Analysis of the heavy mesons in the nuclear matter with the QCD sum rules
aa r X i v : . [ h e p - ph ] D ec Analysis of the heavy mesons in the nuclear matter with the QCDsum rules
Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R. China
Abstract
In this article, I calculate the contributions of the nuclear matter induced condensatesup to dimension 5, take into account the next-to-leading order contributions of the nuclearmatter induced quark condensate, study the properties of the scalar, pseudoscalar, vectorand axialvector heavy mesons in the nuclear matter with the QCD sum rules in a systematicway, and obtain the shifts of the masses and decay constants. Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct, and obtain the conclusion qualitatively aboutthe possible existence of heavy-meson-nucleon bound states.
PACS numbers: 12.38.Lg; 14.40.Lb; 14.40.Nd
Key Words:
Nuclear matter, QCD sum rules
The suppression of
J/ψ production in relativistic heavy ion collisions is considered as an importantsignature to identify the quark-gluon plasma [1]. The dissociation of
J/ψ in the quark-gluon plasmadue to color screening can result in a reduction of its production. The interpretation of suppressionrequires the detailed knowledge of the expected suppression due to the
J/ψ dissociation in thehadronic environment. The in-medium hadron properties can affect the productions of the open-charmed mesons and the
J/ψ in the relativistic heavy ion collisions, the higher charmonium statesare considered as the major source of the
J/ψ [2]. For example, the higher charmonium statescan decay to the D ¯ D , D ∗ ¯ D ∗ pairs instead of decaying to the lowest state J/ψ in case of the massreductions of the D , D ∗ , ¯ D , ¯ D ∗ mesons are large enough. We have to disentangle the color screeningversus the recombination of off-diagonal ¯ cc (or ¯ bb ) pairs in the hot dense medium versus cold nuclearmatter effects, such as nuclear absorption, shadowing and anti-shadowing, so as to draw a definiteconclusion on appearance of the quark-gluon plasma [3, 4]. The upcoming FAIR (Facility forAntiproton and Ion Research) project at GSI (Institute for Heavy Ion Research) in Darmstadt(Germany) provides the opportunity to study the in-medium properties of the charmoniums orcharmed hadrons for the first time. The CBM (Compressed Baryonic Matter) collaboration intendsto study the properties of the hadrons in the nuclear matter [5], while the ¯PANDA (anti-ProtonAnnihilation at Darmstadt) collaboration will focus on the charm spectroscopy, and mass andwidth modifications of the charmed hadrons in the nuclear matter [6]. However, the in-mediummass modifications are not easy to access experimentally despite the interesting physics involved,and they require more detailed theoretical studies. On the other hand, the bottomonium statesare also sensitive to the color screening, the Υ suppression in high energy heavy ion collisions canalso be taken as a signature to identify the quark-gluon plasma [7]. The suppressions on the Υproduction in ultra-relativistic heavy ion collisions will be studied in details at the RelativisticHeavy Ion Collider (RHIC) and Large Hadron Collider (LHC).Extensive theoretical and experimental studies are required to explore the hadron propertiesin nuclear matter. The connection between the condensates and the nuclear density dependenceof the in-medium hadron masses is not straightforward. The QCD sum rules provides a powerfultheoretical tool in studying the in-medium hadronic properties [8, 9], and has been applied exten-sively to study the light-flavor hadrons and charmonium states in the nuclear matter [10, 11, 12].The works on the heavy mesons and heavy baryons are few, only the D , B , D , B , D ∗ , B ∗ , D , B , Λ Q , Σ Q , Ξ QQ and Ω QQ are studied with the QCD sum rules [13, 14, 15, 16, 17, 18]. The E-mail, [email protected]. c (2800) andΛ c (2940) can be assigned to be the S -wave DN state with J P = − and the S -wave D ∗ N statewith J P = − respectively based on the QCD sum rules [19].The article is arranged as follows: I study in-medium properties of the heavy mesons with theQCD sum rules in Sec.2; in Sec.3, I present the numerical results and discussions; and Sec.4 isreserved for my conclusions. I study the scalar, pseudoscalar, vector and axialvector heavy mesons in the nuclear matter withthe two-point correlation functions Π( q ) and Π µν ( q ), respectively. In the Fermi gas approximationfor the nuclear matter, I divide the Π( q ) and Π µν ( q ) into the vacuum part Π ( q ) and Π µν ( q ) andthe static one-nucleon part Π N ( q ) and Π Nµν ( q ), and expand the Π N ( q ) and Π Nµν ( q ) up to the order O ( ρ N ) at relatively low nuclear density [11, 13],Π( q ) = i Z d x e iq · x h T (cid:8) J ( x ) J † (0) (cid:9) i ρ N = Π ( q ) + ρ N m N T N ( q ) , Π µν ( q ) = i Z d x e iq · x h T (cid:8) J µ ( x ) J † ν (0) (cid:9) i ρ N = Π µν ( q ) + ρ N m N T Nµν ( q ) , (1)where the ρ N is the density of the nuclear matter, and the forward scattering amplitudes T N ( q )and T Nµν ( q ) are defined as T N ( ω, q ) = i Z d xe iq · x h N ( p ) | T (cid:8) J ( x ) J † (0) (cid:9) | N ( p ) i ,T Nµν ( ω, q ) = i Z d xe iq · x h N ( p ) | T (cid:8) J µ ( x ) J † ν (0) (cid:9) | N ( p ) i , (2)2here the J ( x ) and J µ ( x ) denote the isospin averaged currents η ( x ), η ( x ), η µ ( x ) and η µ ( x ),respectively, η ( x ) = η † ( x ) = ¯ c ( x ) q ( x ) + ¯ q ( x ) c ( x )2 ,η ( x ) = η † ( x ) = ¯ c ( x ) iγ q ( x ) + ¯ q ( x ) iγ c ( x )2 ,η µ ( x ) = η † µ ( x ) = ¯ c ( x ) γ µ q ( x ) + ¯ q ( x ) γ µ c ( x )2 ,η µ ( x ) = η † µ ( x ) = ¯ c ( x ) γ µ γ q ( x ) + ¯ q ( x ) γ µ γ c ( x )2 , (3)which interpolate the scalar, pseudoscalar, vector and axialvector mesons D , D , D ∗ and D ,respectively. I choose the isospin averaged currents since the D , D , D ∗ and D mesons areproduced in pairs in the antiproton-nucleon annihilation processes. The q denotes the u or d quark, the q µ = ( ω, q ) is the four-momentum carried by the currents J ( x ) and J µ ( x ), the | N ( p ) i denotes the isospin and spin averaged static nucleon state with the four-momentum p = ( m N , h N ( p ) | N ( p ′ ) i = (2 π ) p δ ( p − p ′ ) [13].I can decompose the correlation functions T Nµν ( ω, q ) as T Nµν ( ω, q ) = T N ( ω, q ) (cid:18) − g µν + q µ q ν q (cid:19) + T N ( ω, q ) q µ q ν + T N ( ω, q ) ( q µ u ν + q ν u µ ) + T N ( ω, q ) u µ u ν , (4)according to Lorentz covariance, where the T N ( ω, q ) denotes the contributions of the vector andaxialvector charmed mesons, and the T / / N ( ω, q ) are irrelevant in the present analysis.In the limit q → , the forward scattering amplitude T N ( ω, q ) can be related to the DN ( D N , D ∗ N and D N ) scattering T -matrix, T D/D /D ∗ /D N ( m D/D /D ∗ /D ,
0) = 8 π ( m N + m D/D /D ∗ /D ) a D/D /D ∗ /D , (5)where the a D/D /D ∗ /D are the D/D /D ∗ /D N scattering lengths. I can parameterize the phe-nomenological spectral densities ρ ( ω,
0) with three unknown parameters a, b and c near the polepositions of the charmed mesons D , D , D ∗ and D according to Ref.[13], ρ ( ω,
0) = − π Im T D/D N ( ω, ) (cid:16) ω − m D/D + iε (cid:17) f D/D m D/D m c + · · · , = a ddω δ (cid:16) ω − m D/D (cid:17) + b δ (cid:16) ω − m D/D (cid:17) + c δ (cid:0) ω − s (cid:1) , (6)for the pseudoscalar and scalar currents η ( x ) and η ( x ), ρ ( ω,
0) = − π Im T D ∗ /D N ( ω, ) (cid:16) ω − m D ∗ /D + iε (cid:17) f D ∗ /D m D ∗ /D + · · · , = a ddω δ (cid:16) ω − m D ∗ /D (cid:17) + b δ (cid:16) ω − m D ∗ /D (cid:17) + c δ (cid:0) ω − s (cid:1) , (7)for the vector and axialvector currents η µ ( x ) and η µ ( x ).3ow the hadronic correlation functions Π( ω,
0) and Π µν ( ω,
0) at the phenomenological side canbe written asΠ( ω,
0) = (cid:0) f D/D + δf D/D (cid:1) (cid:0) m D/D + δm D/D (cid:1) m c (cid:0) m D/D + δm D/D (cid:1) − ω + · · · = f D/D m D/D m c m D/D − ω + · · · + ρ N m N a (cid:16) m D/D − ω (cid:17) + bm D/D − ω + · · · , (8)Π µν ( ω,
0) = (cid:0) f D ∗ /D + δf D ∗ /D (cid:1) (cid:0) m D ∗ /D + δm D ∗ /D (cid:1) (cid:0) m D ∗ /D + δm D ∗ /D (cid:1) − ω (cid:18) − g µν + q µ q ν q (cid:19) + · · · , = f D ∗ /D m D ∗ /D m D ∗ /D − ω (cid:18) − g µν + q µ q ν q (cid:19) + · · · + ρ N m N a (cid:16) m D ∗ /D − ω (cid:17) + bm D ∗ /D − ω + · · · (cid:18) − g µν + q µ q ν q (cid:19) + · · · . (9)In Eqs.(6-7), the first term denotes the double-pole term, and corresponds to the on-shell effectof the T -matrix, a = − π ( m N + m D/D ) a D/D f D/D m D/D m c , (10)for the currents η ( x ) and η ( x ) and a = − π ( m N + m D ∗ /D ) a D ∗ /D f D ∗ /D m D ∗ /D , (11)for the currents η µ ( x ) and η µ ( x ); the second term denotes the single-pole term, and correspondsto the off-shell effect of the T -matrix; the third term denotes the continuum term or the remainingeffects, where the s is the continuum threshold parameter. In general, the continuum contribu-tions are approximated by ρ QCD ( ω, θ ( ω − s ), where the ρ QCD ( ω,
0) are the perturbative QCDspectral densities, and θ ( x ) = 1 for x ≥
0, else θ ( x ) = 0. In this article, the QCD spectral densitiesare of the type δ ( ω − m Q ), which include both the ground state and continuum state contribu-tions, I have attributed the excited state contributions to the continuum state contributions, sothe collective continuum state contributions can be approximated as c δ ( ω − s ), then I obtain theresult c/ (cid:0) s − ω (cid:1) in the hadronic representation, see Eq.(15). The doublet ( D (2550) , D (2600))or ( D J (2580) , D ∗ J (2650)) is assigned to be the first radial excited state of the doublet ( D, D ∗ ) [20].The single-pole contributions come from the doublet ( D (2550) , D (2600)) or ( D J (2580) , D ∗ J (2650))are of the form 1 / (cid:16) m D (2550) /D (2600) − ω (cid:17) , so the approximation c/ (cid:0) s − ω (cid:1) is reasonable.Then the shifts of the masses and decay constants of the charmed-mesons can be approximatedas δm D/D /D ∗ /D = 2 π m N + m D/D /D ∗ /D m N m D/D /D ∗ /D ρ N a D/D /D ∗ /D , (12)4 f D/D = m c f D/D m D/D bρ N m N − f D/D m D/D δm D/D m c ! ,δf D ∗ /D = 12 f D ∗ /D m D ∗ /D (cid:18) bρ N m N − f D ∗ /D m D ∗ /D δm D ∗ /D (cid:19) . (13)In calculations, I have used the following definitions for the decay constants of the heavy mesons, h | η (0) | D + ¯ D i = f D m D m c , h | η (0) | D + ¯ D i = f D m D m c , h | η µ (0) | D ∗ + ¯ D ∗ i = f D ∗ m D ∗ ǫ µ , h | η µ (0) | D + ¯ D i = f D m D ǫ µ , (14)with summations of the polarization vectors P λ ǫ µ ( λ, q ) ǫ ∗ ν ( λ, q ) = − g µν + q µ q ν q .In the low energy limit ω →
0, the T N ( ω, ) is equivalent to the Born term T Born N ( ω, ). Now Itake into account the Born terms at the phenomenological side, T N ( ω ) = T Born N ( ω ) + a (cid:16) m D/D /D ∗ /D − ω (cid:17) + bm D/D /D ∗ /D − ω + cs − ω , (15)with the constraint am D/D /D ∗ /D + bm D/D /D ∗ /D + cs = 0 . (16)The contributions from the intermediate spin- charmed baryon states are zero in the soft-limit q µ → charmed baryon states incalculating the Born terms,( D/D /D ∗ /D ) ( c ¯ u ) + p ( uud ) or n ( udd ) −→ Λ + c , Σ + c ( cud ) or Σ c ( cdd ) , ( D/D /D ∗ /D ) + ( c ¯ d ) + p ( uud ) or n ( udd ) −→ Σ ++ c ( cuu ) or Λ + c , Σ + c ( cud ) , (17)where M Λ c = 2 .
