (cid:19) For a non interacting Fermi gas of nucleons with mass 1 1 = m + ρ(a+b)+ ρI(b−a) . (8) 0 m ,calculatingtheaveragenucleonenergywillgiveE = 4 4 N (cid:18) (cid:19) 3E , where E is the nucleon Fermi energy. Following 5 F F the procedure described above and extracting the term Combining Eq. (8) with Eq. (2), we obtain the sym- proportional to I2 gives a nuclear symmetry energy of metry energy. That is, (En −Ep) = 1(En(ρ ,ρ ) − V V 2 V n p 1E . Ep(ρ ,ρ )), hence, 3 F V n p Going back to finite nuclei, assuming a ‘Fermi well’ with constant energy difference ∆ between adjacent nu- Esym = 1 (En(ρ ,ρ )−Ep(ρ ,ρ )), (9) cleonenergylevel,the symmetryenergycanbe obtained V 4I V n p V n p 3 is similar to the relation given in Eq. (4). Therefore, III. QCD SUM RULE FOR NUCLEONS IN THE to this order, the symmetry energy comes only from ASYMMETRIC NUCLEAR MEDIUM the energy difference in the proton and neutron at the Fermi surface in an asymmetric nuclear matter as given A. Operator Product Expansion and Borel sum inEq.(9). However,foroperatorswithhigherdimension rule and higher density dependence, the factors appearing in Eq. (8) should be modified and symmetry energy will ToexpresstheselfenergiesintermsofQCDvariables, have contributions fromboth the sum andthe difference we start with analyzing the correlation function via the of the nucleon energies. The quantity of interest, namely, (En(ρ ,ρ ) − operatorproductexpansion(OPE).TheCorrelatorisde- Ep(ρ ,ρ )) can be obtained by looking atVthenpople of fined as V n p the nucleon propagator in nuclear medium: Π(q)≡i d4xeiqxhΨ |T[η(x)η¯(0)]|Ψ i, (18) 0 0 G(q)=−i d4xeiqxhΨ0|T[ψ(x)ψ¯(0)]|Ψ0i, (10) Z Z whereη(x)isaninterpolatingfieldofnucleonand|Ψ iis where|Ψ iisthenuclearmediumgroundstate,andψ(x) 0 0 the ground state of the asymmetric nuclear medium la- is a nucleon field. Relativistic mean field type of contri- beledwiththerestframemediumdensityρ,themedium bution willthen appear inthe self energies. The nucleon four-velocity uµ and the asymmetric factor I. This propagator can be decomposed as groundstateisassumedtobeinvariantunderparityand G(q)=G (q2,q·u)+G (q2,q·u)q/+G (q2,q·u)u/, timereversal. Wewillbe usingthe Ioffenucleoninterpo- s q u lating field given as [11, 19], (11) where uµ is the four-velocity of the nuclear medium η(x)=ǫ [uT(x)Cγ u (x)]γ γµd (x). (19) abc a µ b 5 c ground state [12]. The nucleon self energy can be decomposed similarly Asinthecaseofthenucleonpropagator,usingLorentz as [11–14]: covariance,parity and time reversal, one can decompose Σ(q)=Σ˜ (q2,q·u)+Σ˜µ(q)γ , (12) the correlator into three invariants [12]: s v µ where Π(q)≡Π (q2,q·u)+Π (q2,q·u)q/+Π (q2,q·u)u/. s q u Σ˜µ(q)=Σ (q2,q·u)uµ+Σ (q2,q·u)qµ. (13) (20) v u q In the mean field approximationΣ and Σ are real and s v Thethreeinvariantsarefunctionsofq2 andq·u,while momentum independent, and Σ is negligible. Hence, q vacuum invariants depends only on q2. For convenience, we set the rest frame for the nuclear medium, which Σv ≡ 1−ΣuΣq ∼Σu, MN∗ ≡ M1N−+ΣΣq˜s ∼MN +Σ˜s. mcoemanessauµfu→nct(i1o,n~0)o;faql0soonwlye,fiwxh|i~qch|.mΠeia(qn2s,Πqi·(uq)2,tqhe·nu)b→e- (14) Π (q ,|~q|→fixed) (i={s,q,u}). i 0 As mentioned before, we will follow the formalism The phenomenological representation of the nucleon adoptedinRef.[11]andwritetheenergydispersionrela- propagator can then be written as tion for these invariant functions at fixed three momen- 1 q/+M∗−u/Σ tum |~q|: G(q)= →λ2 v , (15) q/−M −Σ(q) (q −E )(q −E¯ ) n 0 q 0 0 1 ∞ ∆Π (ω,|~q|) i λ is unity in this discussion, but if one did not neglect Πi(q0,|~q|)= dω +polynomials, 2πi ω−q Σ , effects of Σ should be accounted in λ2 as the factor Z−∞ 0 q q (1 − Σ )−1. E and E¯ are the positive and negative (21) q q q energy pole: ∆Πi(ω,|~q|)≡ lim [Πi(ω+iǫ,|~q|)−Πi(ω−iǫ,|~q|)] ǫ→0+ =2Im[Π (ω,|~q|)]. (22) E =Σ + ~q2+M∗2, (16) i q v N q E¯ =Σ − ~q2+M∗2. (17) The lowest energy contribution to the discontinuity q v N q will be saturated by a quasi-nucleon and quasi-hole con- Withfixed|~q|,G(q)onlydependsonq . Onecanextract tribution in the positive and negative energy domain re- 0 self energy near ∼ E with analytic properties of the spectively. Theircontributiontothespectraldensitywill q nucleon propagator. begivenasinEq.(15),whichwillhavethefollowingcon- 4 tribution to the invariant functions: where we have defined invariants as follow for conve- nience M∗ Π (q ,|~q|)=−λ∗2 N +··· , (23) s 0 N (q −E )(q −E¯ ) 0 q 0 q 1 Π (q ,|~q|)=ΠE(q2,|~q|)+q ΠO(q2,|~q|). (27) Π (q ,|~q|)=−λ∗2 +··· , (24) i 0 i 0 0 i 0 q 0 N (q −E )(q −E¯ ) 0 q 0 q Σ Π (q ,|~q|)=+λ∗2 v +··· , (25) u 0 N (q −E )(q −E¯ ) The OPE of the three invariants of both the even and 0 q 0 q odd parts can be expressed as whereλ∗2 isthe residueatthe quasi-nucleonpole,which N accountsforthe coupling ofthe interpolatingfieldtothe quasi-nucleonexcitationstate,andtheomittedpartsare Π (q2,q2)= Ci(q2,q2)hOˆ i , (28) i 0 n 0 n ρ,I thecontributionsfromthehigherexcitationstates,which n willbeaccountedforthroughthecontinuumcontribution X after the Borel transformation. The even and odd parts of the invariant functions are where hOˆ i is the ground state expectation value of n ρ,I respectively related to the following parts of the