Quark running mass and vacuum energy density in truncated Coulomb gauge QCD for five orders of magnitude of current masses
aa r X i v : . [ h e p - ph ] D ec Quark running mass and vacuum energydensity in truncated Coulomb gauge QCD forfive orders of magnitude of current masses
P. Bicudo
CFTP, Departamento de F´ısica, Instituto Superior T´ecnico,Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
Abstract
We study in detail the effect of the finite current quark mass on chiral symmetrybreaking, in the framework of truncated Coulomb gauge QCD with a linear con-fining quark-antiquark potential. In the chiral limit of massless current quarks, thebreaking of chiral symmetry is spontaneous. But for a finite current quark mass,some dynamical symmetry breaking continues to add to the explicit breaking causedby the quark mass. Moreover, using as order parameter the mass gap, i. e. the quarkmass at vanishing moment or the quark condensate, a finite quark mass transformsthe chiral symmetry breaking from a phase transition into a crossover. For the studyof the QCD phase diagram it thus is relevant to determine how the current quarkmass affects chiral symmetry breaking. Since the current quark masses of the sixstandard flavours u, d, s, c, b, t span over five orders of magnitude from 1.5 MeVto 171 GeV, we develop an accurate numerical method to study the running quarkmass gap and the quark vacuum energy density from very small to very large currentquark masses.
Key words: elsart , quark mass, chiral symmetry breaking, vacuum energydensity
PACS:
Chiral symmetry breaking has been studied in detail with chiral invariant andconfining quark models in the chiral limit of a vanishing current quark mass m . For a finite current quark mass m = 0, studies exist with an approximate Email address: [email protected] (P. Bicudo).
Preprint submitted to Elsevier 23 November 2018 onfinement [1,2,3,4] or with a quadratic confining potential [7,6,5], but veryfew studies have been performed [8,9] with an exactly linear confining poten-tial. Since the current quark masses of the six standard flavours u, d, s, c, b, t span over five orders of magnitude from 1.5 MeV to 171 GeV, here we developa new numerical method to study the quark mass gap and the quark vacuumenergy density, with a linear exactly confining potential, from very small tovery large quark masses.Notice that even in the chiral limit of m = 0, the quark has a constituentrunning mass m ( p ) function of the momentum p , solution of the mass gapequation (equivalent to the Schwinger-Dyson equation) for the quark. Recentlywe have shown how to measure in the excited hadron spectra the running mass m ( p ) [10]. The chiral invariant and confining quark models have also beenapplied to phase studies at finite temperature T and chemical potential µ [11,12,13,14,15,16,17,18]. A finite quark mass is relevant both for the study ofhadrons which have been investigated for decades, and for the study the QCDphase diagram which will be explored in the future at RHIC, LHC and FAIR.In the phase diagram, a finite current quark mass m affects the position ofthe critical point between the crossover at low chemical potential µ and thephase transition at higher µ . Moreover the current quark mass affects the QCDvacuum energy density E , relevant for the dark energy of cosmology. This alloccurs in the dynamical generation of the quark mass m ( p ). While the quarkcondensate h ¯ ψψ i is a frequently used order parameter for chiral symmetrybreaking, the mass gap, i. e. the quark mass at vanishing momentum m (0)is another possible order parameter for chiral symmetry breaking. However,due to technical difficulties, m (0) has not been computed in detail previouslyin confining and chiral invariant quark models.Here we study in detail how a finite current quark mass m = 0 affects thedynamically generated quark mass m ( p ) . We utilize the linear confining poten-tial for the quark-antiquark interaction, in the chiral quark model or CoulombGauge quark model, including both confinement and chiral symmetry. Whilethis model, in the framework Coulomb gauge Hamiltonian formalism is notyet full QCD, it is presently the only model able to include both the quark-antiquark confining potential and quark-antiquark vacuum condensation. Im-portantly, since our model is well defined and solvable, it can be used as asimpler model than QCD, and yet qualitatively correct, to address differentaspects of hadronic physics. Is is adequate to study the QCD phase diagrammicroscopically [11,12,13,14,15,16,17,18]. We scan the current quark masses,from the light quarks to the heavy quarks, computing the running quark mass m ( p ) with detail, including the infrared limit of m (0), i. e. the mass gap.In Section II we derive in detail the mass gap equation. In Section III we reviewthe numerical difficulties of this non-linear integral equation, with cancellinginfrared divergences. We solve the mass gap equation with a new numerical2ethod, dedicated to determine in detail the difficult infrared region of smallmomentum p ≃
