In-Medium Modifications of Scalar Charm Mesons in Nuclear Matter

T. Hilger, B. Kampfer

Forschungszentrum Dresden-Rossendorf, PF 510119, D-01314 Dresden, GermanyTU Dresden, Institut fur Theoretische Physik, 01062 Dresden, Germany

AbstractEmploying QCD sum rules the in-medium modifications of scalar charm mesons in a cold nuclear matter environmentare estimated. The mass splitting of D∗ − D∗ is quantified.

1. Introduction

In-medium modifications of hadrons, embedded instrongly interacting matter, are of considerable interest,since they test the non-perturbative sector of QCD. Asdensity and temperature of the ambient medium can bechanged in a controllable manner, the changes of hadronproperties, compared to the vacuum, may serve as an in-depth check of our understanding of hadron physics.

Previous investigations focussed on light vectormesons and light pseudoscalar strange mesons [1, 2],thus being essentially restricted to the light-flavorSU f (3) sector of QCD. The starting FAIR project willenable the extension to charm degrees of freedom. Infact, two major collaborations [3, 4] plan to investi-gate charm mesons and baryons in proton-nucleus, anti–proton-nucleus and heavy-ion collisions. Given thismotivation, we are going to extend our recent study [5]of the in-medium modifications of pseudoscalar opencharm mesons to the lowest excitations of scalar opencharm mesons. In doing so we employ QCD sum rules[6, 7] and restrict ourselves to estimates of the masssplitting of scalar exciations which can be related to thecurrents jD∗ = dc and jD∗ = cd for small densities.

2. QCD sum rules

The Borel transformed in-medium sum rulesfor the current-current correlator Π(q) =

i∫

d4x eiqx〈〈T[j(x)j†(0)

]〉〉 (here, T [· · ·] means the

time-ordered product and 〈〈· · · 〉〉 stands for Gibbs

average) may be cast in the form (cf. [5] for details)⎛⎜⎜⎜⎜⎝∫ ω+0ω−0+

∫ ω−0−∞+

∫ +∞ω+0

⎞⎟⎟⎟⎟⎠ dωΔΠ(ω, �q )ω je−ω2/M2

= πBQ2→M2

[Π

j′OPE(Q2, �q )

] (M2), (1)

where Q2 = −q20, ΔΠ are the discontinuities along

the entire real axis, M is the Borel mass and the inte-gral over the hadronic spectral function Π(s, �q ) is splitinto a part of the low-energy excitations (first term) andthe so-called continuum contributions (second and thirdterms). The latter ones are mapped by a semi-local du-ality hypothesis to expressions corresponding to the op-erator product expansion (OPE) of the correlator whichyields B

[Πn′

OPE

]. Since particles and anti-particles be-

have differently in a nuclear medium at zero tempera-ture, two sum rules emerge – an even one ( j = 1, j′ = e)and an odd one ( j = 0, j′ = o). The OPE’s neededhere can be obtained by combining the OPE’s for pseu-doscalar D mesons in [5] and the OPE’s for differencesum rules in [8]:

BQ2→M2

[Πe

OPE(Q2, �q = 0 )] (

M2)

=1π

∫ ∞

m2c

dse−s/M2ImΠper

D∗ (s, �q = 0 )

+ e−m2c/M

2(mc〈dd〉 − 1

2

(m3

c

2M4 −mc

M2

)〈dgσGd〉

+112〈αs

πG2〉 +

[(718+

13

lnμ2m2

c

M4 − 2γE

3

) (m2

c

M2 − 1)

−23

m2c

M2

]〈αs

π

((vG)2

v2 − G2

4

)〉

+2(

m2c

M2 − 1)〈d†iD0d〉

Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 277–280

0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/npbps

doi:10.1016/j.nuclphysbps.2010.10.071

−4(

m3c

2M4 −mc

M2

) [〈dD2

0d〉 − 18〈dgσGd〉

]), (2a)

BQ2→M2

[Πo

OPE(Q2, �q = 0 )] (

M2)

= e−m2c/M

2(〈d†d〉 − 4

(m2

c

2M4 −1

M2

)〈d†D2

0d〉

− 1M2 〈d†gσGd〉

)≡ e−m2

c/M2〈K(M)〉n, (2b)

where ImΠperD∗ is given in [7] and mc = 1.3 GeV is the

charm quark mass. We define e ≡ ∫ ω+0ω−0

dωωΔΠe−ω2/M2

and o ≡ ∫ ω+0ω−0

dωΔΠe−ω2/M2to obtain, with a pole ansatz

for the lowest excitations within ω− −ω+, which couplewith strenghts F± to the above currents,

e = m+F+e−m2+/M

2+ m−F−e−m2−/M2

, (3a)

o = F+e−m2+/M

2 − F−e−m2−/M2. (3b)

