Fermion mass and the pressure of dense matter

0.07 MB·English

by  Eduardo S. Fraga

Fermion mass and the pressure of dense matter
Eduardo S. Fraga and Letícia F. Palhares
InstitutodeFísica,UniversidadeFederaldoRiodeJaneiro
C.P.68528,RiodeJaneiro,RJ21941-972,Brazil
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Abstract. We consider a simple toy model to study the effects of finite fermion masses on the
0
0 pressure of cold anddense matter, with possible applicationsin the physicsof condensatesin the
2 coreofneutronstarsandcolorsuperconductivity.

n Keywords: Finite-temperaturefieldtheory;Yukawatheory;Equationofstate
a PACS: 11.15.Bt,11.10.Wx
J

1
1 The role of finite quark masses in QCD thermodynamics has received increasing
attention in the last few years. In the case of cold and dense QCD, it was generally
1
believedthat effects of nonzero quark masses on theequation of statewere of theorder
v
2 of 5%, thereby yielding only minor corrections to the mass-radius diagram of compact
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stars [1]. In fact, mass, as well as color superconductivity gap, contributions to the
0
pressure are supressed by two powers of the chemical potential as compared to zero-
1
0 mass interacting quark gas terms. Therefore, assuming a critical chemical potential for
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the chiral transition of the order of a few hundred MeV, naively those terms should
0
/ not matter. However, recent results for the thermodynamic potential to two loops have
h
shownthatcorrectionsaresizable,andmaydramaticallyaffectthestructureofcompact
p
- stars [2]. Moreover, the situation in which mass (as well as gap) effects are significant
p
corresponds to the critical region for chiral symmetry breakdown in the phase diagram
e
h of QCD. Hence, not only the value of the critical chemical potential will be affected,
:
v butalsothenatureofthechiraltransition.Inparticular,ifthelatterisstronglyfirst-order
i theremightbeanewclassofcompactstars,smalleranddenser,withadeconfinedquark
X
matter core [3]. Of course, contributions due to color superconductivity [4] as well as
r
a chiralcondensation[5]willalsoaffect thispicture.
In what follows, we study a simple toy model – cold and dense Yukawa theory – to
investigatetheinfluenceoffermionmassesonthepressure.Here,wepresentatwo-loop
calculation of the pressure with massive fermions in the modified minimal subtraction
(MS) renormalization scheme [6], and briefly comment on possible implications to
the physics of condensates in the core of neutron stars and effective models for color
superconductivity.Higher-order corrections and a thorough analysis of renormalization
groupeffects willbepresentedelsewhere[7].
We consideragas ofmassivefermionswhoseinteractionismediated bya real scalar
field, f , with an interaction Lagrangian of the Yukawa form, L =g yyf , where g is
I
the coupling constant. In the zero-temperature limit,the perturbativepressure results in
a power series of a ≡ g2/4p . 1 Up to O(a ), the first non-trivial contributions to the
Y Y
pressure are given by the free massive gas term, P , and the “exchange diagram”, P .
0 1
Using standard methods of field theory at finite temperature and density [8], one can
derive the free gas pressure for fermions of mass m, obtaining in the zero-temperature
limitthefollowingform:
1 5 3 m +p
lim P = m p m 2− m2 + m4 ln f , (1)
T→0 0 12p 2(cid:20) f (cid:18) 2 (cid:19) 2 (cid:18) m (cid:19)(cid:21)
where m is the chemical potential and p = m 2−m2 denotes the Fermi momentum.
f
TheO(a Y) renormalizedcorrection reads [6]p:
a 3 L 2
lim P =− Y u2−p4+m2 3+2ln u , (2)
T→0 1 4p 3(cid:20)4 f (cid:18) m2(cid:19) (cid:21)
where u = m p −m2ln[(m + p )/m] and L is the renormalization scale in the MS
f f
scheme.
Fig. 1 illustrates the effect of modifying the mass on the total pressure to O(a ),
Y
P = P +P . The choice of range for m , and accordingly for the masses, are inspired
0 1
by the scales found in the case of QCD [2]. In the same vein, the coupling is fixed to
a =0.3.Itisclearfromthefigurethatmasscorrectionsbringsignificantchangestothe
Y
pressure, even in the absence of renormalization group (RG) running for the coupling
andthemass.ThefigurealsoshowsthedependenceontherenormalizationscaleL .The
valueschosen are motivatedby the ones which appear in QCD, as before. Althoughthe
effectsofvaryingL appeartoberelativelysmall,itwouldbeprematuretoconcludethat
thisfeature willremainafter implementingtheRG flow.In fact, theresultspresented in
Fig.2mostprobablyunderestimatethescaledependenceofthefullcorrection,sincenot
only the coupling but also the mass will run with L . In the Yukawa theory, in contrast
to QCD, the effect will become larger as we increase the chemical potential. For fixed
coupling,largervaluesofL yieldlargermodificationsinthepressure.However,afterthe
inclusionofRG running,this behavior can be mantained,as should bethe case here, or
becometheopposite,as is the casein QCD, depending on thesignof thebeta function.
Since the L -dependence comes from the term ∼ m2a ln(L /m) in (2), there will be a
Y
competition between the behavior of the renormalization scale L and that of m and a
Y
asfunctionsof m .
Evenattwoloopordermasseffectsbringintoplaylogarithmiccorrectionsoriginated
intheMSsubtractionscheme.Asusual,theybringaboutanon-physicaldependenceon
the renormalization scale L , since one has to cut the perturbative series at some order.
Higher-order computations in this framework are in progress [7], and will give a better
handle on the choice of this scale, which in our case should be a function of m and m.
On the other hand, one can also choose the scale in a phenomenologicalway in a given
model,imposingphysicalconstraintstotheequationofstate, as was donein Ref. [3]to
modelthenon-idealityofQCD atfinitedensitywithmasslessquarks.
1 Sinceweareconcernedonlywiththezero-temperaturelimit,therearenooddpowersofgcomingfrom
resummedcontributionsofthezeroMatsubaramodeforbosonsintheperturbativeseries.
1.2 1.1
1.0
1.0
0.8
0.9
P₀ P₀
P/0.6 P/ Λ = µ
m = 0 MeV 0.8 Λ = 2µ
0.4 m = 100 MeV Λ = 3µ
m = 200 MeV 0.7
0.2
0.0 0.6
300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000
µ [MeV] µ [MeV]
FIGURE1. Pressurenormalizedbythefreefermiongaspressureasafunctionofthefermionchemical
potential.Left:L =2m anddifferentvaluesofthefermionmass.Right:m=100MeVanddifferentvalues
oftherenormalizationscaleL .
The points discussed above might be relevant in the study of effective models for
the cold and dense matter found in the interior of compact stars, especially because
the effects seem to be significant near the critical region. In the context of the NJL
model, e.g., it was shown that a self-consistent treatment of quark masses strongly
affects the competition between different phases [5]. And the mechanism of pairing
in color superconductivity will certainly be influenced [9] by the running of nonzero
quarkmasses.Theinvestigationoftheseissues,aswellastheeffectofnonzerofermion
massesintheformationofothercondensatesin neutronstarmatter,is underway [7].
ACKNOWLEDGMENTS
We thank R. D. Pisarski, J. Schaffner-Bielich and C. Villavicencio for fruitful discus-
sions.Thiswork was partiallysupportedby CAPES, CNPq, FAPERJ andFUJB/UFRJ.
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