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arXiv:hep-ph/9307302v2 21 Dec 1994MSUCL-898December 19941

ρ, ω, φ-Nucleon Scattering Lengthsfrom QCD Sum RulesYuji Koike 1National Superconducting Cyclotron Laboratory, Michigan State UniversityEast Lansing, MI 48824-1321, USAAbstractThe QCD sum rule method is applied to derive a formula for the ρ, ω, φ meson-nucleonspin-isospin-averaged scattering lengths aρ,ω,φ. We found that the crucial matrix elementsare ⟨¯qγµDνq⟩N (q = u, d) (twist-2 nucleon matrix element) for aρ,ω and ms⟨¯ss⟩N for aφ, andobtained aρ = 0.14 ± 0.07 fm, aω = 0.11 ± 0.06 fm and aφ = 0.035 ± 0.020 fm.

Thesesmall numbers originate from a common factor 1/(mN +mρ,ω,φ). Our result suggests a slightincrease (< 60 MeV for ρ, ω, and < 15 MeV for φ) of the effective mass of these vectormesons in the nuclear matter (in the dilute nucleon gas approximation).

The origin of thediscrepancy with the previous study was clarified.PACS numbers: 13.75.-n, 12.38.Lg, 11.50.Li, 24.85.+p1Present address: Department of Physics, Niigata University, Ikarashi, Niigata 950-21, Japan2

The operator product expansion (OPE) provides us with a convenient tool to decompose avariety of correlation functions into the perturbatively calculable c-number coefficients andthe nonperturbative matrix elements. In its application to the QCD sum rules (QSR) [1,2], the OPE supplies an expression for the resonance parameters in terms of the vacuumcondensates representing the nonperturbative dynamics in the correlators.

In the applicationto the deep inelastic scattering (DIS) [3], the OPE isolates the quark-gluon distributionfunctions of the target from the short distance cross sections. In this paper, we investigatethe vector meson (ρ, ω, φ)-nucleon scattering lengths utilizing these two aspects of the OPE.These scattering lengths can be measured through the photo-production of these vectormesons.Furthermore, they determine the mass shift of the vector mesons in the dilutenuclear medium.

This will be discussed in the last part of this paper. A similar idea wasrecently presented for the nucleon-nucleon scattering length in ref.

[4].We start our discussion with the forward scattering amplitude of the vector current JVµ(V = ρ, ω, φ) offthe nucleon target with the four momentum p = (p0, p) and the polarizations:Tµν(ω, q) = iZd4x eiq·x⟨ps|TJVµ (x)JVν (0)|ps⟩,(1)where q = (ω, q) is the four-momentum carried by JVµ and the nucleon state is normalizedcovariantly as ⟨p|p′⟩= (2π)32p0δ(p −p′). We set p = (mN, 0) throughout this work andsuppress the explicit dependence on p and s. The vector current JVµ is defined as Jρ,ωµ (x) =(1/2)(¯uγµu(x) ∓¯dγµd(x)), Jφµ(x) = ¯sγµs(x).

Near the pole position of the vector meson, Tµνcan be associated with the T-matrix for the forward V −N helicity amplitude, ThH,h′H′(ω, q)(h(h′) and H(H′) are the helicities of the initial (final) vector meson and the initial (final)nucleon, respectively, and they take the values of h, h′ = ±1, 0 and H, H′ = ±1/2) asǫ(h)µ (q)Tµν(ω, q)ǫ(h′)∗ν(q) ≃−f 2V m4V(q2 −m2V + iη)2ThH,h′H′(ω, q),(2)where we introduced the coupling fV and the mass mV of the vector meson V by therelation ⟨0|JVµ |V (h)(q)⟩= fV m2V ǫ(h)µ (q) with the polarization vector ǫ(h)µ (q) normalized asPpol. ǫ(h)µ (q)ǫ(h)ν (q) = −gµν + qµqν/q2.

As is well known in DIS, Tµν can be decomposed intothe four scalar components respecting the current conservation and the invariance underparity and time-reversal. (Two of them correspond to the spin-averaged structure functionsW1 and W2, and the other two to the spin-dependent ones G1 and G2.) Correspondingly,there are 4 independent helicity amplitudes for the vector current-nucleon scattering; T1 12,1 12,T1 −12 ,1 −12 , T0 12,0 12, T1 −12 ,0 12, all the rest being obtained by time-reversal and parity from thesefour.