The mass splitting Δm and mass center m are related bym± = m±Δm. The following equations determine thesequantities:

Δm =12

oe′ − eo′

e2 + oo′, (4a)

m2 = Δm2 − ee′ + (o′ )2

e2 + oo′, (4b)

where a prime denotes the derivative w.r.t. M−2 and e aswell as o are given by (2a) and (2b) minus the contin-uum parts from (1).

3. Low-density expansion

An expansion in the density n gives the leading termfor the mass splitting

Δm(n) ≈ 12

dodn

∣∣∣0 e′(0) − e(0) do′

dn

∣∣∣0

e(0)2 n ≡ αΔmn , (5)

since e(n = 0) � 0 and o(n = 0) = 0.The primary goal of the present sum rule analysis

is to find the dependence of Δm and m on changes ofthe condensates entering (2). However, also the contin-uum thresholds ω±0 can depend on the density. To studythis influence, we consider the asymmetric splitting ofthe continuum thresholds Δω2

0 = ((ω+0 )2 − (ω−0 )2)/2and parameterize its density dependence by Δω2

0(n) =αΔωn + O(n2), which leads to

dodn

∣∣∣∣∣0=

⎛⎜⎜⎜⎜⎝e−ω20/M

2

πω0ImΠper

D∗ (ω20)

dΔω20

dn

⎞⎟⎟⎟⎟⎠n=0

+ e−m2c/M

2〈K(M)〉 , (6a)

do′

dn

∣∣∣∣∣0=

⎛⎜⎜⎜⎜⎝−e−ω20/M

2

πω0ImΠper

D∗ (ω20)

dΔω20

dn

⎞⎟⎟⎟⎟⎠n=0

+ e−m2c/M

2〈K′(M) − m2c K(M)〉 . (6b)

The perturbative terms stem from the continuum con-tribution in case of unequal thresholds for particle andantiparticle. In linear density approximation of the con-densates the last terms become o/n and o′/n, respec-tively. In this case, αΔm is given as

αΔm = − 12e(0)

(on

m2(0) +o′

n

+e−ω2

0/M2

πω0ImΠper

D∗ (ω20)[m2(0) − ω2

0

]αΔω

⎞⎟⎟⎟⎟⎠ ,(7)

which is dominated by the non-perturbative terms. Wechoose the Borel mass range and the thresholds accord-ing to [9].

As an estimate for the order of αΔω we rely on thesplitting of the thresholds for the pseudoscalar channelin [5] and obtain αΔω ≈ 0.25 · 103 GeV−1. It is anoverestimation of the pseudoscalar O(n) threshold split-ting as it would correspond to a linear interpolation ofΔω2

0 from the vacuum to nuclear saturation density and,hence, includes higher order terms in the density. Wechoose αΔω ≈ 102 . . . 103 GeV−1.

The results are depicted in Fig. 1 for αΔω = ±102

GeV−1 and for αΔω = ±103 GeV−1 in Fig. 2. In Fig. 3αΔm as a function of αΔω for M = 1.37 GeV, the mini-mum of the vacuum Borel curve for the scalar D meson,is displayed.

�

�

Figure 1: αΔm as a function of the Borel mass forαΔω from −102 (lower bundle of curves) GeV−1 to+102 (upper bundle of curves) GeV−1 and for threshold valuesω2

0 = 6.0 GeV2 (solid black), 7.5 GeV2 (dashed red) and 9.0 GeV2

(dotted blue).

Considering the results for the mass splitting ofheavy-light pseudoscalar mesons, e.g. D and B, one

T. Hilger, B. Kämpfer / Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 277–280278

�

�

Figure 2: αΔm as a function of the Borel mass forαΔω from −103 (lower bundle of curves) GeV−1 to+103 (upper bundle of curves) GeV−1. For line code see Fig. 1.

