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Valid QCD Sum Rules for Vector Mesons in Nuclear Matter


Submitted to Phys. Rev. C

TRI-PP-95-40

Valid QCD Sum Rules for Vector Mesons in Nuclear Matter
arXiv:nucl-th/9510064v1 31 Oct 1995
Xuemin Jin? and Derek B. Leinweber?
TRIUMF, 4004 Wesbrook Mall, Vancouver, B. C., V6T 2A3, Canada

Abstract
QCD sum rules for vector mesons (ρ, ω, φ) in nuclear matter are reexamined with an emphasis on the reliability of various sum rules. Monitoring the continuum contribution and the convergence of the operator product expansion plays crucial role in determining the validity of a sum rule. The uncertainties arising from less than precise knowledge of the condensate values and other input parameters are analyzed via a Monte-Carlo error analysis. Our analysis leaves no doubt that vector-meson masses decrease with increasing density. This resolves the current debate over the behavior of the vector-meson masses and the sum rules to be used in extracting vector meson properties in nuclear matter. We ?nd a ratio of ρ-meson masses of mρ? /mρ = 0.78± 0.08 at nuclear matter saturation density.

? E-mail: ? Present

jin@alph02.triumf.ca ? Telephone: (604) 222-1047 ext. 6446 ? Fax: (604) 222-1074

address: Dept. of Physics, Box 351560, University of Washington, Seattle, WA 98195. E-mail: derek@phys.washington.edu ? Telephone: (206) 616–1447 ? Fax: (206) 685–0635 WWW: http://www.phys.washington.edu/?derek/Welcome.html

1

I. INTRODUCTION

Whether the properties of vector mesons might change signi?cantly with increasing nuclear matter density is of considerable current theoretical interest. This interest is motivated by its relevance to the physics of hot and dense matter and the phase transition of matter from a hadronic phase to a quark-gluon plasma at high density and/or temperature. In particular, the modi?cations of vector-meson masses in nuclear matter have been studied extensively. At least three experimentally based studies have been cited as supporting the picture of decreasing vector-meson masses in nuclear matter. These include the quenching of the longitudinal response (relative to the transverse response) in quasi-elastic electron scattering [1], (e, e′ p) reactions [2], and the discrepancy between the total cross section in K + -nucleus scattering on 12 C and that predicted from an impulse approximation calculation using K + nucleon scattering amplitudes (extracted from K + -D elastic scattering) [3–7]. More direct investigations of vector-meson masses in the nuclear medium have also been proposed. One proposal is to study dileptons as a probe of vector mesons in the dense and hot matter formed during heavy-ion collisions [8]. The dilepton mass spectra should allow one to reconstruct the masses of vector mesons decaying electromagneticly. Theoretical investigations of vector-meson masses in nuclear matter have used various approaches and models that include the scaling ansatz of Brown and Rho [9], Nambu– Jona-Lasinio model [10], Walecka model [11–17], quark model [18], and the QCD sum-rule approach [19–22]. Previous studies of vector mesons at ?nite density via the QCD sum-rule approach have been made by Hatsuda and Lee [19], and subsequently by Asakawa and Ko [20,21]. It was found that the vector-meson masses decrease with increasing density. This ?nding is consistent with the scaling ansatz proposed in Ref. [9], the quark model [18] and predictions obtained from the Walecka model provided the polarizations of the Dirac sea are included [13–17]. However, more recently, another QCD sum-rule analysis has shed considerable doubt on the former conclusions. Ref. [22] claims that the previous QCD sum-rule analyses of the in-medium vector-meson masses are incorrect and that the vector-meson masses should in fact increase in the medium. In response, Hatsuda, Lee and Shiomi [23] have argued that the scattering-length approach used in Ref. [22] is conceptually erroneous. In this paper, we reexamine the QCD sum rules for vector mesons in nuclear matter. Our focus is on the reliability and validity of various sum rules. We will show that careful consideration of the validity of the sum rules used to extract the phenomenological results is crucial to resolving this debate. Taking the three momentum to be zero in the rest frame of the nuclear medium, one can only obtain one (direct) sum rule in each vector channel. By taking the derivative of this sum rule with respect to the inverse Borel mass, one may get an in?nite series of derivative sum rules. Hatsuda and Lee [19] used the ratio of the ?rst derivative sum rule and the direct sum rule in their analysis while Koike [22] argued that one should use the 2