Since the information on G1 and G2 is still lacking, we shall focus on the combinationT = T1 + (1 −(p · q)2/m2Nq2)T2 (Im Ti ∼Wi, i = 1, 2), which projects the V −N spin-averaged T-matrix, T (ω, q). In the low energy limit (q →0), T is reduced to the V −Nspin-averaged scattering length aV = (1/3)(a1/2 + 2a3/2) (a1/2 and a3/2 are the scatteringlengths in the spin-1/2 and 3/2 channels, respectively) as T (mV , 0) = 24π(mN + mV )aV [5].A useful quantity for the dispersion analysis is the retarded correlation function defined asT Rµν(ω, q) = iZd4x eiq·xθ(x0)⟨ps|hJVµ (x), JVν (0)i|ps⟩= 1πZ ∞−∞du Im T Rµν(u, q)u −ω −iη ,(3)3

which is analytic in the upper half ω-plane with a fixed q. Noting the crossing symmetry,the V −N scattering contribution to the spin-averaged spectral function at q = 0 can bewritten as1πImhT R(ω, 0)i= θ(ω) 1πIm [T(ω, 0)] −θ(−ω) 1πIm [T(ω, 0)]= −24πf 2V m4V (mN + mV )aVhθ(ω)δ′(ω2 −m2V ) −θ(−ω)δ′(ω2 −m2V )i+ (S. P.), (4)where δ′(x) is the first derivative of the δ-function (double pole term) and (S. P.) denotes thesimple pole term representing the off-shell energy dependence of the T-matrix.

Equation (4)can also be derived starting from the spectral representation. Using this form of the spectralfunction in eq.

(3), and noting that the retarded correlation function T Rµν becomes identicalto the causal correlation function Tµν in the deep Euclidean region ω2 = −Q2 →−∞, onegetsT(ω2 = −Q2) = −24πf 2V m4V (mN + mV )aV1(m2V + Q2)2 + Ros(Q2) + Rc(Q2),(5)where we have used the fact that T becomes a function of ω2 in this limit. In eq.

(5), weassumed the spectral function is saturated by the V −N scattering (with its off-shell effect)and the ”continuum” contribution [6]: Ros(Q2) denotes the simple pole term correspondingto the off-shell part of the V −N T-matrix (∼1/(m2V + Q2)) and Rc(Q2) stands for the”continuum” contribution with its threshold S′0 (∼1/(S′0 +Q2)) [7]. The sum of the residuesof Ros(Q2) and Rc(Q2) is constrained by the 1/Q2-term in the OPE side of the correlator.

(See eq. (6) below.

)We now proceed to the OPE side of T(ω2 = −Q2) (l.h.s. of eq.

(5)). Unlike in DIS, ourOPE is the short distance expansion and hence all the operators with the same dimensioncontribute in the same order with respect to 1/Q2 (= −1/ω2 at q = 0).

The complete OPEexpression for T(Q2) including the operators up to dimension=6 is given in ref. [8] in thecontext of the finite temperature QSR.

For the ρ and ω mesons, it reads from eq. (2.13) of[8] as (−for ρ and + for ω)T ρ,ω(Q2)=14"−2mqQ2 ⟨¯uu + ¯dd⟩N −16Q2⟨αsπ G2⟩N + 2παsQ4⟨Q∓5 + 29Q+⟩N#−m2N2Q2Au+d2+ 5m4N6Q4 Au+d4−m2N2Q4B1∓+ 14B2 + 58B3,(6)where ⟨·⟩N denotes the spin-averaged nucleon matrix element, and Q∓5 and Q+ are thescalar four-quark operators familiar in the QSR for the ρ and ω mesons; Q∓5 = (¯uγµγ5λau ∓¯dγµγ5λad)2, Q+ =¯uγµλau + ¯dγµλad Pu,d,sq¯qγµλaq.In eq.

(6), Au+dn≡Aun + Adn (n =2, 4) are related to the twist-2 operators and are given as the n-th moment of the partondistribution function (q = u, d, s); ⟨ST (¯qγµ1Dµ2 · · · Dµnq(µ))⟩N = (−i)n−1Aqn(µ)(pµ1 · · ·pµn −traces), Aqn(µ) = 2R 10 dx xn−1 (q(x, µ) + (−1)n¯q(x, µ)) with the renormalization scaleµ.Bi (i = 1∓, 2, 3) are associated with the twist-4 matrix elements as ⟨Oiµν(µ)⟩N =(pµpν −m2Ngµν/4)Bi(µ) with O1∓µν = (g2/4)ST(¯uγµγ5λau ∓¯dγµγ5λad)(µ →ν), O2µν =4

