Hyperon threshold and stellar radii

0.31 MB·

by  Luiz L. Lopes

APS/123-QED
Hyperon threshold and stellar radii
Luiz L. Lopes1,2,∗ and Debora P. Menezes1
1Universidade Federal de Santa Catarina; C.P. 476, CEP 88.040-900, Floriano´polis, SC, Brasil
2Centro Federal de Educa¸c˜ao Tecnolo´gica de Minas Gerais Campus VIII; CEP 37.022-560, Varginha - MG - Brasil
(Dated: January 13, 2017)
In this work we show how the emergence of a new degree of freedom in the nuclear bulk not
onlysoftenstheEoS butreducestheradiiofstarswith highcentraldensity. Ifanenoughrepulsive
channel,asthestrangevectorφmeson,isaddedtothescheme,weareabletosimulateverymassive
and compact stars. Indeedweare able toconstruct an equation of state (EoS) that predictsa 2.05
M⊙ as maximum mass and a radius of 11.51 km for the canonical 1.4M⊙ in a thermodynamical
consistent way. We also link the radii of the canonical stars to a soft EoS for densities not much
7 abovethenuclearsaturationpoint. Moreover,comparingourEoSwithresultsobtainedfromheavy
1 ion collisions, we show that the presence of a new degree of freedom allows a better agreement
0 between theory and experiment.
2
PACSnumbers: 24.10.Jv,26.60.Kp
n
a
J I. INTRODUCTION ref. [10] constrained the radius as R = 10.93 ± 2.09 km
2 for a mass M = 1.33 ± 0.33 M⊙.From Fig. 4 of ref. [11]
1
The physics of cold strong interacting matter at ex- and Tab. 8 of ref. [12] a limit of 12.45 km is found for a
] treme densities is still not available in terrestrial labora- neutron star of mass 1.4M⊙.
h tories. So far, the only place believed to reach such Inthis workweshowthatthe onsetofanew degreeof
t
- condition is the interior of neutron stars. The observa- freedom,notonly causesthe knownsofteningof the EoS
cl tions ofpulsarsevolvedinthe lastyearsandprovidedus but alsoreduces the radiiofthe starswhose centralden-
u more precise information about the macroscopic charac- sity is higher than the density of the hyperon threshold
n teristic of these objects. ( what we call here subsequent stars), when compared
[ For instance, the observations of two hyper massive with the EoS without this new degree of freedom. We
1 pulsars, PSR J1614-2230 [1] and PSR J0348+0432 [2], use here the Λ hyperon as this new degree of freedom,
v showus that the EoSofbeta-equilibriumnuclearmatter once it is the lowest mass baryon beyond the nucleons.
1 should be very stiff, being able to produce a two solar Neverthelessitis importanttobearinmindthatthena-
1 masses neutronstar. This indicates that hadroninterac- tureofthenewdegreeoffreedomisnotrelevant. Similar
2
tion at very small distances is strongly repulsive. results can be obtained using Σ or Ξ hyperons, or even
3
On other hand, the physics of neutron stars radii also ∆ resonances, by adjusting the strength of the coupling
0
. improved in this decade. Recent observations indicate constant for those particles. Even more exotic particles,
1 that the radii of pulsars of mass around the canonical as dark matter [13–16] can be used without significantly
0
7 1.4M⊙ arelowerthanmostoftheprevisionsfoundinthe affect the results.
1 literature[3]. Todayweknowthatthe radiusofthe neu- We show that if the hyperon onset happens at rela-
: tron stars are somehow related to the symmetry energy tively low density, and a channel, repulsive enough, as
v
slope, L [4–7], although L cannot be the ultimate infor- the strange vector φ meson is added, we can produce
i
X mation about neutron stars radii, once the same model very massive and compact neutron stars in agreement
r with the same slope, predicts neutron stars whose dif- with the works above mentioned. Moreover, if an at-
a ference in radius reaches 1.7 km [7]. Theoretical works tractive channel, as the strange scalar σ∗ meson is also
andastrophysicalobservationshavestronglyconstrained added,wecanproduceevenmorecompactneutronstars.
the radius of the canonical mass in the last couple of Indeed,weareabletosimulateanEoSthatpredicts2.05
years. For example, based on a chiral effective theory, M⊙ as maximum mass with a radius of 11.51km for the
ref.[8]constrainedthe radiiofthe canonical1.4M⊙ neu- canonical 1.4M⊙.
tronstarto9.7-13.9km. Tofitexperimentalinformation Wealsocompareourresultswithexperiencesofheavy-
fromneutronskins,heavyioncollisions,giantdipoleres- ion collisions (HIC). In ref. [17], the authors determine
onances, and dipole polarizabilities, ref. [9] constrained the pressure of the symmetric nuclear matter up to five
the neutron star radius of a canonical mass in the nar- times nuclear saturation density. They neither exclude
row window 10.7 km < R < 13.1 km (90% confidence). the onset of hyperons nor of more exotic behaviour, as
Usingtime-resolvedspectroscopyofthermonuclearX-ray the quark-hadron phase transition. We show that the
burstsobservedfromandobjectcalledSAXJ1748.9-2021 emergenceofanewdegreeoffreedomalso allowsabetter
agreement of theory with experience.