286 GeV and M Σ c = 2 .
454 GeV [22]. I can take M H ≈ . H means either Λ + c , Σ + c , Σ ++ c or Σ c . In the case of the bottom baryons, I takethe approximation M H = M Σ b + M Λ b ≈ . T Born N ( ω, ) = 2 m N ( m H + m N )[ ω − ( m H + m N ) ] [ ω − m D ] (cid:18) f D m D g DNH m c (cid:19) , (18)for the current η ( x ), T Born N ( ω, ) → T Born N ( ω, ) (with m N → − m N , D → D ) , (19)for the current η ( x ), T Born N ( ω, ) = 2 m N ( m H + m N )[ ω − ( m H + m N ) ] [ ω − m D ∗ ] ( f D ∗ m D ∗ g D ∗ NH ) , (20)for the current η µ ( x ), T Born N ( ω, ) → T Born N ( ω, ) (with m N → − m N , D ∗ → D ) , (21)5or the current η µ ( x ), where the g D/D /D ∗ /D NH denote the strong coupling constants g D/D /D ∗ /D N Λ c and g D/D /D ∗ /D N Σ c . On the other hand, there are no inelastic channels for the ( ¯ D/ ¯ D / ¯ D ∗ / ¯ D ) N and ( ¯ D/ ¯ D / ¯ D ∗ / ¯ D ) − N interactions, and T Born N (0) = 0. In calculations, I have used the followingdefinitions for the hadronic coupling constants, h Λ c / Σ c ( p − q ) | D ( − q ) N ( p ) i = g Λ c / Σ c DN U Λ c / Σ c ( p − q ) iγ U N ( p ) , h Λ c / Σ c ( p − q ) | D ( − q ) N ( p ) i = g Λ c / Σ c D N U Λ c / Σ c ( p − q ) U N ( p ) , h Λ c / Σ c ( p − q ) | D ∗ ( − q ) N ( p ) i = U Λ c / Σ c ( p − q ) g Λ c / Σ c D ∗ N ǫ + i g T Λ c / Σ c D ∗ N M N + M Λ c / Σ c σ αβ ǫ α q β ! U N ( p ) , h Λ c / Σ c ( p − q ) | D ( − q ) N ( p ) i = U Λ c / Σ c ( p − q ) g Λ c / Σ c D N ǫ + i g T Λ c / Σ c D N M N + M Λ c / Σ c σ αβ ǫ α q β ! γ U N ( p ) , (22)where the U N and U Λ c / Σ c are the Dirac spinors of the nucleon and the charmed baryons Λ c / Σ c ,respectively. In the limit q µ →
0, the strong coupling constants g T Λ c / Σ c D ∗ N and g T Λ c / Σ c D N have nocontributions.For example, near the thresholds, the D ∗ N can translate to the DN , D ∗ N , π Σ c , η Λ c , etc,we can take into account the intermediate baryon-meson loops or the re-scattering effects withthe Bethe-Salpeter equation to obtain the full D ∗ N → D ∗ N scattering amplitude, and generatehigher baryon states dynamically [23]. We can saturate the full D ∗ N → D ∗ N scattering amplitudewith the tree-level Feynman diagrams describing the exchanges of the higher resonances Λ c (2595),Σ c ( − ), etc. While in other coupled-channels analysis, the Λ c (2595) emerges as a DN quasi-bound state rather than a D ∗ N quasi-bound state [23]. The translations D ∗ N to the groundstates Λ c and Σ c are favored in the phase-space, as the Λ c (2595) and Σ c ( − ) with J P = − havethe average mass m H ′ ≈ . m H ′ > s , I can absorb the high resonancesinto the continuum states in case the high resonances do not dominate the QCD sum rules. Incalculations, I observe that the mass-shift δm D ∗ does not sensitive to contributions of the groundstates Λ c and Σ c , the contributions from the spin- higher resonances maybe even smaller. In thisarticle, I neglect the intermediate baryon-meson loops, their effects are absorbed into continuumcontributions.At the low nuclear density, the condensates hOi ρ N in the nuclear matter can be approximatedas hOi ρ N = hOi + ρ N m N hOi N , (23)based on the Fermi gas model, where the hOi and hOi N denote the vacuum condensates andnuclear matter induced condensates, respectively [11]. I neglect the terms proportional to p F , p F , p F , · · · at the normal nuclear matter with the saturation density ρ N = ρ = p F π , as the Fermimomentum p F = 0 .