(g2/4)ST(¯uγµλau + ¯dγµλad)Pu,d,sq¯qγνλaq, O3µν = igST¯u{Dµ,∗Gνλ}γλγ5u + (u →d),where the color matrix λa is normalized as Tr(λaλb) = 2δab and the symbol ST makes theoperators symmetric and traceless with respect to the Lorentz indices.To get an expression for the V −N scattering length, we first make a Borel transformof eqs. (5) and (6) with respect to Q2, and then eliminate the unknown parameter whichdetermines the ratio between the coefficient of Ros(Q2) and Rc(Q2) , using the sum ruleobtained by taking the derivative with respect to the Borel mass M2.

We also eliminate theunknown coupling constant fV by taking the ratio between the obtained sum rule and theQSR expression for the vector current correlators in the vacuum. We thus get an expressionfor the spin-averaged scattering length aV asaρ,ω =πM23m2ρ,ω(mN + mρ,ω)rβ/(αM2) + tγ/(αM4)(1 + αsπ )(1 −e−S0/M2) + b/M4 −c/M6,(7)withr=mNΣπN −227m20 + m2N2 Au+d2,t=−112παs81⟨¯uu⟩⟨¯uu⟩N + ⟨¯dd⟩⟨¯dd⟩N−56m4NAu+d4+ m2N2B1∓+ 14B2 + 58B3,b=4π2mq⟨¯uu + ¯dd⟩+ π23 ⟨αsπ G2⟩,c=448π3αs81⟨¯uu⟩2,where ⟨·⟩denotes the vacuum condensate and S0 is the continuum threshold in the vacuumsum rule.

The factors α, β and γ appeared through the process of eliminating the parameterwhich determines the ratio between the residues of Ros(Q2) and Rc(Q2), and they are definedas α = 1 −e−(S′0−m2)/M2(1 + S′0−m2M2 ), β =m2M2 + S′0−m2M2 e−S′0/M2 −S′0M2e−(S′0−m2)/M2, γ = 1 +m2M2 −(1 + S′0M2)e−(S′0−m2)/M2 with m = mρ,ω. If we ignore Rc(Q2) from the beginning, thecorresponding formula is obtained by the replacement; α →1, β →1 −e−m2/M2, γ →1.In eq.

(7), we have used the following relations for the matrix elements as has been used inthe study of QCD sum rules in the nuclear matter [9, 10]: (i) πN σ-term ΣπN is introducedthrough the relation mq⟨¯uu+ ¯dd⟩N = 2mNΣπN. (ii) The nucleon mass in the chiral limit, m0,is introduced in favor of ⟨αsπ G2⟩N through the QCD trace anomaly: ⟨αsπ G2⟩N = −(16/9)m20.

(iii) Factorization is assumed for the vacuum four-quark condensates, ⟨Q∓5 ⟩and ⟨Q+⟩, as isusually adopted in the literature [1, 2]. (iv) Factorization is also employed to estimate thenucleon matrix elements of the scalar four-quark operators ⟨Q∓5 ⟩N and ⟨Q+⟩N after makingthe Fierz transform [10], i.e.

⟨(¯qΓλq)2⟩N →⟨(¯qq)2⟩N ≃2⟨¯qq⟩⟨¯qq⟩N.By repeating the same steps as above for Jφµ, one gets the spin-averaged φ−N scatteringlength asaφ =πM23m2φ(mN + mφ)rsβ/(αM2) + tsγ/(αM4)(1 + αsπ )(1 −e−S0/M2) −6m2s/M2 + bs/M4 −cs/M6,(8)5

withrs=ms⟨¯ss⟩N −227m20 + m2NAs2,ts=−224παs81⟨¯ss⟩⟨¯ss⟩N −53m4NAs4 + m2NBs1 + 14Bs2 + 58Bs3,bs=8π2ms⟨¯ss⟩+ π23 ⟨αsπ G2⟩,cs=448π3αs81⟨¯ss⟩2,where the strange twist-4 matrix elements Bsi (i = 1 −3) are defined similarly to the case ofthe ρ and ω mesons. For the vacuum condensates and the quark masses in eqs.

(7) and (8),we use the standard values at the renormalization scale µ = 1 GeV [2]: αs = 0.36, mq = 7MeV, ms = 110 MeV, ⟨¯uu⟩= ⟨¯dd⟩= (−0.28 GeV)3, ⟨¯ss⟩= 0.8⟨¯uu⟩. With these vacuumcondensates and the continuum threshold S0 = 1.75 GeV2 for ρ, ω and S0 = 2.0 GeV2 forφ, the experimental values for mρ,ω,φ are well reproduced.