�

Figure 3: αΔm as a function of αΔω for M = 1.37 GeV. For line codesee Fig. 1.

could raise the question if the splitting is mainly causedby a splitting of the thresholds and, hence, might be anartifact of the method which determines Δω2

0. From theabove study we find that αΔω indeed influences the masssplitting. A direct correlation in the sense of a correla-tion in sign can not be confirmed. Furthermore, the re-sults for Ds mesons [5] allow for a positive mass split-ting if the net strange quark density falls below a criticalvalue. As the strange quark density enters through thevector quark condensate, this already points to a sup-pressed influence of the threshold splitting on the masssplitting.

4. Beyond low-density approximation

In [5], the threshold splitting is not considered as afree parameter but determined by the requirement thatthe minima of the Borel curves for particle and antipar-ticle are at the same Borel mass.

�

�

Figure 4: Mass splitting parameter Δm of scalar D∗ − D∗ mesonswith the pole + continuum ansatz as a function of density at zerotemperature. For a charm quark mass parameter of mc = 1.3 GeVand mean threshold value ω2

0(0) = 7.5 GeV2. The curves are forω2

0(n) = ω20(0) + ξn/n0 with ξ = 0 (solid), ξ = 1 GeV2 (dotted) and

ξ = −1 GeV2 (dashed).

�

�

Figure 5: As in Fig. 4 but for the mean mass m.

Following the same analysis strategy of [5] and em-ploying the condensates listed in [5], one obtains theresults exhibited in Figs. 4 and 5. This analysis goesbeyond the strict linear density expansion of m and Δmand also takes into account quadratic terms in the den-sity. The medium dependent part of the chiral conden-sate (the density dependence of which is only in lineardensity approximation, as the other condensates too) en-ters the mass splitting next to leading order of the den-sity. The determination of the mass center m dependsstrongly on the chosen center of continuum thresholdsω2

0 = ((ω+0 )2 + (ω−0 )2)/2 as indicated by the broad rangein Fig. 5 when varying their medium dependence. Incontrast, the splitting is fairly robust as evidenced byFig. 1, where the difference between curves of differentthresholds is negligible.

A linear interpolation of the threshold splitting fromvacuum to a density of n = 0.01 f m−3 gives an estimate

T. Hilger, B. Kämpfer / Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 277–280 279

for the O(n) term αΔω ≈ 7 · 102 GeV−1, which justifiesthe range chosen in the previous section.

5. Conclusions

In summary we extend the recent analysis [5] of theQCD sum rules to lowest scalar D∗ mesons. The im-portance of the density dependence of the continuumthresholds is exposed. Going beyond the strict linearapproximation in density one obtains a robust patternof the splitting of scalar D∗ − D∗ mesons which resem-bles the one for pseudoscalars. A firm prediction of theabsolute values of the respective scalar mesons is ham-pered by uncertainties in the determination of the meanmass. With the employed values of the condensates en-tering the truncated sum rule and within the employedanalysis strategy a tendency of a ”mass drop” may bededuced.For the sake of simplicity we restricted our analysis toa pole + continuum ansatz. One can go beyond suchsimplified treatment of the hadronic spectral functionsby considering moments. The latter ones do not longerallow for a simple interpretation but seem more appro-priate for broad resonances. With respect to the re-search programme at FAIR, where precision measure-ments of various charm hadrons are envisaged, moremodel-independent studies are required.

Acknowledgements

The work is supported by GSI-FE and BMBF06DR5059.

References[1] S. Leupold, V. Metag, U. Mosel, Int. J. Mod. Phys. E 19, 147

(2010).[2] R. Rapp, J. Wambach, H. van Hees, arXiv:0901.3289.[3] CBM Collaboration: www.gsi.de/fair/experiments/CBM/index-

e.html.[4] PAND Collaboration: www.gsi.de/fair/experiments/hesr-

panda/pandagsi-e.html.[5] T. Hilger, R. Thomas, B. Kampfer, Phys. Rev. C 79, 025202

(2009).[6] M.A. Shifman, A.I. and Vainshtein and V.I. Zakharov, Nucl.

Phys. B147 (1979) 385.[7] For a review and references to original works, see e.g., S. Nar-

ison, QCD as a theory of hadrons, Cambridge Monogr. Part.Phys. Nucl. Phys. Cosmol. 17 (2002) 1-778 [hep-h/0205006].

[8] T. Hilger, R. Schulze, B. Kampfer, J. Phys. G: Nucl. Part. Phys.37, 094054 (2010).

[9] S. Narison, Phys. Let. B 605, 319 (2005).

T. Hilger, B. Kämpfer / Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 277–280280