ratio of the second and ?rst derivative sum rules. We point out that in practical sum-rule applications, the derivative sum rules are much less reliable than the direct sum rule and eventually become useless as the number of derivatives taken increases. The ratio method used by both previous authors does not reveal the validity of each individual sum rule, and hence can lead to erroneous results. QCD sum rules relate the phenomenological spectral parameters (masses, residues, etc.) to the fundamental properties of QCD. To maintain the predictive power of the sum-rule approach, the phenomenological side of the sum rule is typically described by the vector meson pole of interest plus a model accounting for the contributions of all excited states. By working in a region where the pole dominates the phenomenological side, one can minimize sensitivity to the model and have assurance that it is the spectral parameters of the ground state of interest that are being determined by matching the sum rules. In practice, these considerations e?ectively set an upper limit in the Borel parameter space, beyond which the model for excited states dominates the phenomenological side. At the same time, the truncated OPE must be su?ciently convergent1 as to accurately describe the true OPE. Since the OPE is an expansion in the inverse squared Borel mass, this consideration sets a lower limit in Borel parameter space, beyond which higher order terms not present in the truncated OPE are signi?cant and important. Monitoring OPE convergence is absolutely crucial to recovering nonperturbative phenomena in the sum-rule approach, as it is the lower end of the Borel region where the nonperturbative information of the OPE is most signi?cant. This information must also be accurate and this point will be further illustrated in Sec. III. In short, one should not expect to extract information on the ground state spectral properties unless the ground state dominates the contributions on the phenomenological side and the OPE is su?ciently convergent. In this paper, we will analyze each individual sum rule with regard to the above criteria. A sum rule with an upper limit in Borel space lower than the lower limit is considered invalid. As a measure of the relative reliability of various sum rules we consider the size of the regime in Borel space where both sides of the sum rules are valid. In addition, the size of continuum contributions throughout the Borel region can also serve as a measure of reliability, with small continuum model contributions being more reliable. The uncertainties in the OPE are not uniform throughout the Borel regime. These uncertainties arise from an imprecise knowledge of condensate values and other parameters appearing in the OPE. As such, uncertainties in the OPE are larger at the lower end of the Borel region. To estimate these uncertainties we adopt the Monte-Carlo error analysis

and in the following, “convergence” of the OPE simply means that the highest dimension terms considered in the OPE, with their Wilson coe?cients calculated to leading order in perturbation theory, are small relative to the leading terms of the OPE.

1 Here

3

approach recently developed in Ref. [24]. In turn, these uncertainties provide error estimates for the extracted phenomenological spectral properties. This is the ?rst systematic study of uncertainties for in-medium hadronic properties. In the following we will show in detail how the direct sum rule is valid and the most reliable. The ?rst derivative sum rule su?ers from a small Borel region of validity and relatively large continuum model contributions throughout. It is marginal at best and any predictions from this sum rule are unreliable. Higher derivative sum rules are found to be invalid. Both the direct sum rule and the ?rst derivative sum rule lead to the same conclusion that vector-meson masses decrease as the nuclear matter density increases. This paper is organized as follows: In Sec. II, we sketch the ?nite-density sum rules for vector mesons and discuss the reliabilities of various sum rules. In Sec. III, the sum rules are analyzed and the sum-rule predictions are presented and discussed. Sec. IV is devoted to a conclusion.
II. FINITE-DENSITY SUM RULES

In this section, we brie?y review the QCD sum rules for vector mesons in nuclear matter. We focus on some issues raised in these sum rules and refer the reader to the literature [19–21] for more details of the sum rules. QCD sum rules for vector mesons at ?nite-density study the correlator de?ned by Π?ν (q) ≡ i d4 x eiq·x Ψ0 |T J? (x)Jν (0)|Ψ0 , (2.1)

where |Ψ0 is the ground state of nuclear matter, T is the covariant time-ordering operator [25], and J? represents any of the three conserved vector currents of QCD:
ρ J? ≡ 1 (uγ? u ? dγ? d) , 2 ω J? ≡ 1 (uγ? u + dγ? d) , 2 φ J? ≡ sγ? s .

(2.2)

The nuclear medium is characterized by the rest-frame nucleon density ρN and the fourvelocity u? . We assume that the medium is invariant under parity and time reversal. Lorentz covariance and the conservation of the currents imply that the correlator Π?ν (q) can be decomposed into two independent structures multiplying two invariants, corresponding to the transverse and longitudinal polarizations. The medium modi?cations of these two invariants are in general di?erent. To keep our discussion succinct, we follow the earlier works and take q = 0 in the rest frame of the medium, u? = {1, 0}. Since there is no speci?c spatial direction, the two invariants are related and only the longitudinal part, ΠL = Π? /(?3q 2 )|q=0 , is ? needed. All three currents under consideration are neutral currents. This implies that both time orderings in the correlator correspond to the creation or annihilation of the vector meson. Accordingly, the spectral function is necessarily an even function of the energy variable. One