We thus fixed S0 at these valuesand use mρ,ω = 770 MeV and mφ = 1020 MeV in eqs. (7) and (8).

As a measure of thestrangeness content of the nucleon, we introduce the parameter y = 2⟨¯ss⟩N/(⟨¯uu⟩N +⟨¯dd⟩N)and write ⟨¯ss⟩N = ymNΣπN/mq. For the nucleon matrix elements we use ΣπN = 45 ± 7MeV, y = 0.2 and m0 = 830 MeV obtained by the chiral perturbation theory [11].

Sincewe ignored the twist-2 gluon operators in eqs. (7) and (8), we consistently use the leadingorder (LO) parton distribution functions of Gl¨uck, Reya and Vogt [12] to determine Au+diand Asi (i = 2, 4).

It gives Au+d2= 0.90, Au+d4= 0.12, As2 = 0.05 and As4 = 0.002 at µ2 = 1GeV2. For the twist-4 matrix elements Bi and Bsi , we use our recent result [13] extractedfrom the newest DIS data at CERN and SLAC.

It is based on the SU(2) flavor symmetry(i.e. Bsi = 0 (i = 1 −3)) and a mild assumption on the matrix elements invoked by theflavor structure of the twist-4 operators.Both for the proton and the neutron, it givesB1∓+ B2/4 + 5B3/8 = −0.24 ± 0.15 (−0.41 ± 0.23) GeV2 for the ρ (ω) meson at µ2 = 5GeV2 [14].

[Note that our ρ0 −N (N can be either proton or neutron) scattering lengthcorrespond to the isospin-spin averaged one. ]Using these numbers for the matrix elements, the Borel curves for the ρ, ω, φ-nucleonscattering lengths aρ,ω,φ (eqs.

(7) and (8)) are shown in Fig. 1.

We determined the values ofS′0 in order to minimize the slope of the curves at 0.8 < M2 < 1.3 GeV2. They are 3.32GeV2 for ρ, 3.29 GeV2 for ω and 4.40 GeV2 for φ.

With the above parameters, r in eq. (7)reads r = 0.04 −0.05 + 0.40 = 0.39 GeV2 from the first to the third terms.

Thus r is totallydominated by the twist-2 nucleon matrix element Au+d2and the cancelling contribution fromthe first and the second terms makes the ambiguity in ΣπN and m0 less important. Thet-term in eq.

(7) reads t = 0.42 −0.08 −0.11 ± 0.07 (−0.18 ± 0.10) = 0.23 ± 0.07 (0.16 ± 0.10)GeV4 for ρ (ω), which shows the contribution from the twist-4 matrix elements is sizable.To get an insight on the sensitivity of the results to the variation of t, we also showedaρ,ω without the twist-4 matrix elements in t with S′0 = 3.35 GeV2.One sees that theinclusion of Bi reduces the aρ,ω by about 20 % (30 %) for ρ (ω). With the uncertaintyin Bi in mind, we assign error bars as aρ = 0.14 ± 0.07 fm and aω = 0.11 ± 0.06 fm,taking the values for aρ,ω,φ around M2 = 1 GeV2.

For the case of aφ, the value of ms⟨¯ss⟩Ngoverns the whole result because of large ms, i.e. rs = 0.13 −0.05 + 0.04 = 0.12 GeV2 and6

ts = 0.066 −0.003 + (twist −4 ≡0) = 0.063 GeV4 from the first to the third terms in rs andts. Due to the uncertainty in ms⟨¯ss⟩N, we read from Fig.

1 aφ = 0.035 ± 0.020 fm. Somephenomenological analyses on the nucleon form factor [15] and the nuclear force [16] suggestquite a large OZI violating φNN coupling constant gφNN/gωNN ∼0.4.

Equation (8) suppliesa neat expression for the φN →φN interaction strength in terms of the strangeness contentof the nucleon, showing the importance of ms⟨¯ss⟩N rather than ⟨¯sγµDνs⟩N.If we calculate the scattering lengths without Rc(Q2) in eq. (5), we get even smallernumbers for the scattering lengths: aρ ∼0.1 fm, aω ∼0.08 fm and aφ ∼0.01 fm aroundM = 1 GeV.

This way, the actual numbers for aρ,ω,φ depends on the assumption made inthe spectral function, although their typical order of magnitude does not change.One might be surprised by the smallness of these scattering lengths compared with atypical hadronic size (∼1 fm). From eqs.