Thispaperisorganizedasfollows: insectionIIwedis-
cusstheQHDformalismandpresenttheparametrization
∗ [email protected] of the model alongside some of the physical quantities
2
they foresee for nuclear matter. In section III we expose change of massless gauge bosons called gluons. Since
the results of the threshold of Λ hyperon in the bulk of the QCD has no results for dense cold matter, an effec-
beta-equilibrium nuclear matter and symmetric nuclear tive model is required. In this work we use an extended
matter withinfourdifferentapproaches. Theconclusions version of the relativistic QHD [18], whose Lagrangian
are drawn in section IV. density reads:
II. FORMALISM
The theory of the strong interacting matter is the
QCD, where quarks interact with each other via the ex-
1
L = ψ¯ [γµ(i∂ −g ω −g ~τ ·ρ~ )−(m −g σ)]ψ −U(σ)+
QHD B µ Bω µ Bρ µ B Bσ B
2
XB
1 1 1 1 1
+ (∂ σ∂µσ−m2σ2)− ΩµνΩ + m2ω ωµ+ m2ρ~ ·ρ~µ− Pµν ·P , (1)
2 µ s 4 µν 2 v µ 2 ρ µ 4 µν
in naturalunits. ψ are the baryonic Dirac fields, which further, [4, 30], but for the purpose of the present work,
B
can be the nucleons, or a new degree of freedom, in this the modifications introduced in [7] suffice.
case,Λhyperon. Theσ,ω andρ~ arethemesonicfields. InTableIweshowtheparametersofthemodelandits
µ µ
Theg′saretheYukawacouplingconstantsthatsimulate previsionsforfivenuclearmatterpropertiesatsaturation
the strong interaction, m is the mass of the baryon B density: saturation density point (n ), incompressibility
B 0
and m , m , and m are the masses of the σ, ω, and (K),bindingenergyperbaryon(B/A),symmetryenergy
s v ρ
ρ mesons respectively. The antisymmetric mesonic field (S ) and its slope (L).
0
strength tensors are given by their usual expressions as
presented in [19]. The U(σ) is the self-interaction term Parameters Previsions at n0
introducedinref.[20]toreproducesomeofthesaturation (gNω/mv)2 7.148 fm2 n0 (fm−3) 0.153
properties of the nuclear matter and is given by: (gNσ/ms)2 11.785 fm2 K (MeV) 300
(gNρ/mρ)2 3.880 fm2 B/A (MeV) -16.3
κ/MN 0.005894 S0 (MeV) 30.5
1 1
U(σ)= κσ3+ λσ4. (2) λ -0.006426 L (MeV) 87.9
3! 4!
TABLE I. Slightly modified GM1 parametrization. Parame-
Finally, ~τ are the Pauli matrices. In order to describe
ters of themodel and previsions.
a neutral, chemically stable matter, we add leptons as
free Fermi gases:
L = ψ¯[iγµ∂ −m ]ψ, (3)
lep l µ l l III. RESULTS
Xl
where the sum runs over the two lightest leptons (e and To study the effects of the onset of a new degree of
µ). freedom in detail, we divide this section in four different
Themesonicfieldsareobtainedviameanfieldapprox- approaches. WeconsidertheΛhyperononlybecausethe
imation (MFA) [18, 19, 21] and the EoS by thermody- presenceofotherstrangeparticleswouldmakethisstudy
namic relations [19, 22]. strongly model and parameter dependent.