discon- thephysicaloperatorintheasymmetricnuclearmedium; tinuity: hΨ |Oˆ |Ψ i . We will be adopting the OPE at q2 → 0 n 0 ρ,I −∞ at finite |~q| → fixed; this is equivalent to the limit 1 ∞ ω∆Π (ω,|~q|) ΠE(q2,|~q|)= dω i +polynomials, of q2 → −∞ at finite |~q| → fixed. The Wilson coeffi- i 0 2πiZ−∞ ω2−q02 cien0ts Cni(q2,q0) can thus be calculatedin QCD at short 1 ∞ ∆Π (ω,|~q|) time [11]. ΠO(q2,|~q|)= dω i +polynomials, i 0 2πi ω2−q2 Z−∞ 0 TheOPEofthecorrelatorfortheprotoninterpolating (26) field up to dimension 5 operators are given as follows: 1 4 q2 ΠE(q2,|~q|)= q2ln(−q2)hd¯di + 0hd¯{iD iD }di , (29) s 0 4π2 ρ,I 3π2q2 0 0 ρ,I 1 ΠO(q2,|~q|)=− ln(−q2)hd¯iD di , (30) s 0 2π2 0 ρ,I 1 ΠE(q2,|~q|)=− (q2)2ln(−q2) q 0 64π4 1 4 q2 4 4 q2 + ln(−q2)− 0 hd¯{γ iD }di + ln(−q2)− 0 hu¯{γ iD }ui 9π2 9π2q2 0 0 ρ,I 9π2 9π2q2 0 0 ρ,I (cid:18) (cid:19) (cid:18) (cid:19) 1 α 1 4q2 α − ln(−q2) sG2 − ln(−q2)− 0 s[(u·G)2+(u·G˜)2] , (31) 32π2 π ρ,I 144π2 q2 π ρ,I (cid:18) (cid:19) D E D E 1 2 q2 ΠO(q2,|~q|)= ln(−q2) hu†ui +hd†di − 0 hu¯{γ iD iD }ui q 0 6π2 ρ,I ρI 3π2(q2)2 0 0 0 ρ,I 2 q2 (cid:2) (cid:3) 2 1 1 1 − 0 hd¯{γ iD iD }di − hu¯{γ iD iD }ui + hg u†σ·Gui , (32) 3π2(q2)2 0 0 0 ρ,I 3π2q2 0 0 0 ρ,I 18π2q2 s ρ,I 1 3 q2 1 q2 ΠE(q2,|~q|)= q2ln(−q2) 7hu†ui +hd†di + 0hu¯{γ iD iD }ui + 0hd¯{γ iD iD }di u 0 12π2 ρ,I ρ,I π2q2 0 0 0 ρ,I π2q2 0 0 0 ρ,I 1 (cid:2) 1 (cid:3) − ln(−q2)hg u†σ·Gui + ln(−q2)hg d†σ·Gdi , (33) 6π2 s ρ,I 12π2 s ρ,I 4 16 ΠO(q2,|~q|)=− ln(−q2)hd¯{γ iD }di − ln(−q2)hu¯{γ iD }ui u 0 9π2 0 0 ρ,I 9π2 0 0 ρ,I 1 α + ln(−q2) s[(u·G)2+(u·G˜)2] . (34) 36π2 π ρ,I D E The quark part and their flavor structure of the above OPEcanbeobtainedbysuitablesubstitutionsofthecor- 5 respondingOPEfortheΣgiveninRef.[20];bychanging as phenomenological input. The weighting function will q →u, s →d, and neglecting terms proportional to m . de-emphasize the contribution from the quasi-hole, and s Moreover,whenbothuanddquarksareidentifiedtothe the Boreltransformationsuppress the continuumcontri- genericlightflavorq,ourOPEalsoreducestothatgiven bution. Using Eq. (26), the OPE side of the sum rule in Ref. [14]. can be obtained by taking the Borel transformation of Nexttaskisidentifyingthenucleonselfenergiesinthe Π (q ,|~q|)=ΠE(q2,|~q|)−E¯ ΠO(q2,|~q|). Here, we define i 0 i 0 q i 0 asymmetric nuclear medium. So we have to concentrate differentialoperatorBfortheBoreltransformationofthe on the quasi-nucleon pole and not on the quasi-hole nor OPE side as on the continuum excitations. To this end, we apply the Borel transformation with appropriate weighting func- (−q2)n+1 ∂ n B[f(q2,|~q|)]≡ lim 0 f(q2,|~q|) tion