27 GeV is a small quantity [11].I carry out the operator product expansion to the nuclear matter induced condensates ρ N m N hOi N up to dimension-5 at the large space-like region in the nuclear matter, and take into account theone-loop corrections to the quark condensate h ¯ qq i N . I insert the following term12! ig s Z d D y ¯ ψ ( y ) γ µ ψ ( y ) λ a G aµ ( y ) ig s Z d D z ¯ ψ ( z ) γ ν ψ ( z ) λ b G bν ( z ) , (24)with the dimension D = 4 − ǫ , into the correlation functions T N ( q ) and T Nµν ( q ) firstly, where the ψ denotes the quark fields, the G aµ denotes the gluon field, the λ a denotes the Gell-Mann matrix, thencontract the quark fields with Wick theorem, and extract the quark condensate h ¯ qq i N according tothe formula h N | q iα q jβ | N i = − h ¯ qq i N δ ij δ αβ to obtain the perturbative corrections α s h ¯ qq i N , where6igure 1: The perturbative O ( α s ) corrections to the quark condensate h ¯ qq i N .the i and j are color indexes and the α and β are Dirac spinor indexes. There are six Feynmandiagrams make contributions, see Fig.1. Now I calculate the first diagram explicitly for the current η ( x ) in Fig.1,2 T ( α s , N ( q ) = − Tr( λ a λ b ) h ¯ qq i N g s µ ǫ i (2 π ) D Z d D k Tr (cid:26) iγ i k γ α γ β i k iγ i q + k − m c − iδ ab g αβ k (cid:27) = − Dm c h ¯ qq i N g s µ ǫ π ) D i ∂∂t ( − πi ) πi Z ∞ m c ds R d D kδ (cid:0) k − t (cid:1) δ (cid:0) ( k + q ) − m c (cid:1) s − q | t =0 = − Dm c h ¯ qq i N g s µ ǫ [1 + ǫ (log 4 π − γ E )]12 π Z ∞ m c ds s − q s + m c s − ǫ ( s − m c ) ǫ , (25)where I have used Cutkosky’s rule to obtain the QCD spectral density. There exists infrareddivergence at the end point s = m c . It is difficult to carry out the integral over s , I can performthe Borel transform B M firstly, then carry out the integral over s , B M T ( α s , N ( q ) = − Dm c h ¯ qq i N g s µ ǫ [1 + ǫ (log 4 π − γ E )]12 π M Z ∞ m c ds s + m c s − ǫ ( s − m c ) ǫ exp (cid:16) − sM (cid:17) = m c h ¯ qq i N g s π M exp (cid:18) − m c M (cid:19) (cid:18) ǫ − log 4 π + γ E (cid:19) + m c h ¯ qq i N g s π M Γ (cid:18) , m c M (cid:19) − m c h ¯ qq i N g s π M exp (cid:18) − m c M (cid:19) + m c h ¯ qq i N g s π M exp (cid:18) − m c M (cid:19) log m c µ M , (26)where Γ(0 , x ) = e − x Z ∞ dt t + x e − t . (27)Other diagrams are calculated analogously, I regularize the divergences in D = 4 − ǫ dimension,then remove the ultraviolet divergences through renormalization and absorb the infrared diver-gences into the quark condensate h ¯ qq i N .I calculate the contributions of other condensates at the tree level, the calculations are straight-7orward and cumbersome. In calculations, I use the following formulas, h q α ( x )¯ q β (0) i N = − (cid:20)(cid:18) h ¯ qq i N + x µ h ¯ qD µ q i N + 12 x µ x ν h ¯ qD µ D ν q i N + · · · (cid:19) δ αβ + (cid:18) h ¯ qγ λ q i N + x µ h ¯ qγ λ D µ q i N + 12 x µ x ν h ¯ qγ λ D µ D ν q i N + · · · (cid:19) γ λαβ (cid:21) , (28)and h g s q iα ¯ q jβ G aµν i N = − λ aij n h g s ¯ qσGq i N [ σ µν + i ( u µ γ ν − u ν γ µ ) u ] αβ + h g s ¯ q uσGq i N [ σ µν u + i ( u µ γ ν − u ν γ µ )] αβ − h ¯ quDuDq i N [ σ µν + 2 i ( u µ γ ν − u ν γ µ ) u ] αβ o , (29)where D µ = ∂ µ − ig s λ a G aµ , h ¯ qγ µ q i N = h ¯ q uq i N u µ , h ¯ qD µ q i N = h ¯ quDq i N u µ = 0 , h ¯ qγ µ D ν q i N = 43 h ¯ q uuDq i N (cid:18) u µ u ν − g µν (cid:19) , h ¯ qD µ D ν q i N = 43 h ¯ quDuDq i N (cid:18) u µ u ν − g µν (cid:19) − h g s ¯ qσGq i N ( u µ u ν − g µν ) , h ¯ qγ λ D µ D ν q i N = 2 h ¯ q uuDuDq i N (cid:20) u λ u µ u ν −
16 ( u λ g µν + u µ g λν + u ν g λµ ) (cid:21) − h g s ¯ q uσGq i N ( u λ u µ u ν − u λ g µν ) , (30)and h G aαβ G bµν i N = δ ab h GG i N ( g αµ g βν − g αν g βµ ) + O (cid:0) h E + B i N (cid:1) . (31)Once analytical results at the level of quark-gluon degree’s of freedom are obtained, then I set ω = q , and take the quark-hadron duality below the continuum threshold s , and perform theBorel transform with respect to the variable Q = − ω , finally obtain the following QCD sumrules: a C a + b C b = C f , (32) C a = 1 M exp (cid:18) − m D M (cid:19) − s m D exp (cid:16) − s M (cid:17) ,C b = exp (cid:18) − m D M (cid:19) − s m D exp (cid:16) − s M (cid:17) , (33) C f = 2 m N ( m H + m N )( m H + m N ) − m D (cid:18) f D m D g DNH m c (cid:19) (cid:26)(cid:20) M − m D − ( m H + m N ) (cid:21) exp (cid:18) − m D M (cid:19) + 1( m H + m N ) − m D exp (cid:18) − ( m H + m N ) M (cid:19)(cid:27) − m c h ¯ qq i N (cid:26) α s π (cid:20) − m c M − (cid:18) − m c M (cid:19) log m c µ − (cid:18) , m c M (cid:19) exp (cid:18) m c M (cid:19)(cid:21)(cid:27) exp (cid:18) − m c M (cid:19) + 12 (cid:26) − (cid:18) − m c M (cid:19) h q † iD q i N + 4 m c M (cid:18) − m c M (cid:19) h ¯ qiD iD q i N + 112 h α s GGπ i N (cid:27) exp (cid:18) − m c M (cid:19) , (34)8or the current η ( x ), C i → C i (with m N → − m N , m c → − m c , D → D ) , (35)for the current η ( x ), C a = 1 M exp (cid:18) − m D ∗ M (cid:19) − s m D ∗ exp (cid:16) − s M (cid:17) ,C b = exp (cid:18) − m D ∗ M (cid:19) − s m D ∗ exp (cid:16) − s M (cid:17) , (36) C f = 2 m N ( m H + m N )( m H + m N ) − m D ∗ ( f D ∗ m D ∗ g D ∗ NH ) (cid:26)(cid:20) M − m D ∗ − ( m H + m N ) (cid:21) exp (cid:18) − m D ∗ M (cid:19) + 1( m H + m N ) − m D ∗ exp (cid:18) − ( m H + m N ) M (cid:19)(cid:27) − m c h ¯ qq i N (cid:26) α s π (cid:20) − m c M + 23 (cid:18) m c M (cid:19) log m c µ − m c M Γ (cid:18) , m c M (cid:19) exp (cid:18) m c M (cid:19)(cid:21)(cid:27) exp (cid:18) − m c M (cid:19) + 12 (cid:26) − h q † iD q i N m c h q † iD q i N M + 2 m c h ¯ qg s σGq i N M + 16 m c h ¯ qiD iD q i N M − m c h ¯ qiD iD q i N M − h α s GGπ i N (cid:27) exp (cid:18) − m c M (cid:19) , (37)for the current η µ ( x ), C i → C i (with m N → − m N , m c → − m c , D ∗ → D ) , (38)for the current η µ ( x ), where i = a, b, f . In this article, I neglect the contributions from theheavy quark condensates h ¯ QQ i , h ¯ QQ i = − πm Q h α s GGπ i up to the order O ( α s ) (here I count thecondensate h α s GGπ i as of the order O ( α s )), the heavy quark condensates have practically no effecton the polarization functions, for detailed discussions about this subject, one can consult Ref.[9].In Ref.[25], Buchheim, Hilger and Kampfer study the contributions of the condensates involve theheavy quarks in details, the results indicate that those condensates are either suppressed by theheavy quark mass m Q or by the additional factor α s π (or g s / (4 π ) ). Neglecting the in-mediumeffects on the heavy quark condensates cannot affect the predictions remarkably, as the maincontributions come from the terms h ¯ qq i N .Differentiate above equation with respect to τ = M , then eliminate the parameter b ( a ), I canobtain the QCD sum rules for the parameter a ( b ), a = C f (cid:0) − ddτ (cid:1) C b − C b (cid:0) − ddτ (cid:1) C f C a (cid:0) − ddτ (cid:1) C b − C b (cid:0) − ddτ (cid:1) C a ,b = C f (cid:0) − ddτ (cid:1) C a − C a (cid:0) − ddτ (cid:1) C f C b (cid:0) − ddτ (cid:1) C a − C a (cid:0) − ddτ (cid:1) C b . (39)With the simple replacements m c → m b , D/D /D ∗ /D → B/B /B ∗ /B , Λ c → Λ b and Σ c → Σ b ,I can obtain the corresponding the QCD sum rules for the bottom mesons in the nuclear matter. At the normal nuclear matter with the saturation density ρ N = ρ = p F π , where the Fermimomentum p F = 0 .