To describe the properties of nuclear matter we use
a slightly modified version of the well-known GM1
parametrization [23], a widely accepted parametriza- A. Role of the coupling constant
tion [7, 19, 21, 24–29] that is able to reasonablydescribe
both, nuclear matter and stellar structure, consistent The basic constituents of neutron stars are neutrons
withexperimentalandastrophysicalobservations[28]. In and protons in β-equilibrium. Since both are fermions,
this work we just reduce the strength of the ρ coupling, asthe baryondensityincreases,so dothe Fermimomen-
reducing the symmetry energy slope L from 94 MeV to tum and the Fermi energy, according to the Pauli prin-
87.9 MeV [7], a value closer to what is inferred in recent ciple. Ultimately, the Fermi energy exceeds the masses
observations [9, 11, 12]. This slope can be reduced even of the heavier baryons. The influence of the hyperons in
3
the EoS and neutron star mass-radius relation, strongly 2.5
depend of the coupling constant of the hyperons with Set A
Set B
the mesonic field. It is well known that the Λ potential 2 Set C
Set D
depth U = - 28 MeV [23], so the coupling constants for
Λ
)
ω and σ meson cannot be varied independently in prin- 4 1.5
-m
ciple. So, in this section we fix the potential depth and f
(
vary the values of the coupling constants and see how p 1
they affect the hyperon onset and population, the EoS,
0.5
andconsequentlythemass-radiusrelationoftheneutron
stars. Similar works are found in the literature [19, 23],
0
however, they just analyze the effects on the maximum 0 1 2 3 4 5 6 7
mass. Let’s study, in more detail, the influence on the ε (fm-4)
radiioftheneutronstarsaswell. InTab.IIweshowfour
sets of values for the coupling constants utilized in this FIG. 2. (Color online) EoS for different sets with a fixed
section and in Fig. 1 we plot the Λ fraction YΛ = nΛ/n potential depth UΛ = -28 MeV.
for the four sets.
Set gΛ,ω/gN,ω gΛ,σ/gN,σ UΛ (MeV) Now we plot in Fig. 2 the EoS for the four sets pre-
A 0.25 0.291 -28
sentedinTab.I.Weseethatthehigherthevalueofg ,
Λ,ω
B 0.50 0.483 -28
the stiffer the EoS. Unlike Fig. 1 there is no crossing of
C 0.75 0.675 -28
the curves. This is due to the fact that the main termof
D 1.00 0.865 -28
the pressure is the vector meson ω andit is proportional
tothedensityn. So,indeed,thecurvesdeviatefromeach
TABLE II. Different sets for Λ-mesons coupling constants
other as the density increases.
with a fixed potential depthUΛ = -28 MeV.
Set Mmax/M⊙ RMmax (km) R1.4M⊙ (km)
A 1.45 13.26 13.66
B 1.71 12.68 13.76
0.7
Set A C 2.07 12.06 13.74
0.6 Set B D 2.31 11.95 13.75
Set C
Set D
0.5
TABLE III. Different sets for Λ-mesons coupling constants
Λ 0.4 with a fixedpotential depthUΛ = -28 MeV.
Y
0.3
0.2
To conclude this section we solve the TOV structural
0.1 equations[31]andplotthemass-radiusrelationinFig.3.
0 Here, and in the rest of this work we use the BPS [32]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 equation to simulate the neutron star crust. The most
-3
n (fm )
massivepulsaryetknownisthePSRJ0348+0432[2]with
a mass of 2.01 ± 0.04 M⊙. So, for an EoS to be valid,
FIG.1. (Coloronline)Λthresholdandpopulationforafixed it needs to explain this massive neutron star. Another
potential depth UΛ = -28 MeV. constraint is the radius of the canonical 1.4M⊙ pulsar.
According to ref. [8–12] the maximum radius for these
We see that the higher the value of g , the higher stars lie between 12.45 to 13.9 km. We plot the main
Λ,ω
the density ofthe hyperonthreshold,varying from0.302 results for the TOV solutions in Tab. III.
fm−3 forsetAto0.431fm−3,forsetD,ie,around1.97 By looking at Fig. 3, we can see that according to
and2.82times the nuclearsaturationdensity. Moreover, ref.[2], only sets C andD arevalidas EoSofdense mat-
in general, the Λ population also increases with the in- ter. All the radii are very close, between 13.74 km to
crease of g . We can see that the curves of sets A and 13.76km, except for set A. This is due to the emergence
Λ,ω
Bcrosseachother(setAandCcrossaswellatadensity of a new degree o freedom. A new degree of freedom as
around 1.2 fm−3). This is due to the Pauli blocking of the Λ hyperonnot only reduces the maximum mass, but
the hyperons in set A. Once there are more hyperons in compresses the neutron star. When the hyperon frac-
set A at low densities, the onset of new Λ particles is tionbecomesrelevant,thereis a“turnto the left” inthe
more energetically favorable in set B because the Fermi mass-radius relation, compressing the subsequent neu-
sea for low Fermi moment in set A is already filled. We tronstars, reducing their radii. For a hyperonpotential
can also see that in set D g is so high that Λ is very depthof-28MeV,thethresholdofhyperons appearstoo
Λ,ω
suppressed, YΛ never reaching more than 0.3, while in lateforthis“turntotheleft”affectthecanonical1.4M⊙
other sets this value could reach more than 0.6. to values of radii that agree with the ref. [8–12]. But for
4
2.4 1.55
(a)
1.5
2
PSR J0348+0432
1.45
0 1.6 0
M M
1.4
/ /
M M
1.2
1.35
(b)
Set A
0.8
Set B 1.3
Set C
Set D
0.4 1.25
8 9 10 11 12 13 14 15 13.5 13.6 13.7 13.8
R (km) R (km)
FIG. 3. (Color online) (a) Mass-radius relation obtained via TOV solution, the hatched area comprises the uncertainty about
themass of the PSR J0348+0432. (b) Zoom in themass around 1.4M⊙.