(7) and (8), one sees aV ∼1/(mN + mV ), sincer and rs are dominated by the third and the first terms, respectively. If one applies thepresent method to the axial vector correlator, one can easily get the pion-nucleon scatteringlength in the isospin symmetric channel as aπN ∝mNΣπN/(f 2π(mN + mπ)), which is thesame result as that of the current algebra [17].

[In the chiral limit, aπN = 0, since ΣπN = 0. ]Therefore it is interesting to observe that our method of deriving the vector meson-nucleonscattering length is a generalization of the current algebra technique for the pion-nucleonscattering length.

For the vector meson case, the common factor 1/(mN + mV ) makes aVsmall. We believe this smallness of the V −N scattering lengths somehow sketches the realsituation, although the actual numbers for aV are not trustable because of the simplifiedform for the spectral function in our calculation as was noted before.

A model calculation ofthe ρ −N scattering amplitude based on an effective hadronic lagrangian suggests a similarsmall number for aρ [18].Let us finally discuss the mass shift of the vector mesons in the nuclear medium using theresult for the scattering lengths here. In the dilute nucleon gas approximation, the V -currentcorrelator in the nuclear medium can be written asΠN.M.µν(ω, q) = iZd4x eiq·x⟨TJVµ (x)JVν (0)⟩+Xpol.Z pfd3p(2π)32p0Tµν(ω, q).

(9)By ignoring the Fermi motion of the nucleon, ΠN.M.µν(ω, q = 0) can be approximated near thepole position asΠN.M.µν(ω, 0)≃f 2V m4Vgµν −qµqνω2 1 + O(ρN)ω2 −m2V+ ∆m2V1(ω2 −m2V )2!∼1 + O(ρN)ω2 −m2V −∆m2V+ O(ρ2N),(10)where ∆m2V = 12πaV ρN(mN + mV )/mN with the nucleon density ρN. From this relation,∆m2V can be viewed as a shift of m2V in the nuclear medium [19].

Our values for aρ,ω,φ suggestthat the effective mass for the vector mesons increases by about 27 −57 MeV for ρ, 20 −48MeV for ω and 5 −13 MeV for φ at the nuclear matter density ρN = 0.17 fm−3 [20]. [Notethat the validity of the mass shift discussed here hinges on the assumption that the off-shellenergy dependence and the momentum dependence of the V −N scattering amplitude isweak within the range of the nucleon’s Fermi momentum.

]7

The authors of ref. [10] applied the QSR method to study mass shifts of the ρ, ω andφ mesons in the nuclear medium.

Although their approximation in the OPE side of thecorrelation functions is essentially the same as ours, eq. (9), they found a serious decreaseof these vector meson masses in the nuclear matter.Here we clarify the origin of thisdiscrepancy.

In the recent literature of the QSR method in the nuclear medium for baryonsand mesons [9, 10], the common starting point is that the density dependence of correlationfunctions is ascribed to the density dependent condensates:ΠN.M. (q, ρN) =XiCi(q, µ)⟨Oi(µ)⟩ρN,(11)where Ci(q, µ) and Oi(µ) are the Wilson coefficient and a local operator, respectively, andwe suppressed all the spinor and Lorentz indices.

In the dilute nuclear medium, ⟨Oi(µ)⟩ρNhas been approximated as⟨Oi(µ)⟩ρN=⟨Oi(µ)⟩+Xpol.Z pfd3p(2π)32p0⟨ps|Oi(µ)|ps⟩=⟨Oi(µ)⟩+ ρN2mN⟨Oi(µ)⟩N + o(ρN). (12)As is easily seen by inserting eq.

(12) into eq. (11), the approximation for the condensateeq.

(11) is equivalent to the approximation eq. (9) for the correlator.

Therefore the densitydependent part of the correlator has to be analyzed from the point of view of the forwardcurrent-nucleon scattering amplitude as was done in this paper. In this approximation, whatis relevant for the mass shift is the double pole structure at the pole position appearing inthe forward amplitude.To understand the difference between our result and the one in [10], it is convenientto recall the QSR for the vector meson in the vacuum.

In the vacuum, the vector currentcorrelator can be written as Πµν(q) = (qµqν −gµνq2)Π(q2). As long as one uses a spectralfunction with a single narrow resonance and the continuum, both Π(q2) and q2Π(q2) can beused for the QSR analysis.

