Relation between the mass modification of the heavy-light mesons and the chiral symmetry structure in dense matter
Masayasu Harada, Yong-Liang Ma, Daiki Suenaga, Yusuke Takeda
aa r X i v : . [ h e p - ph ] S e p Relation between the mass modification of the heavy-light mesons and the chiralsymmetry structure in dense matter
Masayasu Harada, ∗ Yong-Liang Ma, † Daiki Suenaga, ‡ and Yusuke Takeda § Department of Physics, Nagoya University, Nagoya, 464-8602, Japan Center for Theoretical Physics and College of Physics, Jilin University, Changchun, 130012, China (Dated: April 16, 2018)We point out that the study of the density dependences of the masses of heavy-light mesons givesome clues to the chiral symmetry structure in nuclear matter. We include the omega meson effect aswell as the sigma meson effect at mean field level on the density dependence of the masses of heavy-light mesons with chiral partner structure. It is found that the omega meson affects the masses ofthe heavy-light mesons and their antiparticles in the opposite way, while it affects the masses ofchiral partners in the same way. This is because the ω meson is sensitive to the baryon number ofthe light degrees included in the heavy-light mesons. We also show that the mass difference betweenchiral partners is proportional to the mean field of sigma, reflecting the partial restoration of chiralsymmetry in the nuclear matter. In addition to the general illustration of the density dependenceof the heavy-light meson masses, we consider two concrete models for nuclear matter, the paritydoublet model and skyrmion crystal model in the sense of mean field approximation. PACS numbers: 11.30.Rd, 14.40.Lb, 21.65.Jk
I. INTRODUCTION
Spontaneous chiral symmetry breaking is one of themost important properties of low energy QCD. It is ex-pected that the spontaneous chiral symmetry breakingcharacterized by non-zero value of the quark condensategenerates a part of hadron masses and causes the masssplitting between chiral partners. Then, schematically,hadron masses can be expressed as a sum of the chiralinvariant mass and the chiral non-invariant mass com-ing from the spontaneous chiral symmetry breaking. Forexample, for the nucleon mass, one has [1–6] m N = m + ∆( h ¯ qq i ) , where m is the chiral invariant mass and ∆ is the partof the mass that vanishes in the chiral symmetric phase.Naturally, it is interesting to ask how much amount of ahadron mass is generated by the chiral symmetry break-ing. An ideal environment to estimate the magnitudeof the hadron mass coming from the spontaneous chi-ral symmetry breaking is QCD at extreme condition inwhich the chiral symmetry is believed to be partially re-stored. In such an environment this can be accessed bystudying the temperature or/and, which will be done inthis work, the density dependence of hadron mass.In the nucleon sector, by using an effective model withparity doublet structure of baryons, it was found that ∼
70% of nucleon mass comes from chiral symmetrybreaking [1]. However, when the baryon as a topolog-ical soliton in the hidden local symmetry Lagrangian ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] is immersed in the dense matter which is treated asskyrmion matter, people found that the chiral invari-ant mass composes ∼