moremassiveneutronstars,thelargeamountofhyperons 0.7
compresses significantly the star. For instance, if some-
0.6
one wonders about the radius of the 2.01M⊙, the PSR
J0348+0432,the answer is 12.91 km, if we assume set C 0.5
asthe bestparametrization,or13.40kmifwechooseset 0.4
Λ
D,becausethe“turntotheleft”happensearlierinsetC Y
0.3
than set D. If we want to compress the canonical 1.4M⊙
neutron star, the Λ threshold needs to take place at ear- 0.2
Set E
lier densities. To accomplish that, the potential depth 0.1 SSeett GF
needs to be lower. Set H
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-3
n (fm )
B. Role of the potential depth
FIG. 4. (Color online) Λ threshold and population for a po-
Now we study how a more attractive potential depth tential depthUΛ varyingfrom -50 MeV to -126 MeV.
affects the neutron stars with hyperons. As mentioned
earlier in this work, although the U is well established
Λ
around around -28 MeV, the nature of the new degree WecanseethatthemoreattractivetheU ,theearlier
Λ
of freedom is not relevant. Any other particle could be thehyperonthreshold. Also,themoreattractivetheU ,
Λ
used, justadjusting the valuesofthe couplingconstants. the higher the Y at high densities. Indeed, the lambda
Λ
Moreover, here we are dealing with highly asymmetric fractioncanreache values higher than 0.7 for sets G and
matter, instead of the symmetric matter where the po- H. The Λ potential depth can be as low as -126 MeV.
tential was inferred, and the hyperon threshold happens This value was not randomly chosen. As we will see in
at higher densities than n0. Because we want the hy- the nexttopic, this valueisthe highestvaluethatis able
peron onset at lower densities we fix the ω−Λ coupling topredictamassiveneutronstarwiththeradiusagreeing
to be very week, as in set A of Tab. II. Now we vary with ref. [8–12].
only the σ −Λ coupling. The parametrization utilized
NowinFig.5weplottheEoSfordifferentvaluesofU .
Λ
are presented in Tab. IV, and we plot the Y fraction in
Λ Due to the extremely attractive potential, and the great
Fig. 4.
amountofhyperons,weexpect verysoftEoS.Indeed, as
one can see, for sets G and H, the dp/dǫ could even be
Set gΛ,ω/gN,ω gΛ,σ/gN,σ UΛ (MeV) negative. That could indicate a phase transition. These
E 0.25 0.396 -50
are deeper waters where we will not navigate into [33].
F 0.25 0.476 -80
Moreover, since a negative value of dp/dǫ, produces un-
G 0.25 0.547 -100
stable neutron stars [19], we concentrate this work only
H 0.25 0.640 -126
inEoSforwhichdp/dǫisgreaterthanzero. Nowweplot
TABLEIV.DifferentsetsforUΛ with afixedω−Λcoupling the TOV solution for sets E and F in Fig. 6.
of gΛ,ω/gN,ω =0.25. Due to the very attractive hyperon potential depth,
and once we have a very weak repulsive channel, these
softEoSproducesverylowmaximummassneutronstar.
5
0.5 UΛpotential,andthe“turntotheleft”willnotaffectthe
SSeett EF canonical 1.4M⊙. One way to overcome this difficulty is
0.4 Set G to add in the Lagrangian a new repulsive channel. We
Set H
use here here the strange vector φ meson [26–28]:
0.3
)
4
-m
0.2
p (f 0.1 LYYφ =−gY,φψ¯Y(γµφµ)ψY + 12m2φφµφµ− 41ΦµνΦµν.
(4)
0
-0.1 Set gΛ,ω/gN,ω gΛ,σ/gN,σ gΛ,φ/gN,ω UΛ (MeV)
0 1 2 3 4 5 6 H-I 0.25 0.640 0.650 -126
ε (fm-4) H-II 0.25 0.640 1.000 -126
H-III 0.25 0.640 1.350 -126
FIG. 5. (Color online) EoS for different sets varying the UΛ H-IV 0.25 0.640 1.754 -126
from -50 MeV to -126 MeV.
TABLE V. Different sets for gΛ,φ/gN,ω with a fixed UΛ =
-126 MeV.
1.2
Set E
Set F
0.7
Set I
M0 0.6 Set II
0.8 Set III
M/ Set IV
0.5
0.4
Λ
Y
0.3
0.4 0.2
10 11 12 13 14
R (km) 0.1
0
FIG. 6. (Color online) Mass-radius relation for UΛ = -50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
MeV and UΛ = -80 MeV. The “Turn to the left” can ocour n (fm-3)
for masses lower than 0.8 M⊙.