to the dispersion relation [12]: 0 −q02,n→∞ n! (cid:18)∂q02(cid:19) 0 −q2/n=M2 1 ω0 0 B[Π (q ,|~q|)]= dω W(ω)∆Π (ω,|~q|), (35) ≡ fˆ(M2,|~q|), (37) i 0 i 2πi Z−ω0 W(ω)= (ω−E¯ )e−ω2/M2, (36) where M is the Borel mass [21]. q The Borel transformed invariants which contain con- where E¯ is the quasi-hole pole which will be assigned tinuum corrections are as follows: q B¯[Πs(q02,|~q|)]= λ∗N2Mp∗e−(Eq2−q~2)/M2 1 4 = − 4π2(M2)2E1hd¯diρ,I − 3π2~q2hd¯{iD0iD0}diρ,IL−49 1 +E¯q − 2π2M2E0hd¯iD0diρ,IL−49 , (38) (cid:20) (cid:21) B¯[Πq(q02,|~q|)]= λ∗N2e−(Eq2−q~2)/M2 1 1 4 = 32π4(M2)3E2L−49 − 9π2M2E0− 9π2~q2 hd¯{γ0iD0}diρ,IL−49 (cid:18) (cid:19) 4 4 − 9π2M2E0− 9π2~q2 hu¯{γ0iD0}uiρ,IL−49 (cid:18) (cid:19) 1 α 1 α + 32π2M2 πsG2 ρ,IE0L−94 + 144π2 M2E0−4~q2 πs[(u·G)2+(u·G˜)2] ρ,IL−49 1 D E (cid:0) 2 (cid:1)D~q2 E +E¯q 6π2M2E0L−49 hu†uiρ,I +hd†diρ,I − 3π2 1− M2 hu¯{γ0iD0iD0}uiρ,IL−94 (cid:20) (cid:18) (cid:19) 2 ~q2(cid:2) (cid:3) − 3π2 1− M2 hd¯{γ0iD0iD0}diρ,IL−94 (cid:18) (cid:19) 2 1 − 3π2hu¯{γ0iD0iD0}uiρ,IL−49 + 18π2hgsu†σ·Guiρ,IL−49 , (39) (cid:21) B¯[Πu(q02,|~q|)]= λ∗N2Σpve−(Eq2−q~2)/M2 1 3 = 12π2(M2)2 7hu†uiρ,I +hd†diρ,I E1L−94 + π2~q2hu¯{γ0iD0iD0}uiρ,IL−49 1 (cid:2) (cid:3) 1 + π2~q2hd¯{γ0iD0iD0}diρ,IL−94 − 6π2M2hgsu†σ·Guiρ,IE0L−94 1 + 12π2M2hgsd†σ·Gdiρ,IE0L−94 4 16 +E¯q − 9π2M2hd¯{γ0iD0}diρ,IE0L−94 − 9π2M2hu¯{γ0iD0}uiρ,IE0L−49 (cid:20) 1 α − 36π2M2 πs[(u·G)2+(u·G˜)2] ρ,IE0L−49 . (40) (cid:21) D E 6 Here, we include the contributions from the anomalous- wheres∗ ≡ω2−~q2,andω istheenergyatthecontinuum 0 0 0 dimensions as threshold. As in many previous studies, we choose ω = 0 1.5 GeV. ln(M/Λ ) −2Γη+ΓOn L−2Γη+ΓOn ≡ QCD , (41) ln(µ/Λ ) (cid:20) QCD (cid:21) where Γ (Γ ) is the anomalousdimensionofthe inter- η On polating field η (Oˆ ), µ is the normalizationpoint of the n OPE, and Λ is the QCD scale [12, 14]. QCD Also,thecontinuumcorrectionsaretakenintoaccount through the factors E0 ≡1−es∗0/M2, (42) E1 ≡1−es∗0/M2 s∗0/M2+1 , (43) E2 ≡1−es∗0/M2(cid:0)s∗02/2M4+(cid:1)s∗0/M2+1 , (44) (cid:0) (cid:1) B. Condensates in the asymmetric nuclear medium To estimate the matrix elements, we will use the linear density approximation in the asymmetric nuclear matter: hOˆi = hOˆi +hn|Oˆ|niρ +hp|Oˆ|piρ ρ,I vac n p 1 1 = hOˆi + hn|Oˆ|ni+hp|Oˆ|pi ρ+ hn|Oˆ|ni−hp|Oˆ|pi Iρ. (45) vac 2 2 (cid:16) (cid:17) (cid:16) (cid:17) The quark flavor of condensate becomes important in the asymmetric nuclear medium. Consider an operator Oˆ u,d composed of either u or d quark respectively. Making use of the isospin symmetry relation, hn|Oˆ |ni=hp|Oˆ |pi, (46) u,d d,u we can convertthe neutron expectationvalue