0. In Section IV we discuss our results and conclude.
Our framework can be approximately derived from QCD, in two differentgauges. In Coulomb gauge ∇ · A ( x , t ) = 0 the interaction potential, has beenderived by Lee, [19], and by Szczepaniak and Swanson [20,21]. In the presentstudy we address the quark fields only and thus V I reduces to the quark partof the density-density term, V I = + 12 g Z d x d y J − ψ † ( x )T a ψ ( x ) h x , a | × ( ∇ · D ) − ( −∇ )( ∇ · D ) − | y , b iJ ψ † ( y )T b ψ ( y ) (1)The covariant derivative in the adjoint representation D = ∇ − g A , and J = Det[ ∇· D ] contribute to the density-density interaction, which is expectedto be confining in QCD.Another approximate path from QCD to our model considers the modifiedcoordinate gauge of Balitsky [22], A ( , t ) = 0 , x · A ( x , t ) = 0 and in theinteraction potential for the quark sector, V I = Z d x (cid:20) ψ † ( x ) ( m β − i~α · ~ ∇ ) ψ ( x ) + 12 g Z d yψ ( x ) γ µ λ a ψ ( x ) h A aµ ( x ) A bν ( y ) i ψ ( y ) γ ν λ b ψ ( y ) + · · · (2)retains the first cumulant order, of two gluons [23,24,25] g h A aµ ( x ) A bν ( y ) i andthis also results in a simple density-density harmonic effective confining inter-action. As in QCD, this only has one scale.Thus our framework is similar to an expansion of the QCD interaction, trun-cated to the leading density-density term. Moreover, to address phenomenol-ogy where the meson spectrum fits in linear Regge trajectories, one also needsto assume that the confining quark-antiquark potential is a linear potential.Notice that the short range Coulomb potential could also be included in theinteraction, but here we ignore it since it only affects the quark mass through3ltraviolet renormalization [26], which is assumed to be already included inthe current quark mass. While this is not exactly equivalent to QCD, ourframework maintains three interesting aspects of non-pertubative QCD, achiral invariant quark-antiquark interaction, [27,28,29,30,31,32] the cancel-lation of infrared divergences [7,6,5] and a quark-antiquark linear potential[8,9,20,33,34,35,36]. We derive the mass gap equation, where constituent quarks acquire the con-stituent mass m ( k ) [37] in the true and stable vacuum, solving the Schwinger-Dyson equation for the quark propagator, S − ( p ) = S − ( p ) − ✛ kp − k ···· · ···· . (3)We utilize the truncated Schwinger-Dyson equation at the Rainbow level,where the dotted line represents the same density-density interaction V I re-sulting identically from the truncation of Coulomb gauge QCD or of Balitskygauge QCD. This leads to the same mass gap equation and quark dispersionrelation obtained assuming a quark-antiquark P condensed vacuum, com-puting the vacuum energy density with the Hamiltonian, and minimizing theenergy density. The relativistic invariant Dirac-Feynman [31] propagators S ,can be decomposed in the quark and antiquark Bethe-Goldstone propagators[37], S ( k , ~k ) = i k − m + iǫ (4)= ik − √ k + m + iǫ X s u s u † s β − i − k − √ k + m + iǫ X s v s v † s β , where m is the quark mass and where the quark spinor u s and antiquark spinor v s are, u s ( k ) = s S s − S b k · ~σγ u s (0) v s ( k ) = s S − s − S b k · ~σγ v s (0) , (5)where S = m/ √ k + m is a function of the quark mass.4mportantly, in the free propagator, the correct quark propagator in the noncondensed vacuum, the quark mass m is equal to the current quark mass m .And it is this current quark mass m which effects in the current quark masswe study in great detail. However when chiral symmetry breaking occurs, m isnot determined from the onset. In the physical vacuum, the