60% of nucleon mass [3, 4] whichroughly the same as that obtained based on the renor-malization group analysis of hidden local symmetry La-grangian with baryons [2]. In this paper, we study themedium modified mass splitting of heavy-light mesonswith chiral partner structure in which, it is widely ac-cepted that the mass splitting arises from the sponta-neous breaking of chiral symmetry [7, 8]. Such a kindof study can be tested in the planned experiments in J-PARC, FAIR and so on.Studying the properties of heavy-light mesons inmedium is also expected to give clues for understand-ing the chiral symmetry structure (see, e.g., Ref. [9] for areview). The medium modified heavy-light meson spec-trum has been studied by several groups in the liter-ature [10–16]. In Ref. [14], it was shown that the D meson ( J P = 0 − ) is mixed with D ∗ meson ( J P = 1 − )in the spin-isospin correlated matter, in which the mix-ing strength reflects the strength of the correlation. InRefs. [10, 15, 16], by regarding the D ∗ ( J P = 0 + ) and D ( J P = 1 + ) mesons as the chiral partners to D and D ∗ mesons, it was shown that the mass splitting of thechiral partner is reduced at high density and tempera-ture. In particular in Ref. [16], by replacing the chiralfield for pions interacting with the heavy mesons with itsmean field value obtained in the nuclear matter createdby the skyrmion crystal approach [3], it was shown thatthe masses of D and D ∗ increase with density while themasses of D ∗ and D decrease, and that their masses ap-proach the average value. In other word, the degeneratedmass (actually, the difference between the degeneratedmass and the heavy quark mass) agrees with the chiralinvariant mass, which is given by the average at vacuum.However, in the analyses of Refs [14, 16], only the pion isincluded in the light hadron sector, and effects of othermesons are not included. In particular, the analysis inRef. [11] shows that the ω meson increases the mass of D meson, while it decreases the ¯ D meson.In this paper, we study the effects of ω meson as wellas the σ meson on the density dependence of effectivemasses of heavy-light mesons. We show that the effectof the σ meson increases the masses of ( D , D ∗ ) heavyquark doublet, while it decreases the masses of the chiralpartners, i.e. ( D ∗ , D ) doublet, similarly to the analysisin Refs. [15, 16]. On the other hand, the effect of the ω meson increases the masses of both doublets. Neverthe-less, the difference between the masses of chiral partnersdecreases proportional to the mean field value of the σ meson, which reflects the partial chiral symmetry restora-tion. As a result, the masses of ( D , D ∗ ) doublet and ( D ∗ , D ) doublet approach a certain degenerate value. Differ-ently from the previous analysis, the degenerate valuedoes not agree with the average value at vacuum whichis the chiral invariant mass of those doublets. In thefollowing analysis, after a general consideration, we con-sider two concrete models, the parity doublet model [17]and skyrmion crystal model based on hidden local sym-metry [3] to give quantitative results. II. FRAMEWORK
For explaining the main point explicitly, we work inthe heavy quark limit and consider a simple chiral ef-fective model for a heavy meson multiplet of charmedmesons with J P = 0 − , 1 − , 0 + and 1 + based on the chi-ral doubling structure [7, 8]. Let H and G denote theheavy-quark doublets of heavy-light mesons with the ex-pression H = 1 + v µ γ µ (cid:2) D ∗ µ γ µ + iDγ (cid:3) , G = 1 + v µ γ µ (cid:2) D ∗ − iγ µ D ′ µ γ (cid:3) , (1)where v µ is the velocity of the heavy-light mesons, and D , D ∗ µ , D ∗ and D ′ µ are corresponding meson fields. Weintroduce the chiral fields H L,R as H R = 1 √ G + iHγ ] , H L = 1 √ G − iHγ ] , (2)which transform linearly under the chiral symmetry: H R,L → H
R,L g † R,L with g R,L ∈ SU(2)