27 GeV is a small quantity, the condensates hOi ρ N in the nuclear matter can9e approximated as hOi ρ N = hOi + ρ N m N hOi N , the terms proportional to p F , p F , p F , · · · can beneglected safely, where the hOi = h |O| i and hOi N = h N |O| N i denote the vacuum condensatesand nuclear matter induced condensates, respectively [11].The input parameters at the QCD side are taken as ρ N = (0 .
11 GeV) , h ¯ qq i N = σ N m u + m d (2 m N ), h α s GGπ i N = − .
65 GeV(2 m N ), σ N = 45 MeV, m u + m d = 12 MeV, h q † iD q i N = 0 .
18 GeV(2 m N ), h ¯ qg s σGq i N = 3 . (2 m N ), h ¯ qiD iD q i N + h ¯ qg s σGq i N = 0 . (2 m N ), m N = 0 .
94 GeV[10], m c = (1 . ± .
1) GeV, m b = (4 . ± .
1) GeV, α s = 0 .
45 and µ = 1 GeV. If we take thenormalization h N ( p ) | N ( p ′ ) i = (2 π ) δ ( p − p ′ ), then hOi ρ N = hOi + ρ N hOi N , the unit 2 m N in the brackets in the values of the condensates h ¯ qq i N , h α s GGπ i N , · · · disappears. I choose thevalues of the nuclear matter induced condensates determined in Ref.[10], which are still widelyused in the literatures. Although the values of some condensates are updated, those condensatesare irrelevant to the present work. The updates focus on the four-quark condensate [26]. In thisarticle, I take into account the condensates up to dimension-5, the four-quark condensates haveno contributions, the dominant contributions come from the nuclear matter induced condensate h ¯ qq i N , h ¯ qq i N = σ N m u + m d (2 m N ). The value m u + m d = 12 MeV is obtained from the famous Gell-Mann-Oakes-Renner relation at the energy scale µ = 1 GeV, while the value σ N = 45 MeV is stillwidely used [26].The parameters at the hadronic side are taken as m D = 1 .
870 GeV, m B = 5 .
280 GeV, m D =2 .
355 GeV, m B = 5 .
740 GeV, m D ∗ = 2 .
010 GeV, m B ∗ = 5 .
325 GeV, m D = 2 .
420 GeV, m B =5 .
750 GeV, f D = 0 .
210 GeV, f B = 0 .
190 GeV, f D = 0 . m c m D GeV, f B = 0 . m b m B GeV, f D ∗ =0 .
270 GeV, f B ∗ = 0 .
195 GeV, f D = 0 .
305 GeV, f B = 0 .
255 GeV, s D = (6 . ± .
5) GeV , s B =(33 . ± .
0) GeV , s D ∗ = (6 . ± .
5) GeV , s B ∗ = (35 . ± .
0) GeV , s D = (8 . ± .
5) GeV , s B = (39 . ± .
0) GeV , s D = (8 . ± .
5) GeV and s B = (39 . ± .
0) GeV , which are determinedby the conventional two-point correlation functions using the QCD sum rules [14, 27]. I neglectthe uncertainties of the decay constants to avoid double counting as the main uncertainties of thedecay constants originate from the uncertainties of the continuum threshold parameters s .The value of the strong coupling constant g DN Λ c is g Λ c DN = 6 .
74 from the QCD sum rules[28], while the average value of the strong coupling constants g Λ c DN and g Σ c DN from the light-coneQCD sum rules is g Λ cDN + g Σ cDN = 6 .
775 [29], those values are consistent with each other. Theaverage value of the strong coupling constants g Λ c D ∗ N and g Σ c D ∗ N from the light-cone QCD sumrules is g Λ cD ∗ N + g Σ cD ∗ N = 3 .
86 [29]. In this article, I take the approximation g DN Λ c ≈ g DN Σ c ≈ g BN Λ b ≈ g BN Σ b ≈ g D N Λ c ≈ g D N Σ c ≈ g B N Λ b ≈ g B N Σ b ≈ .