The formula for the vector meson mass in the Borel sum ruleobtained by using these two sum rules are, respectivelym2VM2=(1 + αsπ )[1 −(1 + S0/M2)e−S0/M2] −D4/M4 + D6/M6(1 + αsπ )[1 −e−S0/M2] + D4/M4 −D6/2M6,(13)m2VM2=2(1 + αsπ )[1 −(1 + S0/M2 + S20/2M4)e−S0/M2] −D6/M6(1 + αsπ )[1 −(1 + S0/M2)e−S0/M2] −D4/M4 + D6/M6 ,(14)where D4 and D6 are the sums of the dim.=4 and dim.=6 condensates, respectively. Bothsum rules give right amount for mV .

However, one should note that the sign of the powercorrection due to the condensates is opposite between the two sum rules. Thus the shift ofthe condensates due to the medium effects (the second term of eq.

(9) or eq. (12)) is expectedto cause opposite physical effects depending on which sum rule one uses.

The authors of[10] analyzed ΠN.M. (ω, 0) = ΠN.M.µµ(ω, 0)/(−3ω2), with a simple pole ansatz for the vectormeson (together with a scattering term) in the spectral function, which picks up the same8

effect of the shift of the condensate as eq. (13).

On the other hand, if we recognize that thesecond term in eq. (9) is associated with the V −N forward amplitude through eq.

(2), we caneasily see that it is ω2ΠN.M. (ω, 0) which has to be analyzed with a simple pole ansatz in theorder O(ρN) as is shown in eq.

(10). In this case, the vector meson mass receives the effectof the change of the condensate as is expected from the formula eq.

(14). In ΠN.M.

(ω, 0),the density dependent part appears as a form of (ρN/2mN)T(ω, 0)/ω2, which brings a factor−∆m2V (m2V /Q2) (Q2 = −ω2 > 0) instead of ∆m2V in the first line of eq. (10).

In this case,due to the additional factor 1/Q2, the double pole term can not be incorporated into themass shift. Thus the use of eq.

(13) in the nuclear medium is simply wrong! Therefore aninadequate form of the spectral function in [10] led to a fictitious “negative” mass shift.Since the second term in eq.

(9) has an unique relation with the V −N T-matrix around thepole position as is shown in eq. (2), we believe the mass shift of the vector mesons in thenuclear medium in the context of the QSR should be understood as presented in this work.Finally, we wish to make a comment on the speculation on the in-medium behaviorof the hadron masses existing in the literature.

From the finite energy sum rule analysis,one gets the ρ meson mass as mρ ∝|⟨¯ψψ⟩|1/3 in the vacuum. Since |⟨¯ψψ⟩| decreases inthe nuclear medium according to the formula eq.

(12), one might naively expect mρ wouldalso decrease [21]. We have illustrated, however, that a consistent organization of the QCDsum rule does not predict such a behavior.It would rather support (within our crudeapproximation) another naive expectation that a tendency of ρ−A1 degeneracy might occurin the nuclear medium, since the application of our method to the A1 meson gives decreasingmA1.

We also remind the readers that (1) |⟨¯ψψ⟩T| decreases as the temperature (T) goes up,while all hadron masses stay constant in the order O(T 2) [22], (2) a consistent organizationof the sum rule at finite temperature certainly gives the same behavior [19] unlike the abovenaive expectation, and (3) this is because the pion-hadron scattering length is zero in thechiral limit.In conclusion, we have derived the ρ, ω, φ -nucleon spin-isospin averaged scatteringlengths aρ,ω,φ from QCD sum rules. We obtained very small positive numbers (correspondingto the repulsive interaction) for aρ,ω,φ, aρ,ω,φ ∼1/(mN +mρ,ω,φ), although the actual numbersdepend on the factorization assumption for the nucleon matrix element of the scalar four-quark operator as well as the simplified form for the spectral function.

This result suggestsslight increase of these vector mesons in the nuclear medium, which is contradictory to theprevious result by Hatsuda and Lee [10]. We have clarified the origin of this discrepancyand pointed out the problem of the analysis in ref.

[10]. The detail of the calculation will bepublished elsewhere.AcknowledgementI would like to thank T. Hatsuda, S. H. Lee, P. Danielewicz, F. Lenz, O. Morimatsu, E. V. Shuryakand K. Yazaki for many valuable discussions and comments.

This work is supported in partby the US National Science Foundation under grant PHY-9017077.9

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Figure CaptionsFig. 1 The Borel curves for the ρ, ω, φ-nucleon scattering lengths.

The dashed line denotesthe one for ρ and ω without the twist-4 matrix elements in eq. (7).12

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