R,L .The relevant Lagrangian used in the present calcula-tion is expressed as [16, 23] L = tr (cid:2) H L ( iv · ∂ ) ¯ H L ] + tr[ H R ( iv · ∂ ) ¯ H R (cid:3) − g ωDD Tr (cid:2) H L v µ ω µ ¯ H L + H R v µ ω µ ¯ H R (cid:3) + ∆ M f π tr (cid:2) H L M ¯ H R + H R M † ¯ H L (cid:3) − i g A f π tr (cid:2) H R γ γ µ ∂ µ M † ¯ H L − H L γ γ µ ∂ µ M ¯ H R (cid:3) , (3)where ∆ M is the mass difference between G and H dou-blets, f π is the pion decay constant, g A is a dimensionlessreal parameter. In the above Lagrangian, the omega me-son field ω µ is introduced as a chiral singlet and the field M is parametrized as M = σ + i P a =1 π a τ a with thePauli matrix τ a , which transforms as M → g L M g † R . Werewrite the effective Lagrangian (3) in terms of H and G fields as L = tr (cid:2) Gv µ ( i∂ µ + g ωDD ω µ ) ¯ G − Hv µ ( i∂ µ + g ωDD ω µ ) ¯ H (cid:3) + ∆ M f π tr (cid:2) G (cid:0) M + M † (cid:1) ¯ G + H (cid:0) M + M † (cid:1) ¯ H − iG (cid:0) M − M † (cid:1) γ ¯ H + iH (cid:0) M − M † (cid:1) γ ¯ G (cid:3) − ig A f π tr (cid:2) Gγ (cid:0) / ∂M † − / ∂M (cid:1) ¯ G − Hγ (cid:0) / ∂M † − / ∂M (cid:1) ¯ H + iG (cid:0) / ∂M † + / ∂M (cid:1) ¯ H − iH (cid:0) / ∂M † + / ∂M (cid:1) ¯ G (cid:3) . (4)Now, we replace the light meson fields by their meanfield values in medium. Here we consider the symmetricmatter only and assume no pion condensation, so that h M i = h σ i and h ∂ µ M i = 0. Note that the mean fieldvalue of σ at vacuum agrees with the pion decay constant, In Ref. [16], another term for the pionic interaction is included.In the present analysis we do not explicitly include the term, sinceone-pion interaction terms do not contribute to the followinganalysis. h σ i = f π . From the above form, we obtain L eff = tr (cid:2) G ( i∂ + g ωDD h ω i ) ¯ G (cid:3) − tr (cid:2) H ( i∂ + g ωDD h ω i ) ¯ H (cid:3) + ∆ M f π h σ i tr (cid:2) G ¯ G + H ¯ H (cid:3) , (5)where we used v µ = (1 ,~ H and G doublets are obtained as m (eff) H = m − ∆ M f π h σ i + g ωDD h ω i , m (eff) G = m + ∆ M f π h σ i + g ωDD h ω i , (6)where m is the average mass of the H and G doubletswith m = ( m H + m G ) /
2. The masses of H and G dou-blets are determined by the spin average of the physicalmasses as m H = m D + 3 m D ∗ , m G = m D ∗ + 3 m D . (7)We should note that, for the anti-charmed mesons ¯ D ,¯ D ∗ , ¯ D ∗ and ¯ D , the sign in front of the coupling to theomega meson is flipped, so that the effective masses arewritten as m (eff)¯ H = m − ∆ M f π h σ i − g ωDD h ω i ,m (eff)¯ G = m + ∆ M f π h σ i − g ωDD h ω i . (8)Now, let us study the density dependence of masses inEqs. (6) and (8). As for the mean field value of ω , wesimply take h ω i = g ωNN m ω ρ B , (9)where g ωNN is the omega meson coupling to the nucleon, m ω is the mass of omega meson and ρ B is the baryonnumber density. As for the mean field of σ we adopt thelinear densty approximation as h σ ih σ i = 1 − σ πN m π f π ρ B , (10)where σ πN is the coefficient of the π - N sigma term.For making a numerical estimation, we use m G =2 .
40 GeV, m H = 1 .
97 GeV, and ∆ M = m G − m H =430 MeV for masses, in addition to m ω = 783 MeV, m π = 137 MeV and f π = 92 . σ πN = 45 MeV, | g ωDD | = 3 . | g ωNN | = 6 . which lead to | g ωNN g ωDD | = 23, as reference values. We note thatthe D meson includes the anti-light quark, and ¯ D mesondoes the light quark. Therefore, it is natural to considerthat the ¯ D meson is affected by the Pauli blocking in adense medium, which is represented by the effect of themean field of the ω meson. Then, for the concreteness ofthe discussion, we take g ωNN g ωDD < g ωNN g ωDD >
0, we just exchange H with ¯ H and G with ¯ G in the following discussion. There are several values listed in literatures. Here, we use a valueobtained in an analysis of nuclear matter based on the paritydoublet model in Ref. [17], in which the saturation density, thebinding energy and the incompressibility are reproduced. Table Iin the paper includes some errors, and the value | g ωNN | = 6 .