to the protonexpectation value, thereby rewriting Eq. (45) for the two quark operators as follows: hOˆ i =hOˆ i +(hp|Oˆ |pi∓hp|Oˆ |piI)ρ. (47) u,d ρ,I u,d vac 0 1 Here, ‘−’ and ‘+’ are for ‘u’ and ‘d’ quark flavor respectively, and the isospin operators are defined as 1 1 Oˆ ≡ (Oˆ +Oˆ ), Oˆ ≡ (Oˆ −Oˆ ). (48) 0 u d 1 u d 2 2 Hence, we will convert all the expectation values in terms of the proton counterparts and denote them as hp|Oˆ|pi→ hOˆi , throughout this paper. The next task is to find hOˆ i and hOˆ i for all operators appearing in our OPE. p 0 p 1 p 1. hq¯Dµ1···Dµnqi type of condensates Let us start by estimating the lowest dimensional operators h[q¯q] i and h[q¯q] i . To find h[q¯q] i , we will use an 0 p 1 p 1 p estimate based on using the QCD energy momentum tensor in the baryonoctet mass relationto leading order in the quark mass[22]; Eq. (A3) in Appendix A. Using Eq. (A4), one finds h[q¯q] i = 1 hp|u¯u|pi−hp|d¯d|pi = 1 (mΞ0 +mΞ−)−(mΣ+ +mΣ−) . (49) 1 p 2 2 2m −2m (cid:18) s q (cid:19) (cid:0) (cid:1) We will use the baryon masses as given in the Particle Data Group [23]: mΞ0 = 1315 MeV, mΞ− = 1321 MeV, mΣ+ =1190 MeV, mΣ− =1197 MeV. Using ms =150 MeV and mq ≡ 12(mu+md)= 5 MeV, Eq. (49) becomes 1 249 MeV h[q¯q] i = ∼0.43. (50) 1 p 2 300 MeV−2m (cid:18) q(cid:19) 7 For h[q¯q] i , we make use of the nucleon σ =45 MeV term: 0 p N 1 σ h[q¯q] i = hp|u¯u|pi+hp|d¯d|pi = N ∼4.5. (51) 0 p 2 2m q (cid:0) (cid:1) For convenience, one can introduce the parameter R (m ) defined as ± q hp|u¯u|pi±hp|d¯d|pi= R (m )·hp|u¯u|pi, (52) ± q which leads to R (m ) − q h[q¯q] i = ·h[q¯q] i . (53) 1 p 0 p R (m ) + q Using the previously selected values with the explicit quark mass dependence, we have σ 249 MeV σ 249 MeV N N R (m )≡ 1± − + , (54) ± q m 300 MeV−2m m 300 MeV−2m (cid:18) (cid:18) q q(cid:19)(cid:30)(cid:18) q q(cid:19)(cid:19) so that R (m =5 MeV))=1±0.68. ± q Using this parametrization, we can express the u quark or d quark condensates as follows, R (m ) − q h[q¯q] i = h[q¯q] i + 1∓ I ·h[q¯q] i ρ, (55) u,d ρ,I u,d vac 0 p R (m ) (cid:18) + q (cid:19) where [q¯q] =u¯u and [q¯q] =d¯d. For hq¯qi , we use the Gellmann-Oakes-Renner relation: u d vac 2m hq¯qi =−m2f2, (56) q vac π π where m =138 MeV and f =98 MeV [14]. For m =5 MeV, we have hq¯qi =−(263 MeV)3 π π q vac Likewise, we will further assume that the ratios between the isospin singlet and triplet operators remain the same for all two quark operator expectation values with any number of covariant derivatives inserted: R (m ) − q h[q¯D ···D q] i = ·h[q¯D ···D q] i . (57) µ1 µn 1 p R (m ) µ1 µn 0 p + q With