FIG. 7. (Color online) Λ threshold and population for differ-
ent values of Λ−φ coupling, with a fixed UΛ = -126 MeV.
Also,onlyforsetsEandFitispossibletosolvetheTOV
equations, since it requires dp/dǫ greater than zero. In-
Theφfieldisanalogoustothe ω fieldandits expected
deed, all EoS in this section are ruled out, since none of
valueisalsoobtainedviaMFA[18,19,21]. Addinganew
them is able to explain the massive PSR J0348+0432.
mesonicfieldhas twoadvantages. First,since the φ does
Nevertheless, the key point in the section is the “turn to
notcoupletothenucleon,itdoesnotaffecttheproperties
the left”. Looking at Fig. 6, we can see that for U =
Λ of the nuclear matter. Second, since the φ field is zero
-80 MeV, the “turn to the left” occurs for masses below
below the hyperon threshold density, it does not affect
0.8M⊙. If anevenmore attractivepotentialis used, ear- the potential depth, and has little influence on the point
lier would be this behavior. The star masses are so low ofthe“turntotheleft”. NowwefixtheU =−126MeV
Λ
due to the very weak repulsive channel. In next section
as in set H. Indeed, all sets in this section are derived
we see how to overcome this issue. from set H of Tab. IV, only varying the Λ−φ coupling
constant. So,wenumberedthe setsfromH-ItoH-IV.In
the figures the H is omitted to not burden the notation.
C. Role of the strange vector meson φ We choosethatsetbecause,as wewillsee,this potential
depth is the less attractive potential able to produce a
We see that the emergenceof a new degree of freedom 2.05M⊙ neutron star, with a radius of 12.45 km. The
causes a “turn to the left” in the mass-radius relation, valuesutilizedinthis calculationarepresentedinTab.V.
compressing the subsequent stars. The earlier the onset Now we plot the Λ fraction in Fig. 7. As we can see,
ofthisnewparticle,thelowerthemassthatisaffectedby since φ field is zero before the hyperon threshold, and
this“turntotheleft”. Butwealsoseethatthethreshold the potential depth are the same for all sets, the hy-
of a new particle softens the EoS. To stiffen the EoS, we peron onset is the same for all values of Λ−φ coupling,
needtoincreasetherepulsivechannel. Thetermrespon- around0.177n . Thisisarelativelowvalue,correspond-
0
sible to this in eq. (1) is the Λ−ω coupling constant. ing to 16% above the nuclear saturation density. Since
However,theincreaseoftheΛ−ω channelwillaffectthe φ is a repulsive channel, the stronger the coupling, the
6
2.5 As discussed in ref. [4, 28], the GM1 is in disagreement
Set I withthis constraint. However,aspointedinref.[28]this
Set II
2 Set III could be due to the fact we are not considering the on-
Set IV
set of hyperons in the bulk of nuclear matter. Now we
)
4 1.5 define the symmetric hypernuclear matter as in ref. [28].
-m
f Consider a symmetric matter. In this regime, by defini-
(
p 1 tion, the density of the protons is equal to the density
of neutrons, n = n . Due to this, the ρ field is zero
p n
0.5
and the chemicalpotential of these particles are also the
same, µ =µ . If we compress this matter, the onset of
0 n p
0 1 2 3 4 5 6 7 strange particles, as Λ, becomes energetically favorable.
ε (fm-4) Since ref. [17] does not rule out neither hyperons nor
even more exotic pictures, such as quark-hadron phase
FIG. 8. (Color online) EoS for different values of Λ−φ cou- transitions, we assume the possibility of Λ onset in the
pling, with a fixed UΛ = -126 MeV. In these cases dp/dǫ is symmetric matter, imposing:
always greater than zero.
µ =µ =µ . (5)
p n Λ
stronger the hyperon suppression at high densities. In-
deedinsetH-IV,YΛ neversurpasses0.3. Also,theφfield This choice implies that only symmetric nuclear mat-
contributes in the pressure, stiffening the EoS. Plotting terexistsuntilthe densityishighenoughsothe creation
the EoS of Tab. V in Fig. 8, we can see that with the φ ofstrangeparticlesbecomesenergeticallyfavorable,soft-
field, dp/dǫ is always positive. ening the EoS. The pressure of GM1 symmetric mat-
terandsymmetric hypernuclearmatter alongsidethe in-
Set Mmax/M⊙ RMmax (km) R1.4M⊙ (km) ferredpressureuptofivetimesnuclearsaturationdensity
H-I 1.42 7.91 8.11 (hatchedarea),obtainedinref.[17]areplottedinFig.10.
H-II 1.72 9.06 10.49
Since only set H-IV is able to predict a 2.0M⊙ we plot
H-III 1.91 9.95 11.70
only this set.