23 isthe corrected value obtained for the chiral invariant mass m =700 MeV. Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L FIG. 1. (Color online) Density dependence of the effectivemasses of H doublet (black curve), ¯ H doublet (blue curve), G doublet (red curve) and ¯ G doublet (green curve) with σ πN =45 MeV and g ωDD g ωNN = − We plot the density dependence of the masses in Fig. 1.This shows that the masses of H and ¯ H doublets as wellas those of G and ¯ G doublets are split by ω contribution.From the ω contribution combined with the σ contri-bution, the mass of G doublet (indicated by red-dashedcurve) decreases with increasing density, and the ¯ H mass(by blue-dotted curve) increases. On the other hand, H mass (by black-solid curve) and ¯ G mass (by green-dotdashed curve) are rather stable. Aa a result, the G mass tends to degenerate with the mass of H doublet atcertain high density. If one measures the mass of H only,one might think that the chiral invariant mass would bealmost same as the mass of H doublet. However, theactual chiral invariant mass is larger than the H mass atvacuum, which can be obtained by averaging the massesof the particles ( G and H ) and anti-particles ( ¯ H and ¯ G ),as shown in Fig. 2. We should note that the sums of Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L H m G H eff L + m G H eff L L(cid:144) H m H H eff L + m H H eff L L(cid:144) FIG. 2. (Color online) Density dependence of the effectivemasses of charmed mesons. The red-dashed and black-solidcurves show the sums of m (eff) G + m (eff) G and m (eff) H + m (eff) H divided by two, respectively, which do not depend on the signof g ωDD g ωNN . masses of particle and anti-particle are actually indepen-dent of the sign of g ωDD g ωNN .Mass difference between the chiral partners, i.e., the H doublet and the G doublet, is caused by the spontaneouschiral symmetry breaking. This structure is seen by sub-tracting the mass of H doublet from that of G doubletwith Eq. (6) as m (eff) G − m (eff) H = ∆ M f π h σ i . (11)So the mass difference is expected to give a clue for thechiral condensate. In the mean field approximation, itis actually proportional to the mean field h σ i as shownin Fig. 3. This figure clearly shows that, with the in-creasing of the nuclear matter density, chiral symmetryis (partially) restored. Ρ B (cid:144) Ρ E ff ec ti v e M a ss D i ff e r e n ce H M e V L FIG. 3. Density dependence of the difference of the effectivemasses of charmed mesons defined by Eq. (11).
For checking the π - N sigma term dependence of theeffective masses, we vary the value of σ πN as 30 and60 MeV, which are plotted in Fig. 4. This shows that thedifference between the masses of H and G as well as thatbetween ¯ H and ¯ G decreases more rapidly for larger valueof σ πN . As a result, the chiral symmetry restores morerapidly for the larger σ πN .We next check the dependence on the value of | g ωDD g ωNN | in Fig. 5, by taking 30% deviation from theestimated value. This shows that the masses change morerapidly for larger value of | g ωDD g ωNN | . III. MODEL ANALYSIS
After the above general discussion, let us study thedensity dependences of the effective masses based onsome specific models. Here we use the the nuclear mat-ter described by the parity doublet model [17] and bythe skyrmion crystal model based on the hidden localsymmetry [3].
A. Parity doublet model
In Ref. [17], the parity doublet model based on thelinear σ model [1], in which an excited nuclear with neg-ative parity, N ∗ (1535), is regarded as the chiral partnerto the ordinary nucleon, was extended by including a six-point interaction for σ field and interactions to the ω and ρ mesons based on the hidden local symmetry, to study Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L FIG. 4. (Color online) Density dependence of the effectivemasses of charmed mesons for with σ πN = 30 MeV (upperpanel) and σ πN = 60 MeV (lower panel). Notations are thesame as in Fig. 1. Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L FIG. 5. (Color online) Density dependence of the effec-tive masses of charmed mesons for σ πN = 45 MeV with g ωDD g ωNN = −
16 (upper panel), and g ωDD g ωNN = − the nuclear matter. It was shown that, for wide range ofthe chiral invariant mass for the nucleon, the model re-produces the saturation density, binding energy, incom-pressibility and symmetry energy. In Refs. [18] and [19],it is shown that the ratio of the mean field h σ i at normalnuclear matter density to the one at vacuum obtainedfor the chiral invariant mass of nucleon m = 500 MeV isconsistent with the experimental value of the one for thepion decay constant [20, 21]. Here we use the density de-pendences of h σ i and h ω i obtained from the model with m = 500 MeV.We show the resultant density dependence of massesin Fig. 6. Here we use | g ωDD | = 3 . Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L FIG. 6. (Color online) Density dependence of the effectivemasses of charmed mesons in parity doublet model. Notationsare the same as in Fig. 1. pendix A as a typical value. This shows that the densitydependence of all masses in the very low density region ρ B /ρ . . h σ i and h ω i in the parity doublet model areconsistent with those obtained in the linear density ap-proximation as can be seen in Ref. [18]. However, aroundthe density region ρ B /ρ ∼ .