this assumption, hq¯D ···D qi can be written as µ1 µn ρ,I R (m ) − q h[q¯D ···D q] i = h[q¯D ···D q] i + 1∓ I ·h[q¯D ···D q] i ρ. (58) µ1 µn u,d ρ,I µ1 µn u,d vac R (m ) µ1 µn 0 p (cid:18) + q (cid:19) The symmetric and traceless part of the above type of expectation values, constitute the moments of the twist-3 e (x,µ2) structure function defined as follows[36]: n h[q¯{D ···D }q] i ≡(−i)ne (µ2){p ···p }, (59) µ1 µn 0 p n µ1 µn e (µ2)≡ dx xne (x,µ2), (60) n n Z where {µ ···µ } means symmetric and traceless indices. Then the two quark twist-3 condensates in our sum rule 1 n can be written as follows: h[q¯iDµ′q]u,diρ,I = h[q¯iD0q]u,diρ,I ·u′µ = mqh[q†q]u,diρ,I =0, (61) 4 1 h[q¯{iDµ′iDν′}q]u,diρ,I = 3h[q¯{iD0iD0}q]u,diρ,I · u′µu′ν − 4gµν , (62) (cid:18) (cid:19) where the in-medium rest frame u′ ≡(1,~0) has been taken and the matrix element is estimated as µ 1 h[q¯{iDµ′iDν′}q]0ip = MN2e2(µ2) u′µu′ν − 4gµν , (63) (cid:18) (cid:19) 8 where one can identify that M2e (µ2)= 4h[q¯{iD iD }q] i and h[q¯{iD iD }q] i can be written as N 2 3 0 0 0 p 0 0 u,d ρ,I R (m ) h[q¯{iD iD }q] i ≃ 1∓ − q I ·M2e (µ2) ρ. (64) 0 0 u,d ρ,I R (m ) N 2 (cid:18) + q (cid:19) Sincetherearenomeasurementsonthetwist-3structurefunction,wewilltaketheestimateforM2e (µ2)∼0.3GeV2 N 2 given in [14, 26]. When spin indices are contracted, the operator becomes 1 1 R (m ) h[q¯D2q] i = h[g q¯σ·Gq] i = 1∓ − q I ·h[g q¯σ·Gq] i ρ, (65) u,d ρ,I s u,d ρ,I s 0 p 2 2 R (m ) (cid:18) + q (cid:19) where h[g q¯σ·Gq] i is chosen to be 3 GeV2 as in Ref. [14, 26]. s 0 p 2. hq¯γµ1Dµ2···Dµnqi type of condensates The simplest condensate of this type is hq¯γ qi =hq¯u/′qi u′ →hq†qi u′. (66) λ ρ,I ρ,I λ ρ,I λ For this, the ratio hu†ui /hd†di =2, and the iso-spin relation for hq†qi can be written as p p ρ,I 1 h[q†q] i = h[q†q] i , (67) 1 p 0 p 3 which leads to the following matrix elements appearing in the sum rule: 1 3 1 h[q†q] i = 1∓ I ·h[q†q] i ρ= ∓ I ρ. (68) u,d ρ,I 0 p 3 2 2 (cid:18) (cid:19) (cid:18) (cid:19) Whencovariantderivativesareincluded,onecanestimatethetwoquarktwist-2condensatesfromthecorresponding parton distribution function: (−i)n−1 hq¯{γ D ···D }qi ≡ Aq(µ2){p ···p }, (69) µ1 µ2 µn p 2M n µ1 µn N where Aq(µ2)=(Au(µ2)+Ad(µ2))/2 is the reduced matrix element [37, 38]: n n n 1 Aq(µ2)=2 dx xn−1[q(x,µ2)+(−1)nq¯(x,µ2)], (70) n Z0 where q(x,µ2) and q¯(x,µ2) are the distribution functions for quarks and antiquarks in the proton respectively. µ2 is the renormalization scale. For the distribution functions, we used the leading order (LO) parametrization given in Ref. [39]. Specifically, the spin-2 part can be written as 