H-IV 2.05 10.57 12.45
The hyperonthreshold softens the EoS making exper-
iment results from HIC and theory reach an agreement
TABLE VI. Main properties of neutron stars with the inclu-
again. While inbeta-equilibriummatterthe onsetofthe
sion of theφ meson for UΛ = -126 MeV.
new degree of freedom is at densities about 16% above
the nuclear saturation density, in symmetric matter this
Now we plot the mass-radius relation in Fig. 9 with onset happens latter, at densities about 51% above the
themainpropertiesresumedinTab.VI.Wecanseethat saturationpoint. This is due to the fact that the ρ field,
due to the onset of hyperons at low densities, the “turn thatisarepulsivechanneliszero,andthenuclearmatter
to the left” occursat verylowmasses. Indeed insetH-I, has a lowerchemicalpotential comparedwith the stellar
ithappens atmass around0.5M⊙. The higher the Λ−φ one. Indeed, the nuclear matter is bounded for densities
coupling, the less pronounced is the compression of the belowthesaturationdensity. Weseethat,theemergence
subsequent neutron stars. Also, in set H-I, we see that ofanew degreeoffreedom,besidesbeing ableto explain
the canonical 1.4M⊙ is only 8.11 km, a very compact the astrophysical observations of a very massive pulsar
neutron star. Nevertheless, its maximum mass of only and the inferred low radius for the canonical mass, rec-
1.42M⊙ indicates that this EoS must be ruled out. onciles these results with those obtained in laboratory.
Increasing the Λ − φ coupling, we also increase the
maximum mass. In set H-IV, for a strong coupling, the
maximum mass reaches 2.05M⊙, the upper limit of the D. Role of the strange scalar meson σ∗
2.01 ± 0.04 M⊙, PSR J0348+0432 pulsar. But, unlike
sets C andDofFig.3,now wehavea muchlowerradius Nowwegiveonemorestepandaddanewstrangeme-
forthecanonical1.4M⊙. WiththismodifiedGM1model, son,the scalarσ∗. As the vectorφ meson,the scalarone
the radius was around 13.76 km. Now we have a radius just couples to the Λ particles, and does not affect any
of only 12.45 km. In other words, an EoS that is able of the properties of nuclear matter presented in Tab. I.
to explain the massive PSR J0348+0432 [2], and also Also, the σ∗ field is zero in the absence of Λ particles,
in agreement with the inferred measures of the pulsar therefore it would not affect the hyperon threshold. The
radii[8–12]. Weshowthattheemergenceofanewdegree Lagrangianof σ∗ is analogous to the σ and reads:
offreedomis abletocompressthe neutronstarsandstill
produce very massive pulsars.
ofAsynmotmheetrrgicomodatctoenrsftorraidnetntsoitibeesauspsetsosefidveistitmheesptrhesesnuure- LYYσ∗ =gY,σ∗(ψ¯YψY)σ∗+ 21(cid:18)∂µσ∗∂µσ∗−m2σ∗σ∗2(cid:19).
clear saturation density, as inferred in ref. [17] via HIC. (6)
7
2.1 1.55
PSR J0348+0432
1.8 1.5
1.5 1.45
0 0
M M
1.2 1.4
/ /
M M
0.9 1.35
Set I (b)
0.6 Set II 1.3
(a) Set III
Set IV
0.3 1.25
8 10 12 14 16 11.2 11.6 12 12.4 12.8
R (km) R (km)
FIG. 9. (Color online) (a) Mass-radius relation obtained via TOV solution with the inclusion of the φ meson. The hatched
area comprises the uncertainty about the mass of the PSR J0348+0432. (b) Zoom in the mass around 1.4M⊙, although sets
H-I and H-IIare not showed, it is not relevant since its maximum mass is lower than the PSR J0348+0432.
0.4
100
) 0.3
3
m
f
eV/ YΛ 0.2
M 10
(
p 0.1 Set V
Set VI
GM1 Set VII
Set IV Set VIII
1 0
1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n/n0 n (fm-3)
FIG.10. (Coloronline)Pressureofsymmetricnuclearmatter FIG. 11. (Color online) Λ threshold and population for dif-
of the GM1 model and the symmetric hypernuclear matter ferent values of Λ−σ∗ and Λ−φ coupling, with a fixed UΛ
with the onset of a new degree of freedom compared with = -126 MeV.
experimental constraint (hatched area).
Set gΛ,σ∗/gN,σ gΛ,φ/gN,ω UΛ (MeV) able to reproduce a 2.05M⊙ as a maximum mass. And
H-V 0.65 1.80 -126 again, in all cases the set H of Tab. IV is used to fix the
H-VI 1.00 1.88 -126 hyperon potential depth at -126 MeV.