4

2 2 2 can write the invariant function as ΠL (q0 ) = Π? (q0 )/(?3q0 ),2 which satis?es the following ? dispersion relation3 [19] 2 ΠL (Q2 ≡ ?q0 ) =

1 π

∞ 0

ds

Im ΠL (s) + subtractions . s + Q2

(2.3)

To facilitate our discussion of derivative sum rules, it is useful to derive the following dis(n) persion relation for ΠL (Q2 ) ≡ (Q2 )n ΠL (Q2 ) ΠL (Q2 ) =
(n) (n)

1 π

∞ 0

ds

Im ΠL (s) + subtractions , s + Q2

(n)

(n ≥ 1) ,

(2.4)

with ImΠL (s) = (?1)n sn ImΠL (s). For large Q2 , one can evaluate the correlator by expanding the product of currents according to the operator product expansion (OPE). The result can in general be expressed as ΠL (Q2 ) = ?c0 ln(Q2 ) + c1 c2 c3 + 4 + 6 +··· , 2 Q Q Q (2.5)

where we have omitted the polynomials in Q2 , which vanish under Borel transform. The ?rst term corresponds to the perturbative contribution and the rest are nonperturbative power corrections. The coe?cients, c0 , c1 , c2 , and c3 , have been given in literature. For ρ and ω mesons, one has [19,20] c0 = 1 αs , 1+ 2 8π π
ρN

c1 = 0,
ρN

c2 = mq qq c3 = ?

+

1 αs 2 G 24 π
2 ρN

+

1 u+d A MN ρN , 4 2 (2.6)
ρN

112 π αs qq 81

?

5 u+d 3 A MN ρN , 12 4

? ? where O ρN ≡ Ψ0 |O|Ψ0 denotes the in-medium condensate, mq ≡ 1 (mu + md ), qq 2 uu ρN = dd ρN , and Aq is de?ned as n Aq ≡ 2 n
1 0

=

dx xn?1 q(x, ?2 ) + (?1)n q(x, ?2 ) .

(2.7)

In the baryon case, the correlation function, considered in the rest frame as a function of q0 , has both even and odd parts. The reason is that, at ?nite baryon density, a baryon in medium propagates di?erently than an antibaryon, yielding a correlation function that is asymmetric in the energy variable (see Refs. [26,27]).
3 Here 2 we have omitted the in?nitesimal as we are only concerned with large and space-like q0 .

2

5

Here q(x, ?2 ) and q(x, ?2 ) are the scale dependent distribution functions for a quark and antiquark (of ?avor q) in a nucleon. We follow the standard linear density approximation ? ? ? ? to the in-medium condensates, O ρN = O 0 + O N ρN , with O 0 the vacuum condensate ? and O N the nucleon matrix element. The corresponding result for the φ meson is [21] c0 = 1 4π 2 1+ αs , π + 1 αs 2 G 12 π
2 ρN

c1 = 0,
ρN

c2 = 2 ms ss c3 = ?

ρN

+ As MN ρN , 2 (2.8)

224 π αs ss 81

?

5 s 3 A M ρN . 3 4 N

We adopt the usual pole plus continuum ansatz for the spectral density. This ansatz was recently tested in the Lattice QCD investigation of Ref. [28] where it was found to describe nucleon correlation functions very well. The phenomenological spectral density for vector mesons in medium takes the form [19–21] 1 ρsc ? ImΠL (s) = 2 δ(s) + Fv δ s ? m?2 + c0 θ(s ? s? ) , v 0 π 8π (2.9)

where the ?rst term denotes the contribution of the Landau damping, m? is the vector-meson v mass in the medium, and s? is the continuum threshold. In the calculations to follow, we 0 use mv , Fv and s0 to denote the corresponding vacuum (zero density limit) parameters. Substituting Eqs. (2.5) and (2.9) into the dispersion relation of (2.3) and applying the Borel transform to both sides, one obtains the direct sum rule ρsc ?2 2 ? Fv e?mv /M = ? 2 + c0 M 2 E0 8π c2 c3 cm + c1 + 2 + +···+ +··· . (2.10) 4 M 2! M (m ? 1)! (M 2 )m?1 From the dispersion relation of (2.4), one ?nds
? Fv (m?2 )n e?mv v
?2 /M 2

= n! c0 (M 2 )n+1 En + (?1)n cn+1 + cn+2 cm +···+ + · · · , (2.11) 2 M (m ? n ? 1)! (M 2 )m?n?1

where we have de?ned Ek ≡ 1 ? e
s? /M 2 0

1 s? 0 2 i=0 (k ? i)! M

k

k?i

.