4 1 h[q¯{γµiDν}q]u,diρ,I → h[q¯{γµ′iDν′}q]u,diρ,I = 3h[q¯{γ0iD0}q]u,diρ,I · u′µu′ν − 4gµν , (71) (cid:18) (cid:19) where the in-medium rest frame has been taken. The matrix elements for each flavor in h[q¯{γ iD }q] i can be 0 0 u,d p identified as 1 1 hu¯{γµ′iDν′}uip = 2MNAu2(µ2)· u′µu′ν − 4gµν , (72) (cid:18) (cid:19) 1 1 hd¯{γµ′iDν′}dip = 2MNAd2(µ2)· u′µu′ν − 4gµν , (73) (cid:18) (cid:19) where Au(µ2)≃0.74 and Ad(µ2)≃0.36 at µ2 =0.25 GeV2 (LO) [39]. 2 2 9 One can introduce a ratio factor for hOˆ i as 1 p h[q¯{γµ′iDν′}q]1ip = RA2(µ2)·h[q¯{γµ′iDν′}q]0ip, (74) where R (µ2)=(Au−Ad)/(Au+Ad)≃0.35 so that h[q¯{γ iD }q] i can be written as A2 2 2 2 2 0 0 u,d ρ,I h[q¯{γ iD }q] i = 1∓R (µ2)I ·h[q¯{γ iD }q] i ρ 0 0 u,d ρ,I A2 0 0 0 p 1 = (cid:0)1∓R (µ2)I(cid:1)· M Aq(µ2)ρ. (75) A2 2 N 2 (cid:0) (cid:1) The spin-3 part can be written as 1 h[q¯{γλ′iDµ′iDν′}q]u,diρ,I =2h[q¯{γ0iD0iD0}q]u,diρ,I · u′λu′µu′ν − 6(u′λgµν +u′µgλν +u′νgλµ) , (76) (cid:18) (cid:19) where the matrix elements for each flavor in h[q¯{γ iD iD }q] i can be identified with 0 0 0 u,d p 1 1 hu¯{γλ′iDµ′iDν′}uip = 2MNAu3(µ2)· u′λu′µu′ν − 6(u′λgµν +u′µgλν +u′νgλµ) , (77) (cid:18) (cid:19) 1 1 hd¯{γλ′iDµ′iDν′}dip = 2MNAd3(µ2)· u′λu′µu′ν − 6(u′λgµν +u′µgλν +u′νgλµ) , (78) (cid:18) (cid:19) where Au(µ2)≃0.22 and Ad(µ2)≃0.07 atµ2 =0.25 GeV2 (LO) [39]. Similar to the spin-2 condensate case,one can 3 3 write hOˆ i for spin-3 condensate as 1 p h[q¯{γλ′iDµ′iDν′}q]1ip = RA3(µ2)·h[q¯{γλ′iDµ′iDν′}q]0ip, (79) where R (µ2)=(Au−Ad)/(Au+Ad)≃0.51 and h[q¯{γ iD iD }q] i can be written as A3 3 3 3 3 0 0 0 u,d ρ,I h[q¯{γ iD iD }q] i = 1∓R (µ2)I ·h[q¯{γ iD iD }q] i ρ 0 0 0 u,d ρ,I A3 0 0 0 0 p 1 = (cid:0)1∓R (µ2)I(cid:1)· M Aq(µ2)ρ. (80) A3 2 N 3 (cid:0) (cid:1) Operators with contracted spin indices are h[q¯D/q] i = 0, (81) u,d ρ,I 1 1 h[q†D2q] i = h[g q†σ·Gq] i ≃ (1∓R I)·h[g q†σ·Gq] i ρ, (82) u,d ρ,I 2 s u,d ρ,I 2 A3 s 0 p where h[g q†σ·Gq] i is chosen to be −0.33 GeV2 [14, 26]. s 0 p 3. Gluon condensates As for the gluon operators, because they do not carry quark flavors, the expectation values do not depend on I. These operators can be written as [13, 14] α α α sG2 = sG2 −2 s(E~2−B~2) ρ, (83) π ρ,I π vac π p Dα E D E D α E s[(u·G)2+(u·G˜)2] =− s(E~2+B~2) ρ, (84) π ρ,I π p D E D E where E~ and B~ are the color electric and color magnetic fields. For the expectation values of the gluon operators we take; h(α /π)G2i = (0.33 GeV)4 [27], h(α /π)(E~2 −B~2)i = 0.325±0.075 GeV and h(α /π)(E~2 +B~2)i = s vac s p s p 0.10±0.01 GeV [13]. 