3, the mass of H doublet(black curve) starts to decrease and that of G doublet(green curve) to increase, differently from the linear den-sity approximation. In this model, the mean field h ω i isproportional to the density similarly to the linear densityapproximation in Eq. (9). Then, the different density de-pendence of the masses is originated in h σ i . In Fig. 7, weplot the difference of two masses of H and G doublets.This shows that the difference decreases more slowly thanthat in the linear approximation shown in Fig. 3. B. Skyrme model
In Refs. [3, 4], the skyrmion crystal model is used tostudy the qualitative structure of nuclear matter by re-garding the skymrion matter as nuclear matter in thesense of large N c limit of QCD. A robust conclusiondrown in the skyrmion crystal approach is that, when thedensity of the nuclear matter is increased, the skyrmionmatter undergoes a topological phase transition to the Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L FIG. 7. Density dependence of the difference of the effectivemasses of charmed mesons defined by Eq. (11) in the paritydoublet model. matter made of half-skyrmions in which the space av-erage of the chiral condensate vanishes although it is lo-cally non-zero and the chiral symmetry is still broken [22].Since the half-skyrmion phase is not observed in nature,the critical density should be higher than the normal nu-clear density. Recently, the description of nuclear mat-ter from the skyrmion crystal and the implication of thetopological phase transition in the equation of state ofneutron star have gotten great progress (see, e.g., Ref. [6]for a recent review).In the present analysis, we calculate the mean fields h σ i and h ω i in the skyrmion crystal model, and substitutethe values into Eqs. (6) and (8) to obtain the densitydependence of the charmed meson masses. We plot in Ρ B (cid:144) Ρ E ff ec ti v e M a ss H M e V L m G H eff L m H H eff L m G H eff L m H H eff L FIG. 8. (Color online) Density dependence of the effectivemass of charmed mesons in skyrmion crystal model with g ωDD = − .
7. Notations are the same as in Fig. 1.
Fig. 8 the density dependence of the effective masses ofcharmed mesons by using h σ i and h ω i with g ωDD = − . G and ¯ G masses decrease with densitywhile both H and ¯ H increase with density. Because thedensity dependence of ¯ H mass and G mass is strongerthan that of ¯ G mass and H mass, H and G as well as ¯ H and ¯ G become degenerate at density ∼ . ρ at which theskyrmion phase transits to half-skyrmion phase. This isbecause, the mass difference between H and G as well as¯ H and ¯ G is proportional to h σ i which vanishes in the half-skymrion phase. Moreover, we find that the degeneratedmass of H and G and that of ¯ H and ¯ G linearly depend ondensity in the half-skyrmion phase. The reason is that,the h ω i is a linear function of density and this lineardependence agrees Eq. (9). We plot in Fig. 9 the densitydependence of h σ i and h ω i . Note that, as stressed above,the analysis shows just a qualitative structure, and thedegeneracy of chiral partners does not imply the chiralrestoraton but due to the vanish of the space average ofchiral condensate in the half-skyrmion matter. Ρ B (cid:144) Ρ < Σ > & < Ω > <Ω> H MeV L <Σ>´ FIG. 9. (Color online) Density dependence of h σ i and h ω i calculated in skyrmion crystal model. IV. A SUMMARY AND DISCUSSION