H-VII 1.50 2.00 -126
Now we plot in Fig. 11 the hyperon fraction for the
H-VIII 2.00 2.02 -126 sets of Tab. VII. We see that the σ∗ has little influence
if the Λ−σ∗ is not high. Sets H-V to H-VII are very
TABLEVII.DifferentsetsforgΛ,σ∗/gN,σ andgΛ,φ/gN,ω with similar. For set H-VIII the coupling constant of Λ−σ∗
a fixed UΛ = -126 MeV.
is two times the N −σ one, and the difference is higher.
Butinallcases,the Y islow,andagainneversurpasses
Λ
0.3.
The σ∗ is an attractive scalar meson, while the φ is The differences in the EoS are even smaller. Since all
a repulsive vector meson. Therefore, the σ∗ will domi- the EoS are adjusted to reproduce a 2.05M⊙ as maxi-
nate at low densities, softening the EoS and increasing mum mass,they are allvery similar. Due to this we just
the amount of hyperons. At high densities, the φ meson plot the set H-V and set H-VIII, that have the weakest
dominates, causing a strong suppressionof the hyperons and strongest coupling constants. Of course we could
andstiffening the EoS.This behaviorisanalogoustothe increase the repulsive channel and simulate even higher
δ −ρ competition in the symmetry energy reported in mass neutron stars, however this would imperatively in-
ref. [7]. In Tab. VII we show the values of the coupling crease the radius of the canonical 1.4M⊙. We also plot
constants for the Λ−σ∗ and Λ−φ mesons. All these the resultsobtainedwithsetCfromTab.IIsinceitpre-
values have been chosen in such a way that the EoS is dicts similar maximum mass. The results are presented
8
2.4 σ∗ meson we are able to simulate neutron stars agreeing
Set C with the PSRJ0348+0432[2], andthe radiiin the range
Set V
2 Set VIII proposed in ref. [8–12] . Comparing the TOV solution
with the EoS of Fig. 12 we can also link the radius of
1.6
)
4 canonicalstarstoasoftEoSatdensitiesnotmuchabove
-m
f 1.2 the saturation point. This seems a more fundamental
(
p 0.8 relation than linking the radii to the slope. The cause
and effect follows: the hyperon onset softs the EoS (at
0.4 low densities), and this soft EoS produces the “turn to
the left” in the mass-radius relation. The strong Λ−φ
0
0 1 2 3 4 5 couplingproducesastiffEoS(athighdensities),andthis
ε (fm-4) stiff EoS produces very massive neutron stars.
To conclude this section we compare our results for
FIG. 12. (Color online) EoS for sets C, H-V and H-VIII. a symmetric hypernuclear matter with the constrainim-
posedbyref.[17]. Again,duetothefactthattheEoSare
very similar we plot only set H-V and set H-VIII. Since
in Fig. 12. We see that set H-V produces a lower value setH-VIandH-VIIhaveintermediatevaluesforthecou-
ofΛ−σ∗ andΛ−φcoupling. Duetothis,theEoSisless pling constants, their results will be alwaysbetween sets
softatlowdensityandalsolessstifferathighdensity. In H-V and H-VIII. We see in Fig. 14 that the same effects
setH-III,Λ−σ∗ andΛ−φcouplingareveryhigh. Soσ∗ that happen in β-equilibrium matter are reproduced in
meson strongly dominates at low density, softening the symmetrichypernuclearmatter. ThestrongertheΛ−σ∗
EoS, while φ stiffens it at high densities. Set C has no coupling, the softer the EoS. But due to the high value
hyperons at low densities, so it is stiffer in this regime. of the Λ−φ coupling needed to reproduce very massive
Also,due tothe absenceoftheφmeson,thissetissofter pulsars,thisEoSwillbecomestifferathighdensities. We
at high densities. Sets H-VI and H-VII are between set concludethatourmodelwiththeonsetofanewdegreeof
H-V and H-VIII. The stronger the interaction with both freedom agrees with astrophysicalobservations and HIC
mesons, the softer it will be at low density and stiffer experiments.
at higher ones due to the different nature of the mesons
(scalar and vector).
IV. FINAL REMARKS
Set Mmax/M⊙ RMmax (km) R1.4M⊙ (km)
H-V 2.05 10.54 12.43
In this work we show how the onset of a new degree
H-VI 2.05 10.50 12.38
of freedom is able to reconcile the recent measurements
H-VII 2.05 10.37 12.20
of very massive pulsars [1, 2] with compact ones [8–12],
H-VIII 2.05 10.57 11.51
and also satisfy HIC experimental constraints [17]. Al-
though the true nature of this new degree of freedom is
TABLE VIII. Main properties of neutron stars with the in-
clusion of theσ∗ and φ mesons for UΛ = -126 MeV. anopenpuzzle,weshowthataparticlenotmuchheavier
than the nucleon and a new repulsive channel is enough
to accomplish this goal. Also, we are able to construct
Now, lets solve the TOV equations and obtain a thermodynamical consistent model with no ad hoc in-
the mass-radius relation. The results are plotted in puts.