10 C. Dimension 6 four-quark operators In many previous QCD sum rule studies, dimension 6 four-quark condensates are assumed to have the factorized form as huau¯bucu¯di ≃huau¯bi hucu¯di −huau¯di hucu¯bi , (85) α β γ δ ρ,I α β ρ,I γ δ ρ,I α δ ρ,I γ β ρ,I huau¯bdcd¯di ≃huau¯bi hdcd¯di . (86) α β γ δ ρ,I α β ρ,I γ δ ρ,I WhilelargeN argumentscanbemadetojustifyfactorizationinthevacuum,nosuchargumentexistsinthemedium. c Using the following steps, one can classify the four-quark condensates in terms of the independent operators and of different twist. 1. Twist-4 operators with pure quark flavor The first type of four-quarkoperator appearing in the OPE of the nucleonsum rule, involves quark operatorswith the same flavor, and is of the color anti-triplet diquark times triplet anti diquark form. Using the following Fierz transformation,one can identify the independent four-quarkoperatorsin terms of products of quark-antiquarkpairs. 1 ǫabcǫa′b′c(uTaCγµub)(u¯Tb′γνCu¯a′)= ǫabcǫa′b′c 16 (u¯a′Γoua)(u¯b′Γkub)·Tr γµΓkγνCΓToC 1 (cid:2) (cid:3) = ǫabcǫa′b′c (u¯a′ua)(u¯b′ub)·(−4gµν) 16 (cid:26) +(u¯a′γ5ua)(u¯b′γ5ub)·(4gµν) +(u¯a′γαua)(u¯b′γβub)·(4Sµβνα) −(u¯a′γαγ5ua)(u¯b′γβγ5ub)·(4Sµβνα) 1 +(u¯a′σαα¯ua)(u¯b′σββ¯ub)· 4Tr γµσββ¯γνσαα¯ +(u¯a′γαua)(u¯b′γβγ5ub)·(8iǫµ(cid:2)ανα) (cid:3) −(u¯a′ua)(u¯b′σαα¯ub)·(8igαµgα¯ν) −(u¯a′γ5ua)(u¯b′σαα¯ub)·(4ǫµναα¯) , (87) (cid:27) where Γ={I,γ ,iγ γ ,σ ,γ } and S =g g +g g −g g . α α 5 αβ 5 µανβ µα νβ µβ αν µν αβ When quarks of the same flavor combine into a diquark, certain combinations are not allowed due to Fermi statis- tics. From these conditions, one can extract constraints among four-quark operators that can be used to identify independent operators. Among several conditions, the most suitable constraint for our OPE can be obtained from the zero identity used in Ref. [35]. With the constraint Eq. (B2) in Appendix B, Eq. (87) can be simplified as 1 ǫabcǫa′b′c(uTaCγµub)(u¯Tb′γνCu¯a′)= ǫabcǫa′b′c 16 (u¯a′γαua)(u¯b′γβub)−(u¯a′γαγ5ua)(u¯b′γβγ5ub) ·(8Sµανα) (cid:26) (cid:2) (cid:3) +(u¯a′γαua)(u¯b′γβγ5ub)·(16iǫµβνα) . (88) (cid:27) The last term in Eq. (88) will be dropped as one should take expectation value with respect to parity-even nuclear medium ground state. Then, only two types of four-quark operators remain. Each type can be written as 1 2 ǫabcǫa′b′c(q¯a′Γαqa)(q¯b′Γβqb)=ǫabcǫa′b′c 9δa′aδb′b(q¯Γαq)(q¯Γβq)+ 3tAaa′δb′b(q¯ΓαtAq)(q¯Γβq) (cid:26) 2 + 3δa′atBbb′(q¯Γαq)(q¯ΓβtBq)+4tAaa′tBbb′(q¯ΓαtAq)(q¯ΓβtBq) , (89) (cid:27) whereΓα ={γα,iγαγ }andtA isthegeneratorofSU(3)normalizedasTr[tAtB]= 1δAB. Combinedwiththeproduct 5 2 of epsilon tensors ǫabcǫa′b′c = δbb′δaa′ −δba′δab′, one finds that the second and third term in the R.H.S. of Eq. (89)