(2.12)

One recognizes that Eq. (2.11) corresponds to the derivative sum rules as they may also be obtained by taking derivatives of Eq. (2.10) with respect to 1/M 2 . Hatsuda and Lee used Eq. (2.10) and the ?rst derivative sum rule (n = 1) of (2.11) in their calculations [19], while Koike claimed that one should use the ?rst and second (n = 2) 6

derivative sum rules [22]. Of course, if one could carry out the OPE to arbitrary accuracy and use a spectral density independent of the model for excited state contributions, the predictions based on Eq. (2.10) and those based on the derivative sum rules should be the same. In practical calculations, however, one has to truncate the OPE and use a simple phenomenological ansatz for the spectral density. Thus it is unrealistic to expect the sum rules to work equally well. The question is, which sum rules give the most reliable predictions. To answer this question, let us compare the nth derivative sum rule with the direct sum rule of (2.10). We observe the following: 1. The perturbative contribution in the derivative sum rule has an extra factor n! relative to the corresponding term in Eq. (2.10), implying that the perturbative contribution is more important in the derivative sum rules than in the direct sum rule, and becomes increasingly important as n increases. Since the perturbative term mainly contributes to the continuum of the spectral density, maintaining dominance of the lowest resonance pole in the sum rule will become increasingly di?cult as n increases. 2. In Eq. (2.10), the term proportional to cm is suppressed by a factor of 1/(m ? 1)!, while it is only suppressed by 1/(m ? n ? 1)! in the derivative sum rule (m > n). This implies that the convergence of the OPE is much slower in the derivative sum rule than in the direct sum rule. This arises because the convergence of the OPE for (n) ΠL (Q2 ) = (Q2 )n ΠL (Q2 ) is obviously much slower than that for ΠL (Q2 ) for large Q2 . Consequently, the high order power corrections are more important in the derivative sum rule than in Eq. (2.10), and become more and more important as n increases. If one would like to restrict the size of the last term of the OPE to maintain some promise of OPE convergence, the size of the Borel region in which the sum rules are believed to be valid is restricted. 3. The power corrections proportional to c1 , c2 , · · · , cn do not contribute to the nth derivative sum rule but do contribute to Eq. (2.10) [29]. If one truncates the OPE, part or all of the nonperturbative information will be lost in the derivative sum rules. It is also worth noting that the leading power corrections are the most desirable terms to have. They do not give rise to a term in the continuum model and they are not the last term in the OPE, whose relative contribution should be restricted to maintain OPE convergence. In practice, the predictions based on the direct sum rule of (2.10) are more reliable than those from the derivative sum rules, which become less and less reliable as n increases. This can also be demonstrated by analyzing the sum rules numerically.
III. SUM-RULE ANALYSIS AND DISCUSSION

If the sum rules were perfect, one would expect that the two sides of the sum rules overlap for all values of the auxiliary Borel parameter M. As mentioned above, one has to 7

truncate the OPE and use a phenomenological model for the spectral density in practical calculations. Hence, the two sides of the sum rules overlap only in a limited range of M (at best). All the previous works have used the ratio method of two sum rules. There, one chooses the continuum threshold to make the ratio of the two sum rules as ?at as possible as a ? function of Borel mass (the residue Fv drops out in the ratio). Although it has also been used in various sum-rule calculations in vacuum, we note that the ratio method has certain drawbacks. First, the ratio method does not check the validity of each individual sum rule. It may happen that individual sum rules are not valid while their ratio is ?at as function of Borel mass. Secondly, the ratio method cannot account for the fact that sum rules do not work equally well. The Borel region where a sum rule is valid can vary from one sum rule to another. Finally, the continuum contributions to the sum rules are not monitored in the ratio method. If the continuum contribution is dominant in a sum rule, one should not expect to get any reliable information about the lowest resonance.
A. Outline of the method

To overcome these shortcomings of the ratio method, we adopt here the optimizing procedure originated in Ref. [30], which has been extensively used in analyzing various vacuum sum rules [31] and ?nite-density sum rules [26]. In this method, one optimizes the ?t of the two sides of each individual sum rule in a ?ducial Borel region, which is chosen such that the highest-dimensional condensates contribute no more than ? 10% to the QCD side while the continuum contribution is less than ? 50% of the total phenomenological side (i.e., the sum of the pole and the continuum contribution). The former sets a criterion for the convergence of the OPE while the latter controls the continuum contribution. While the selection of 50% is obvious for pole dominance, the selection of 10% is a reasonably conservative criterion that has not failed in practice.4 This point is further illustrated in the discussions to follow. The sum rule should be valid in this ?ducial Borel region as the pole contribution dominates the phenomenological side and the QCD side is reliable. We then select 51 points in the ?ducial region and use a χ2 ?t to extract the spectral parameters. The reader is referred to Refs. [30,24] for more details of the method. Since QCD sum rules relate the spectral parameters to the properties of QCD, any imprecise knowledge of the condensates and related parameters will give rise to uncertainties in the extracted spectral parameters. These uncertainties have not been analyzed systematically in the previous works. Here we follow Ref. [24] and estimate these uncertainties via a Monte-Carlo error analysis. Gaussian distributions for the condensate values and related

alternatives to the 10% and 50% criteria are automatically explored in the MonteCarlo error analysis, as the condensate values and the continuum threshold change in each sample.