Fig 13 and the main relevant properties are resumed in We startby reviewingthe effects ofhyperonthreshold
Tab.VIII.The presenceofthe σ∗ softensthe EoSatlow indensenuclearmatter. Theconsequentsofteningofthe
densities. This behavior causes the “turn to the left” in EoSandreductionofthemaximummassisawell-known
mass-radius relation to be more pronounced, compress- theme in the literature [19, 21, 23], but the “turn to be
ing the subsequent neutronstars and producing a signif- left” when the hyperon fraction becomes relevant seems
icantlylowervalueforthe radiusofthe1.4M⊙ star. The to have passed unnoticed. We show that for the well-
strongertheΛ−σ∗coupling,thelowertheradius. Indeed established Λ potential depth of -28 MeV, there is little
this radius can reach 11.51 km for a very strong Λ−σ∗ influenceofthehyperononsetintheradiiofstarsaround
couplingconstant,oftwotimestheN−σ one. Thislimit the canonical mass of 1.4M⊙.
of two times the N −σ one presented in this work is not NextwemodifytheΛpotentialdepthallowingittoas-
arbitrary. This is the highest value that produces a neu- sume more attractive values. Therefore,the onset of the
tronstarof2.05M⊙ withdp/dǫalwaysgreaterthanzero. newdegreeoffreedomappearsatlowerdensitiesandthe
If we increase the Λ−σ∗ coupling, this condition will be ‘turn to be left” occurss for less massive stars, reaching
broken. Of course we can increase the Λ−φ coupling thecanonicalmass. Duetotheveryweakrepulsivechan-
constant to avoid a negative value of dp/dǫ, however it nel, all resulting neutron stars have very low maximum
willalsoincreasethemaximummaxbeyond2.05M⊙ and mass and the corresponding EoS must be ruled out.
enlargetheradiusofthecanonicalstar. Withthehelpof To overcome this issue we add a new repulsive chan-
9
2.1 1.55
PSR J0348+0432 (b)
1.8 1.5
1.5 1.45
0 0
M M
1.2 1.4
/ /
M M
0.9 1.35
Set V
0.6 Set VI 1.3
(a)
Set VII
Set VIII
0.3 1.25
9 10 11 12 13 14 15 11.2 11.6 12 12.4 12.8
R (km) R (km)
FIG.13. (Coloronline)a)Mass-radiusrelation consideringbothσ∗ andφmesonsconstrainedtoamaximummassof2.05M⊙.
b) Zoom in themass around 1.4M⊙. Stronger is the Y −Y interaction, lower are theradii.
channel, the strange scalar σ∗ meson. This new attrac-
tive field softens the EoS at low densities, making the
100 “turntobeleft”morepronounced. Thiscompresseseven
3) more the subsequent stars, allowing us to construct an
m
V/f EoS that predicts a maximum mass of 2.05M⊙, while
e the canonical 1.4M⊙ has a radius of only 11.51 km. We
M 10 are also able to link the low radius to a soft EoS at low
(
p densities, what seems a more fundamental reason than
relating it to the symmetry energy slope. Moreover, the
Set V
Set VIII emergence of a new degree of freedom again reconciles
1
1 1.5 2 2.5 3 3.5 4 4.5 5 this EoS with HIC experiments.
n/n0 We finish this work by pointing out that the nature of
thenewdegreeoffreedomneedstobebetterstudied,but
FIG.14. (Coloronline)Roleofthetheσ∗andφmesonsinthe itseemsthatthe EoSneeds tobe softatlowdensitiesto
symmetric hypernuclearmatter compared with experimental predict very compact neutron stars. However, the radii
constraint (hatched area). of canonical neutron stars is not a closed subject. Some
studies indicate that their values could be very low, not
surpassing11.1km[34],closebutstillbelowthe11.51km
nel, through the strange vector φ meson. This allows us found in this work. Nevertheless, since it is still an open
to stiffen the EoSand increasethe maximum masswith- puzzle, a relatively large lower limit of 10.7 km for the
outchangeany propertiesof nuclearmatter andwith no canonicalneutronstarispresentedinref.[35]toaccount
effect on the hyperon threshold. This also allows us to for the causality.
constructa veryrealisticEoS,that is able to explainthe
massive PSR J0348+0432 and the measurements of low
radii pulsars. The result is even better when compared
with the results of HIC. Now we are able to reconcile Acknowledgments; DPM acknowledges partial su-
astrophysicalobservation and lab constraints. uport from CNPq and LLL from CAPES and CE-
In the last part of our work we add a new attractive FET/MG.
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