In this work, by regarding the ( D ∗ , D ) heavy quarkdoublet as the chiral partner of the ( D , D ∗ ) doublet,we explicitly showed that the effect of the ω meson de-creases the masses of both doublet, while ( ¯ D ∗ , ¯ D ) and( ¯ D , ¯ D ∗ ) meson masses are increased. We explicitly pointout that the ω meson effect is significant for understand-ing the density dependence of effective hadron massesin medium. Even though the qualitative dependence ismodel dependent, the tendency that the masses of theheavy-light mesons and their antiparticles are split due tothe ω meson effect is robust. We hope this medium mod-ified masses of the heavy-light mesons can be detectedin the future experiments at J-PARC and FAIR throughthe strong and weak channels, such as ψ (3770) → D ¯ D , J/ψ → ¯ De + ν e and so on. (See, e.g., Ref. [24].)We would like to note that the result of the omegameson effect to D and ¯ D mesons are consistent with theresult obtained in Ref. [11]. In our analysis, we furtherintroduce p -wave excited D and ¯ D mesons by using thechiral doubling model, and we found that the differencebetween the masses of chiral partners decreases in pro-portion to the mean field value of the σ meson, whichreflects the partial chiral symmetry restoration even ifthe ω meson contribution enters. In our calculation, we simply take the mean field ap-proach. An extension of the present work to include someloop contributions will be reported in [25]. In the presentwork, we only discussed the medium modified charmedmesons. The results presented here are intact for theirbottom cousins except the average mass m should betaken the value of bottom mesons. Appendix A: Estimation of g ωDD In the present analysis, we estimate a reference valueof g ωDD defined by Eq. (3) in the heavy hadron limit byusing the following naive scaling property: (cid:12)(cid:12)(cid:12)(cid:12) ˜ g ωD ¯ D ˜ g ωK ¯ K (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˜ g D ∗ + D π − ˜ g K ∗ K − π + (cid:12)(cid:12)(cid:12)(cid:12) , (A1)where the coupling constants are defined in the rela-tivistic form of the interaction Lagrangian among a vec-tor meson V and two pseudoscalar mesons P and P ′ espressed as L V P P ′ = i ˜ g V P P ′ V µ ( ∂ µ P P ′ − ∂ µ P ′ P ) . (A2)The coupling ˜ g D ∗ + D π + appears in the decay width of D ∗ + → D π + asΓ( D ∗ + → D π + ) = ˜ g D ∗ Dπ | ~p | πm H = 56 . , (A3)which with | ~p | = 39 . m H = 1 .
97 GeV leads to | ˜ g D ∗ Dπ | = 16 . . (A4)In a class of three-flavor chiral models for vector mesons g ωK ¯ K and g K ∗ K − π + are related to the vector mesonmasses as [26] g ωK + K − = g ωK ¯ K = g ωK ¯ K = 14 m ω gf K ,g K ∗ K − π + = 12 √ m K ∗ gf K f π , (A5)where m ω and m K ∗ are the masses of ω and K ∗ mesons, f π and f K are the pion and kaon decay constants, and g is the gauge coupling constant of the hidden local sym-metry [27]. Using m ω = 783 MeV, m K ∗ = 896 MeV, f π = 92 . f K = 110 MeV, the ratio of two cou-plings in Eq. (A5) is estimated as (cid:12)(cid:12)(cid:12)(cid:12) g ωK ¯ K g K ∗ K − π + (cid:12)(cid:12)(cid:12)(cid:12) = 0 . , (A6)which with Eq. (A4) leads to | ˜ g ωD ¯ D | = (cid:12)(cid:12)(cid:12)(cid:12) ˜ g ωK ¯ K ˜ g K ∗ K − π + ˜ g D ∗ + D π − (cid:12)(cid:12)(cid:12)(cid:12) = 7 . , (A7)From the heavy quark Lagrangian in Eq. (4), the ω - D -¯ D interaction is written as L ωDD = 2 g ωDD D ω µ v µ ¯ D , (A8)with a scaling factor of the mass of heavy meson M H .Comparing this with Eq. (A1), we estimate g ωDD as | g ωDD | = 12 | ˜ g ωD ¯ D | = 3 . , (A9)which is the value used in the present work. ACKNOWLEDGMENTS
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