4 Reasonable

8

parameters are generated via Monte Carlo. The distributions are selected to re?ect the spread of values assumed in previously published QCD sum-rule analyses and uncertainties such as the factorization hypothesis. These distributions provide a distribution for the OPE and thus uncertainty estimates for the QCD side which will be used in the χ2 ?t. In ?tting the sum rules taken from the samples of condensate parameters one learns how these uncertainties are mapped into uncertainties in the extracted spectral parameters. As in the previous works [19–21], we truncate the OPE at dimension six and keep only the terms considered in the literature. In the linear density approximation, the quark and gluon condensates can be written as mq qq ms ss αs 2 G π
ρN

ρN

σN ρN , 2 ms σN ρN , = ms ss 0 + y mq 2 = mq qq
0

+

(3.1) (3.2) (3.3)

ρN

=

αs 2 G π

0

+

αs 2 G π

N

ρN ,

where y ≡ ss N / qq N . The values of vacuum condensates we use are a = ?4π 2 qq 0 = 0.62±0.05 GeV3 [24], b = 4π 2 (αs /π)G2 0 = 0.4±0.15 GeV4 [24], and ss 0 / qq 0 = 0.8±0.2 [32,30]. The quark mass mq is chosen to satisfy the Gell-Mann–Oakes–Renner relation, 2 2 mq qq 0 = ?m2 fπ , and the strange quark mass is taken to be ms = (26 ± 2.5) mq [33]. We π adopt σN = 45 ± 7 MeV [33], (αs /π)G2 N = ?650 ± 150 MeV [27], and y = 0.2 ± 0.1 [19]. For the moments of the parton distribution functions, Aq ’s, we quote the values given in n Ref. [19] and assign a 20% uncertainty to each value, Au+d = 0.9 ±0.18, Au+d = 0.12 ±0.024, 2 4 As = 0.05 ± 0.01, As = 0.002 ± 0.0004. The strong coupling constant is taken to be 2 4 αs /π = 0.117 ± 0.014 at 1 GeV scale [24]. The values of both vacuum and in-medium four-quark condensates are not well determined. Early arguments placed the values of vacuum four-quark condensates within 10% of the vacuum factorized values [34]. However, later analyses suggested that factorization underestimates the four-quark condensates signi?cantly [35,36]. Parameterizing the condensate as κ qq 2 , we will consider values of κ = 2 ± 1 and 1.0 ≤ κ ≤ 3.5 [35,24]. As for the 0 in-medium four-quark condensates, previous authors have adopted the in-medium factorized values (mean ?eld approximation) [19–21]. In the study of ?nite-density baryon sum rules [26,27], it was found that the in-medium factorized values of certain four-quark condensates led to results in contradiction with experiment. However, it should be pointed out that the four-quark operators appearing in the baryon sum rules are di?erent from those in the vector meson sum rules. Here we parameterize the in-medium four-quark condensates as κ qq 2N . ρ The Gaussian distributions for the condensate values and various parameters are generated using the values given above. The error bars in the extracted ?t parameters (see below) are given by the standard deviation of the distribution after 100 condensate values generated via Monte Carlo. It is perhaps worth emphasizing that the error bars do not represent the standard error of the mean, which is 10 times smaller for this case. Hence the error bars 9

are representative of the spread of input parameter values. In addition, the uncertainty estimates become insensitive to the number of Monte Carlo samples after about 50 samples. ? We normalize all ?nite-density spectral parameters (m? , Fv , and s? ) to their corresponding v 0 values in vacuum (i.e., zero density limit). Thus, the error bars in the ratios are dominated by the uncertainties in the density dependent terms of the in-medium condensates since the errors in the vacuum sum rules and ?nite-density sum rules are correlated.
B. Numerical results