74 and g Λ c D ∗ N ≈ g Σ c D ∗ N ≈ g Λ c D N ≈ g Σ c D N ≈ g Λ b B ∗ N ≈ g Σ b B ∗ N ≈ g Λ b B N ≈ g Σ b B N ≈ . M , respectively. From the figures, I cansee that there appear platforms. In this article, I choose the Borel parameters M according tothe criterion that the uncertainties originate from the Borel parameters M are negligible. Thevalues of the Borel parameters M are shown explicitly in Table 1. From Figs.2-3 and Table 1,I can see that the Borel parameters M in the QCD sum rules for the mass-shift δm and decay-constant-shift δf of the same meson are different. It is not un-acceptable, as the mass-shift δm and decay-constant-shift δf come from different QCD sum rules, not coupled QCD sum rules, seeEq.(39), the platforms maybe appear in different places in different QCD sum rules.I can obtain the shifts of the masses and decay constants of the heavy mesons in the nuclearmatter in the Borel windows, which are shown explicitly in Table 2. From the Table 2, I canobtain the fractions of the shifts δm D/D ∗ /D /D m D/D ∗ /D /D ≤ δf D/D ∗ /D /D f D/D ∗ /D /D ≤ δm B/B ∗ /B /B m B/B ∗ /B /B =(5 − δf B/B ∗ /B /B f B/B ∗ /B /B = (25 − m c h ¯ qq i N and m b h ¯ qq i N . From Table3, I can see that the next-to-leading order corrections α s h ¯ qq i N are important. In the case ofthe shifts δm B ∗ /B m B ∗ /B and δf B ∗ /B f B ∗ /B , the next-to-leading order contributions α s h ¯ qq i N and the leadingorder contributions h ¯ qq i N are almost equivalent. In this article, I choose the special energy scale10 D m ( M e V ) M (GeV ) I II D * m ( M e V ) M (GeV ) I II D m ( M e V ) M (GeV ) I II D m ( M e V ) M (GeV ) I II
15 20 25 30 35 40 45-600-500-400-300-200-100 B m ( M e V ) M (GeV ) I II
15 20 25 30 35 40 45-800-640-480-320-1600 B * m ( M e V ) M (GeV ) I II
15 20 25 30 35 40 45080160240320400 B m ( M e V ) M (GeV ) I II
15 20 25 30 35 40 450120240360480600 B m ( M e V ) M (GeV ) I II
Figure 2: (Color online) The shifts of the masses of the heavy mesons in the nuclear matter withvariations of the Borel parameter M , the I (II) denotes contributions up to the next-to-leadingorder (leading order) are included. 11 D f ( M e V ) M (GeV ) I II D * f ( M e V ) M (GeV ) I II D f ( M e V ) M (GeV ) I II D f ( M e V ) M (GeV ) I II
10 15 20 25 30 35 40-80-64-48-32-160 B f ( M e V ) M (GeV ) I II
15 20 25 30 35 40 45-120-96-72-48-240 B * f ( M e V ) M (GeV ) I II
15 20 25 30 35 40 4501428425670 B f ( M e V ) M (GeV ) I II
15 20 25 30 35 40 4504080120160 B f ( M e V ) M (GeV ) I II
Figure 3: (Color online) The shifts of the decay constants of the heavy mesons in the nuclearmatter with variations of the Borel parameter M , the I (II) denotes contributions up to thenext-to-leading order (leading order) are included.12 m D δm D ∗ δm D δm D δm B δm B ∗ δm B δm B M . − . . − . . − . . − . −
33 30 −
34 32 −
36 32 − δf D δf D ∗ δf D δf D δf B δf B ∗ δf B δf B M . − . . − . . − . . − . −
29 27 −
31 30 −
34 31 − . δm D δm D ∗ δm D δm D δm B δm B ∗ δm B δm B NLO − −
102 80 97 − −
687 295 522LO − −
70 54 66 − −
340 209 260[13] − − − δf D δf D ∗ δf D δf D δf B δf B ∗ δf B δf B NLO − −
26 11 31 − −
111 56 134LO − −
18 7 21 − −
55 39 67[15] − − µ = 1 GeV. The logarithm log m b µ in the next-to-leading contributions is very large and enhancesthe next-to-leading contributions greatly. Although the nuclear matter induced condensates evolvewith the renormalization group equation, their evolving behaviors with the energy scales are notwell known, as this subject has not been studied in details yet at the present time. A larger energyscale µ can lead to smaller logarithm log m b µ therefore more reasonable predictions. In Table 4, Ipresent the main uncertainties, which originate from the uncertainties of the heavy quark massesand the continuum threshold parameters.The mass-shifts of the negative (positive) parity mesons are negative (positive), the decaysof the high charmonium states to the D ¯ D and D ∗ ¯ D ∗ ( D ¯ D and D ¯ D ) pairs are enhanced(suppressed) in the phase space, and we should take into account those effects carefully in studyingthe production of the J/ψ so as to identifying the quark-gluon plasmas. δm D m D δm D ∗ m D ∗ δm D m D δm D m D δm B m B δm B ∗ m B ∗ δm B m B δm B m B NLO − −
5% 3% 4% − −
13% 5% 9%LO − −
3% 2% 3% − −
6% 4% 5% δf D f D δf D ∗ f D ∗ δf D f D δf D f D δf B f B δf B ∗ f B ∗ δf B f B δf B f B NLO − −
10% 6% 10% − −
57% 24% 53%LO − −
7% 4% 7% − −
28% 17% 26%Table 3: The fractions of the shifts of the masses and decay constants of the heavy mesons inthe nuclear matter, where the NLO (LO) denotes contributions up to the next-to-leading order(leading order) are included. 13 ( δm D ) δ ( δm D ∗ ) δ ( δm D ) δ ( δm D ) δ ( δm B ) δ ( δm B ∗ ) δ ( δm B ) δ ( δm B ) δm Q ± ± ± ± ± ± ± ± δs ± ± ± ± ± ± ± ± δ ( δf D ) δ ( δf D ∗ ) δ ( δf D ) δ ( δf D ) δ ( δf B ) δ ( δf B ∗ ) δ ( δf B ) δ ( δf B ) δm Q ± ± ± ± ± ± ± ± δs ± ± ± ± ± ± ± ± a D a D ∗ a D a D a B a B ∗ a B a B NLO − . − . . . − . − . . . − . − . . . − . − . . . Qq and ¯ Qiγ q (also ¯ Qγ µ q and ¯ Qγ µ γ q ) are mixed with each other under thechiral transformation q → e iαγ q , the currents ¯ Qq , ¯ Qiγ q , ¯ Qγ µ q , ¯ Qγ µ γ q are not conserved inthe limit m q →
0, it is better to take the doublets (
D, D ) and ( D ∗ , D ) as the parity-doubletsrather than the chiral-doublets. The quark condensate h ¯ qq i ρ N serves as the order parameter, andundergoes reduction in the nuclear matter, the chiral symmetry is partially restored; however,there appear new medium-induced condensates, which also break the chiral symmetry. In thisarticle, the hOi N are companied by the heavy quark masses m Q , m Q or m Q , the net effects cannotwarrant that the chiral symmetry is monotonously restored with the increase of the ρ N . Whenthe ρ N is large enough, the order parameter h ¯ qq i ρ N →