Let us start with the sum rules for ρ and ω mesons. We ?rst analyze the direct sum rule of Eq. (2.10). The Landau damping contribution proportional to ρsc is very small at the densities considered here [19]. Treating ρsc as a search parameter, we ?nd that the direct sum rule predicts a value for ρsc in accord with the Fermi-gas approximation, ρsc ? 2π 2 ρN /MN [23,19]. In calculating the uncertainty of this parameter we found that there is insu?cient information in the sum rules to reliably determine this small contribution. As a result we ? use the Fermi-gas relation and treat m? , Fv , and s? as search parameters in the following. v 0 The predictions for the ratio m? /mρ as a function of the nucleon density is plotted in ρ Fig. 1. One can see that the ρ-meson mass decreases with increasing density. At nuclear matter saturation density ρN = ρ0 = (110 MeV)3 , we ?nd m? /mρ = 0.78±0.08. The residue ρ N ? ? Fρ and the continuum threshold s0 also decrease as the density increases. The predictions ? for the ratios Fρ /Fρ and s? /s0 are shown in Figs. 2 and 3, respectively. 0

FIG. 1. Predictions of the direct sum rule for m? /mρ as a function of the medium density. ρ

10

? FIG. 2. Predictions of the direct sum rule for the ratio Fρ /Fρ as a function of the medium density.

FIG. 3. Predictions of the direct sum rule for the ratio (s? /s0 )1/2 as a function of the medium 0 density.

11

FIG. 4. The left- and right-hand sides of the direct sum rule as functions of the Borel mass M at the nuclear matter saturation density.

In Fig. 4 the left- and right-hand sides of the direct sum rule are plotted as functions of M at the nuclear matter saturation density. The near perfect overlap of the two sides of this sum rule is typical of the quality of ?ts seen at other densities. The corresponding valid Borel window and the relative contributions of the continuum and the highest order term in the OPE are displayed in Fig. 5 as dashed curves. One notices that the direct sum rule is valid in a broad Borel regime, where the highest order term contributes less than 10% and the continuum contributes only about 15% at the lower bound and the required 50% at the upper bound. Thus, the pole contribution truly dominates the sum rule in the Borel region of interest, implying that the predictions are reliable. We also ?nd that both lower and upper bounds are functions of the density and decrease as the density increases. The rate of decrease for the upper bound is larger than that for the lower bound, which means that the optimal Borel window shrinks with increasing density. We proceed now to analyze the ?rst derivative sum rule of Eq. (2.11) (with n = 1). It is found that this sum rule is valid in a much smaller Borel regime. The continuum and highest order OPE term contributions are shown in Fig. 5 as dot-dashed curves for ρN = ρ0 . N It can be seen that the continuum contribution exceeds 33% in the entire Borel window and the relative importance of the highest order term increases as compared to the direct sum rule. Thus, the predictions of the ?rst derivative sum rule are less reliable than those from the direct sum rule. At zero density, the ?rst derivative sum rule predicts a very large ρ-meson mass and the continuum threshold is only about 100 MeV above the pole position. ? Nevertheless, the ?rst derivative sum rule also predicts that the ratios m? /mρ , Fρ /Fρ , and ρ 12

FIG. 5. Relative contributions of the continuum and the highest order OPE terms to the sum rules as functions of the Borel mass at nuclear matter saturation density. The dashed and dot-dashed curves correspond to the direct sum rule and the ?rst derivative sum rule, respectively. Note the relatively broad regime of validity for the direct sum rule.

s? /s0 all decrease as the density increases, which agrees qualitatively with the predictions of 0 the direct sum rule. The reasons for the failure of the ?rst derivative sum rule to reproduce the ρ-meson mass obtained from the direct sum rule are discussed at length in Ref. [24]. In the second derivative sum rule of Eq. (2.11) (with n = 2), the contributions of the quark and gluon condensates drop out and the nonperturbative power correction starts with dimension six condensates, the last term of the truncated OPE. The perturbative contribution and hence the continuum contribution is multiplied by an extra factor of two relative to that for the direct sum rule. Numerical analysis indicates that there is no Borel window where the sum rule is valid. Thus, one cannot get information about ground-state vector mesons from this sum rule. This is also true for the third and higher derivative sum rules, where there are no power corrections at the level of the OPE truncation considered here. All of the results above are for the ρ and ω mesons. A similar analysis can be done for the φ meson. Again, we ?nd the same pattern. The direct sum rule gives the most reliable predictions, the ?rst derivative sum rule yields a less reliable result, and the second and higher derivative sum rules are invalid. We ?nd from the direct sum rule m? /mφ = 0.99±0.01 φ at nuclear matter saturation density. This rate of decrease is much smaller than that for the ρ and ω mesons. This is due to the dominance of ms ss ρN and its slow change with the medium density. 13

C. Discussion

The Borel transform plays important roles in making the QCD sum-rule approach viable. It suppresses excited state contributions exponentially on the phenomenological side, thus minimizing the continuum model dependence. It also improves OPE convergence by suppressing the high order power corrections factorially on the QCD side. We observe that taking derivatives of the direct sum rule with respect to 1/M 2 is equivalent to a partial reverse of the Borel transform. It is thus not surprising to ?nd that the continuum contribution becomes more important and the convergence of the OPE becomes slower in the derivative sum rules than in the direct sum rule. Since the excited state contributions are modeled roughly by a perturbative evaluation of the correlator starting at an e?ective threshold, and the higher order OPE terms are not well determined, there are more uncertainties in the derivative sum rules than in the direct sum rule. In fact, a simultaneous ?t of both the direct and ?rst derivative sum rules in vacuum reveals that the ?rst derivative sum rule plays a negligible role in determining the ?t parameters, when a χ2 measure weighted by the OPE uncertainty and relative reliabilities of the sum rules is used [24]. To improve the reliability of the derivative sum rules, one must include more higher order terms in the OPE. However, one usually does not have much control of the values of the higher-dimensional condensates. In addition, the derivative sum rule will always su?er from a factorial enhancement of the terms contributing to the continuum model. Therefore, the direct sum rule will always yield the most reliable results for vector mesons. Hatsuda and Lee invoked both the direct and the ?rst derivative sum rules [19]. Their results for the ratio mρ? /mρ are somewhat larger than those we obtained from the direct sum rule but are somewhat smaller than those from the ?rst derivative sum rule. This discrepancy is obviously attributed to their use of the two sum rules simultaneously without weighing the relative merits of the sum rules. In Ref. [22], Koike adopted both the ?rst and second derivative sum rules. His conclusion of slightly increasing vector-meson masses in the medium depends on the use of the second derivative sum rule, which we have found to be invalid when the OPE is truncated at dimension six. Hatsuda et al. [23] also pointed out some shortcomings of the second derivative sum rule. However, their arguments are based on concerns over the lack of information on the QCD side of the second derivative sum rule and the absence of a “plateau” in the ratio of the second and ?rst derivative sum rules from which a mass is extracted. In this paper, we have extensively explored why these observations come about, and why even Hatsuda et al.’s analysis is less than satisfactory. The use of the ratio method is mainly driven by the expectation that if the sum rules work well one should see a plateau in the predicted quantities as functions of the Borel mass. The usual interpretation of this criterion is that the ratio of two di?erent sum rules, proportional to certain spectral parameter of interest (e.g. mass), should be ?at as function of the Borel mass. Although it is true ideally, this interpretation is potentially problematic in practice. We have seen that the reliabilities and validities of two sum rules are usually di?erent. This feature cannot be revealed in the ratio of the two sum rules. In addition, 14

one can always achieve the ?atness of the ratio in the large Borel mass region, where both sides of the sum rules are dominated by the continuum. However, one learns little about the lowest pole in this case. To make contact with the plateau criterion, we propose that if a sum rule works well, one should see a plateau in the plot of an extracted quantity expressed as a function of the Borel mass. For example, from the direct sum rule of (2.10), one can express the vector-meson mass as m? = M 2 ln v
? Fv ΠS (M) 1/2

,

(3.4)

where ΠS (M) denotes the right-hand side of Eq. (2.10). In Fig. 6, we plot the right-hand side of Eq. (3.4) for the ρ-meson case at four di?erent densities, with optimized values for ? Fv and s? . One indeed sees very ?at curves within the region of validity denoted by the 0 error bars. It should be emphasized that (1) this interpretation only involves one sum rule and is thus di?erent from the ratio method previously used in the literature; (2) the value of m? is only meaningful in the valid region of a sum rule; (3) in this interpretation, the v plateau criterion is a true criterion, measuring the quality of the overlap between the two sides of a valid sum rule. For curiosity, we also display the curve outside the validity region

FIG. 6. The ρ-meson mass as obtained from the right-hand side of Eq. (3.4) as a function of M at various nuclear matter densities. The curves from top down correspond to ρN = 0, ρN = 0.5 ρ0 , ρN = ρ0 , and ρN = 1.5 ρ0 . The error bars denoting the valid regimes are obtained N N N from the relative errors of Fig. 1. Note how the Borel regime shifts and becomes smaller as the density increases. The curve for ρN = ρ0 is plotted outside the valid region to demonstrate the N importance of carefully selecting a Borel region.