0, the chiral symmetry is restored, theFermi gas approximation for the nuclear matter breaks down, and the parity-doublets maybe havedegenerated masses approximately. In this article, I study the parity-doublets at the low ρ N , themass breaking effects of the parity-doublets maybe even larger, see Table 2. We expect that smallermass splitting of the parity-doublets at the high nuclear density is favored, however, larger masssplitting of the parity-doublets at the lower nuclear density cannot be excluded. In Refs.[16, 17],the mass center m P of the pseudoscalar mesons increases in the nuclear matter while the masscenter m S of the scalar mesons decreases in the nuclear matter, the mass breaking effect m S − m P of the parity-doublets is smaller than that in the vacuum.In Table 5, I show the scattering lengths a D , a D ∗ , a D , a D , a B , a B ∗ , a B , a B explicitly, the a D , a D ∗ , a B , a B ∗ are negative, which indicate the interactions DN , D ∗ N , BN , B ∗ N are attractive,the a D , a D , a B , a B are positive, which indicate the interactions D N , D N , B N , B N arerepulsive. It is difficult (possible) to form the D N , D N , B N , B N ( DN , D ∗ N , BN , B ∗ N )bound states. In Ref.[19], Zhang studies the S -wave DN and D ∗ N bound states with the QCDsum rules, the numerical results indicate that the Σ c (2800) and Λ c (2940) can be assigned to bethe S -wave DN state with J P = − and the S -wave D ∗ N state with J P = − , respectively.In the present work and Refs.[13, 14, 15], the correlation functions are divided into a vacuumpart and a static one-nucleon part, and the nuclear matter induced effects are extracted explicitly;while in Refs.[16, 17], the pole terms of (or ground state contributions to) the hadronic spectraldensities of the whole correlation functions are parameterized as ∆Π( ω ) = F + δ ( ω − m + ) − F − δ ( ω + m − ), where m ± = m ± ∆ m and F ± = F ± ∆ F , and QCD sum rules for the mass center m andthe mass splitting ∆ m are obtained. In the leading order approximation, the present predictionsof the δm D and δm B are compatible with that of Refs.[13, 15] and differ greatly from that of14 .6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.20.30.40.50.60.70.80.91.01.11.2 F r a c t i on s M (GeV ) I II III IV
Figure 4: (Color online) The contributions of the perturbative term (I) and quark condensate term(II) in the QCD sum rules for the D mesons in the vacuum. Furthermore, I show the mass m D (III)and decay constant f D (IV) explicitly, which are normalized to be 1 at the value M = 1 . .Refs.[16, 17], see Table 2. The values obtained from the QCD sum rules depend heavily on theBorel windows, the values extracted from different Borel windows especially in different QCD sumrules maybe differ from each other greatly.In Refs.[16, 17], the authors study the masses of the heavy mesons in the nuclear matter directlyby including both the vacuum part and the static one-nucleon part in the QCD sum rules, thenthe continuum contributions are well approximated by ρ QCD ( ω ) θ (cid:16) ω − ω ± (cid:17) , where the ω ± arethe continuum threshold parameters, it is one of the advantages of Refs.[16, 17]. However, theydefine the moments S n ( M ) to study the mass-shifts, S n ( M ) = Z ω +0 ω − dωω n ∆Π( ω ) exp (cid:18) − ω M (cid:19) , (40)the odd moment o = S ( M ) and the even moment e = S ( M ), then obtain dod (1 /M ) = − S ( M )and ded (1 /M ) = − S ( M ) by assuming the F ± and m ± are independent on the Borel parameters atthe phenomenological side. In fact, dod (1 /M ) = − S ( M ) and ded (1 /M ) = − S ( M ) at the operatorproduct expansion side according to the QCD spectral densities ∆Π( ω ), which depend on the Borelparameters explicitly, the approximations dod (1 /M ) = − S ( M ) and ded (1 /M ) = − S ( M ) lead toundetermined uncertainties.In Refs.[16, 17], the perturbative O ( α s ) corrections to the perturbative terms are taken intoaccount. In the QCD sum rules for the pseudoscalar D mesons in the vacuum, if we take intoaccount the perturbative O ( α s ) corrections to the perturbative term and vacuum condensate term,the two criteria (pole dominance and convergence of the operator product expansion) of the QCDsum rules leads to the Borel window M = (1 . − .
8) GeV , the resulting predictions of themass m D and decay constant f D are consistent with the experimental data. In Fig.4, I plotthe contributions of the perturbative term and quark condensate term in the operator productexpansion. From the figure, I can see that the main contributions come from the perturbativeterm, the quark condensate h ¯ qq i plays a less important role.The modifications of the condensates in the nuclear matter are mild, for example, h ¯ qq i ρ N ≈ . h ¯ qq i , while the perturbative contributions are not modified (or modified slightly by introduc-ing a minor splitting ∆ s , ω ± = s ± ∆ s ) by the nuclear matter. If we turn on the in-mediumeffects, the contributions of the quark condensate are even smaller, the Borel windows are deter-15ined dominantly by the perturbative terms [16, 17]. If the perturbative O ( α s ) corrections to theperturbative terms are also included, the contributions of the perturbative are even larger [27], theQCD sum rules are dominated by the perturbative terms, which are not (or slightly) affected bythe nuclear matter. It is not favored to extract the mass-shifts in the nuclear matter, and impairsthe predictive ability.In the present work and Refs.[13, 14, 15], the correlation functions are divided into the vacuumpart and the static one-nucleon part, which are of the orders O (0) and O ( ρ N ), respectively. Wecan obtain independent QCD sum rules from the two parts respectively. The QCD sum rulescorrespond to the orders O (0) and O ( ρ N ) respectively can have quite different Borel parameters.In this article, I separate the nuclear matter induced effects unambiguously, study the QCD sumrules correspond to the order O ( ρ N ), and determine the Borel parameters by the criteria of theQCD sum rules.In the conventional QCD sum rules, we usually choose the Borel parameters M to satisfy thefollowing three criteria: · Pole dominance at the phenomenological side; · Convergence of the operator product expansion; · Appearance of the Borel platforms.In the present work and Refs.[13, 14, 15], the nuclear matter induced effects are extractedexplicitly, the resulting QCD sum rules are not contaminated by the contributions of the vacuumpart, the Borel windows are determined completely by the nuclear matter induced effects, it isthe advantage. As the QCD spectral densities are of the form δ ( ω − m Q ), we have to take thehadronic spectral densities to be the form δ ( ω − m H ) and model the continuum contributionswith the function δ ( ω − s ), and determine the s by some constraints, see Eq.(16), where the H denotes the ground state and excited state heavy mesons. In this article, I attribute the higherexcited states to the continuum contributions, the δ -type hadronic spectral densities make sense. Sothe pole dominance at the phenomenological side can be released as the continuum contributionsare already taken into account. Furthermore, I expect that the couplings of the interpolatingcurrents to the excited states are more weak than that to the ground states, the uncertaintiesoriginate from continuum contributions are very small. For example, the decay constants of thepseudoscalar mesons π (140) and π (1300) have the hierarchy f π (1300) ≪ f π (140) from the Dyson-Schwinger equation [30], the lattice QCD [31], the QCD sum rules [32], etc, or from the experimentaldata [33].In the present work and Refs.[13, 14, 15], large Borel parameters are chosen to warrant theconvergence of the operator product expansion and to obtain the Borel platforms, and small Borelparameters cannot lead to platforms. In the Borel windows, where the platforms appear, the maincontributions come from terms h ¯ qq i N , the operator product expansion is well convergent. Thecriteria and can be satisfied. The continuum contributions are not suppressed efficiently forlarge Borel parameters compared to that for small Borel parameters. In calculations, I observethat the predictions are insensitive to the s , the uncertainties originate from the continuumthreshold parameters s are very small in almost all cases, the large Borel parameters make sense.Furthermore, the continuum contributions are already taken into account. On the other hand, fromEqs.(8-9) and Eqs.(12-13), we can see that the mass-shifts δm D/D /D ∗ /D and decay constant shifts δf D/D /D ∗ /D reduce to zero in the limit ρ N →