15

for the case of ρN = ρ0 in Fig. 6. One notices a deviation from the plateau just outside N the valid Borel region. This feature supports our selection of 10% as the criterion for OPE convergence. In the present analysis as well as the previous works, the linear density approximation has been assumed for the in-medium condensates. For a general operator there is no systematic way to study contributions that are of higher order in the medium density. Modeldependent estimates in Ref. [37] suggest that the linear approximation to qq ρN should be good (higher-order corrections ? 20% of the linear term) up to nuclear matter saturation density. In our analyses, we have assigned generous uncertainties to various condensate values and parameters. We expect these will cover the uncertainties arising from the linear density approximation. As the medium density increases, the deviation from the linear density approximation will increase. One then needs more precise knowledge of the density dependence of various condensates in order to have reliable QCD sum-rule predictions. As in previous works, we have neglected the dimension-six twist-four operators in (2.6), as the nucleon matrix elements of these operators are unknown. However, estimates may be obtained from deep-inelastic-scattering data provided one is willing to make a few additional assumptions [23]. Taking the estimates given in Ref. [23], we ?nd these additional contributions have little e?ect on our results. At saturation density, our present in-medium ρ-meson mass of 0.59 GeV is increased slightly to 0.62 GeV. Taking a 100% uncertainty on the twist-four contributions has no apparent e?ect on the present uncertainty of 0.11 GeV. At saturation density, the ratio mρ? /mρ is shifted from 0.78 to 0.82 which is small relative to the uncertainty of 0.08. Certainly further study of the twist-four contributions is required. However, we do not expect such contributions to signi?cantly alter the conclusions of this paper. As a ?nal remark, we comment on the electromagnetic width of the ρ meson, Γ(ρ0 → e+ e? ). In free space, it is given by [38,34] Γ(ρ0 → e+ e? ) = 1 2 α mρ 3 E 4π 2 gρ = 4π 2 α 3 E Fρ mρ , (3.5)

where αE is the electromagnetic coupling constant. The modi?cation of this result in medium may be estimated by replacing mρ and Fρ with their corresponding values in medium. The ratio of the free space and in-medium widths can be expressed as mρ F ? Γ? (ρ0 → e+ e? ) = ? ρ . Γ(ρ0 → e+ e? ) mρ Fρ (3.6)

? Note that mρ /m? increases while Fρ /Fρ decreases with increasing density. However, the ρ ? rate of decrease for Fρ /Fρ is larger than the rate of increase for mρ /m? . Consequently, the ρ ratio for the widths is less than 1. At ρN = ρ0 , our estimate is Γ? /Γ = 0.85 ± 0.10. This N implies that the ρ-meson electromagnetic width becomes smaller in nuclear matter. This behavior might be observed in the proposed experiment studying dileptons as a probe of vector mesons in the dense and hot matter [8].

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IV. CONCLUSION

In this paper, we have carefully examined the QCD sum rules for vector mesons in nuclear matter. Our primary concern has been on the validity and reliability of various sum rules. We emphasize that the sum rules do not work equally well due to the truncation of the OPE and the use of a model for the phenomenological spectral density. In particular, the derivative sum rules are less reliable than the direct sum rule. This is attributed to: (1) the perturbative contribution and hence the continuum contribution become increasingly important in the derivative sum rules; (2) the high order terms in the OPE become increasingly important in the derivative sum rules; (3) part (or all) of the nonperturbative information is lost in the derivative sum rules. We therefore conclude that any predictions based on (or partially on) second or higher derivative sum rules are incorrect given the level of the OPE truncation adopted in the literature. One should avoid using the derivative sum rules in practical applications. We tested this conclusion numerically by analyzing the sum rules with regard to pole dominance and OPE convergence [30]. A Monte Carlo based error analysis was used to provide reliable uncertainties on our predictions and remove the sensitivity of the results to the input parameters [24]. We found that the direct sum rule satis?es our criteria and leads to reliable predictions. The ?rst derivative sum rule su?ers from a small region of validity and large continuum contributions throughout. The second and higher derivative sum rules are invalid. Our analysis con?rms that the QCD sum-rule approach predicts a decrease of vectormeson masses with increasing density, and resolves the debate between Hatsuda et al. [23] and Koike [22]. The prediction of a slight increase of vector-meson masses in medium is based on an invalid second derivative sum rule. We note that all previous authors have used the ratio method in the analysis of the sum rules, which has many drawbacks and may lead to incorrect results. We encourage the community to adopt the approach developed in Ref. [24] which checks the quality of the overlap between two sides of each individual sum rule by monitoring pole dominance and the convergence of the OPE. This approach also allows one to realistically estimate the uncertainties and reveal the predictive ability of QCD sum rules. The analysis presented here is the most reliable QCD analysis of in-medium vector-meson properties. At nuclear matter saturation density, we predict m? ρ = 0.78 ± 0.08, mρ Γ? (ρ0 → e+ e? ) = 0.85 ± 0.10 , Γ(ρ0 → e+ e? ) (4.1)

and look forward to experimental vindication of these results [8].

17

ACKNOWLEDGMENTS

We would like to thank Tom Cohen for insightful conversations. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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