0, the QCD sum rules correspond to the nuclearmatter induced effects decouple, their Borel parameters (irrespective of large or small) are alsoirrelevant to the ones in the QCD sum rules for the vacuum part of the correlation functions. Sothe present predictions are sensible.The predictions depend on the in-medium hadronic spectral functions [34], for example, thereare two generic prototypes of the in-medium spectral functions for the ρ meson, they differ indetails at the low mass end of the spectrum. The Klingl-Kaise-Weise spectral function emphasizesthe role of chiral in-medium ππ interactions [35], while the Rapp-Wambach spectral functionfocuses on the role of nucleon-hole, ∆(1232)-hole and N ∗ (1520)-hole excitations [36]. Both of thespectral functions account quite well for the low-mass enhancements observed in dilepton spectra16rom high-energy nuclear collisions. However, the QCD sum rules analysis of the lowest spectralmoments reveals qualitative differences with respect to their Brown-Rho scaling properties [34].If the simple spectral densities F δ ( ω − M P/V ) analogous to the ones in Refs.[16, 17] are taken,where the P denotes the pseudoscalar mesons π , η c , the V denotes the vector mesons ρ , ω , φ , J/ψ ,the F denotes the constant pole residues, the in-medium mass-shifts δM P/V are smaller than zeroqualitatively [37]. I expect that the S-wave mesons q ′ ¯ q , c ¯ q , c ¯ c with the spin-parity J P = 0 − (or1 − ) have analogous in-medium mass-shifts, at least qualitatively. Further studies based on moresophisticated hadronic spectral densities are needed.In fact, there are controversies about the mass-shifts of the D and B mesons in the nuclearmatter, some theoretical approaches indicate negative mass-shifts [23], while others indicate pos-itive mass-shifts [38]. The different predictions originate mainly from whether or not the heavypseudoscalar and heavy vector mesons are treated on equal footing in the coupled-channel ap-proaches. If we obtain the meson-baryon interaction kernel by treating the heavy pseudoscalarand heavy vector mesons on equal footing as required by heavy quark symmetry, the mass-shift δM D is negative [23], which is consistent with the present work; furthermore, the attractive D-nucleus interaction can lead to the formation of D -nucleus bound states, which can be confrontedto the experimental data in the future directly [39].The upcoming FAIR project at GSI provides the opportunity to study the in-medium propertiesof the charmoniums or charmed hadrons for the first time, however, the high mass of charmedhadrons requires a high momentum in the antiproton beam to produce them, the conditions forobserving in-medium effects seem unfavorable, as the hadrons sensitive to the in-medium effectsare either at rest or have a small momentum relative to the nuclear medium. We have to findprocesses that would slow down the charmed hadrons inside the nuclear matter, but this requiresmore detailed theoretical studies. Further theoretical studies on the reaction dynamics and on theexploration of the experimental ability to identify more complicated processes are still needed. In this article, I divide the two-point correlation functions of the scalar, pseudoscalar, vector andaxialvector currents in the nuclear matter into two parts, i.e. the vacuum part and the staticone-nucleon part, then study the in-medium modifications of the masses and decay constants byderiving QCD sum rules from the static one-nucleon part of the two-point correlation functions.In the operator product expansion, I calculate the contributions of the nuclear matter inducedcondensates up to dimension 5, especially I calculate the next-to-leading order contributions ofthe in-medium quark condensate and obtain concise expressions, which also have applications instudying the mesons properties in the vacuum. In calculation, I observe that the next-to-leadingorder contributions of the in-medium quark condensate are very large and should be taken intoaccount.All in all, I study the properties of the scalar, pseudoscalar, vector and axialvector heavymesons with the QCD sum rules in a systematic way, and obtain the shifts of the masses anddecay constants in the nuclear matter. The numerical results indicate that the mass-shifts of thenegative parity and positive parity heavy mesons are negative and positive, respectively. For thepseudoscalar meson D , I obtain the prediction δM D <
0, which is in contrast to the predictionin Refs.[16, 17], where the mass-shift is positive δM D >
0. In Refs.[16, 17], the authors studythe masses of the heavy mesons in the nuclear matter directly by including both the vacuum partand static one-nucleon part with the QCD sum rules, and parameterize the spectral density of thewhole correlation functions by a simple function ∆Π( ω ) = F + δ ( ω − m + ) − F − δ ( ω + m − ). I discussthe differences between the QCD sum rules in the present work and that in Refs.[16, 17] in details,and show why I prefer the present predictions. In the present work and Refs.[13, 14, 15, 16, 17],the finite widths of the mesons in the nuclear matter are neglected, further studies based on themore sophisticated hadronic spectral densities by including the finite widths are needed.17s the masses of the heavy meson paries, such as the D ¯ D , D ∗ ¯ D ∗ , D ¯ D , D ¯ D are modifiedin the nuclear environment, we should take into account those effects carefully in studying theproduction of the J/ψ (and Υ) so as to identifying the quark-gluon plasmas. Furthermore, I studythe heavy-meson-nucleon scattering lengths as a byproduct, and obtain the conclusion qualitativelyabout the possible existence of the heavy-meson-nucleon bound states.
Acknowledgements
This work is supported by National Natural Science Foundation, Grant Numbers 11375063, andNatural Science Foundation of Hebei province, Grant Number A2014502017.
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