constituent quarkmass m ( k ), is a function of the momentum, a dynamical solution of the massgap equation.Replacing the propagator of eq. (5) in the Schwinger-Dyson equation andprojecting it with the spinors, we get the mass gap equation and the quarkdispersion relation,0 = u † s ( k ) ( k b k · α + m β − Z dk ′ π d k ′ (2 π ) i e V ( k − k ′ ) S ( k ′ , ~k ′ ) ) v s ′′ ( k ) (6) E ( k ) = u † s ( k ) ( k b k · α + m β − Z dk ′ π d k ′ (2 π ) i e V ( k − k ′ ) S ( k ′ , ~k ′ ) ) u s ( k ) , where the usual notation for Dirac matrices is assumed. Writing the runningmass in terms of a sine and a cosine of ϕ ( k ) = arctan km ( k ) , the chiral angle , S ( k ) = sin ϕ ( k ) = m ( k ) q k + m ( k ) ,C ( k ) = cos ϕ ( k ) = k q k + m ( k ) , (7)the mass gap equation and the quark energy are,0 = + S ( p ) B ( p ) − C ( p ) A ( p ) (8) E ( p ) = + S ( p ) A ( p ) + C ( p ) B ( p ) (9)where the propagator functions A and B , respectively replacing the quarkmass m and quark momentum | p | in the one-loop dressed quark propagatorof eq. (4) are, A ( p ) = m + 12 Z d k (2 π ) e V ( p − k ) S ( k ) ,B ( p ) = p + 12 Z d k (2 π ) e V ( p − k )(ˆ p · ˆ k ) C ( k ) . (10)5quivalently to solve the non-linear integral mass gap equation (8), we canalternatively minimize the vacuum energy density per unit volume, E = − g Z d p (2 π ) S ( p ) [ A ( p ) + m ] + C ( p ) [ B ( p ) + p ] (11)where g = N f N s N c is the degeneracy factor counting the number of differentbut degenerate quarks. N s = 2 is the number of spins and N c = 3 is thenumber of colours. N f is the number of degenerate flavours, but since eachquark has a different current quark mass m one should compute separatelythe vacuum energy difference for each quark flavour. Notice that in the case of a linear potential, divergent in the infrared limit oflarge r , the Fourier transform needs an infrared regulator µ . A possible formof the linear potential, V ( r ) = − σ e − µ r µ ≃ − σµ + σ r − σµ r + · · · (12)corresponds, in the limit of small infrared regulator µ , to a model of linearconfinement where the quark also has an infinite binding energy − σµ . Whileother infrared regulations can be used for the linear potential, say V ( r ) = σr e − µ r which has no infrared divergent binding energy, the infrared divergentconstant of Eq. (12) is exactly cancelled in the mass gap equation by the factorin the integrand [ S ( k ) C ( p ) − S ( p ) C ( k )ˆ k · ˆ p ]. The potential in Eq. (12) has asimple three-dimensional Fourier transform, V ( k ) = σ − π ( k + µ ) , (13)and this is the momentum space potential frequently utilized to account forlinear confinement.The integrals in the angular variable ω of Eq. (10) are, Z − dω − π ( k + p + 2 kpω + µ ) = − π [( k − p ) + µ ][( k + p ) + µ ] , Z − dω − π ω ( k + p + 2 kpω + µ ) (14)6 − π (2 kp ) (cid:26) − kp ( k + p + µ )[( k − p ) + µ ][( k + p ) + µ ] + 12 log " ( k + p ) + µ ( k − p ) + µ . We find for the propagator functions A and B , A ( p ) = m − σp ∞ Z dk π I A ( k, p, µ ) S ( k ) ,I A ( k, p, µ ) = pk ( p − k ) + µ − pk ( p + k ) + µ ,B ( p ) = p − σp ∞ Z dk π I B ( k, p, µ ) C ( k ) ,I B ( k, p, µ ) = pk ( p − k ) + µ + pk ( p + k ) + µ + 12 log ( p − k ) + µ ( p + k ) + µ , (15)leading to the mass gap equation in the two equivalent forms of a non-linearintegral functional equation,0 = pS ( p ) − m C ( p ) − σp ∞ Z dk π [ (16) I A ( k, p, µ ) S ( k ) C ( p ) − I B ( k, p, µ ) S ( p ) C ( k )] , and of a minimum equation of the energy density, E = − g π ∞ Z dp π (cid:20) p C ( p ) + 2 p m S ( p ) + σ × (17) ∞ Z dk π I A ( k, p, µ ) S ( k ) S ( p ) + I B ( k, p, µ ) C ( p ) C ( k ) (cid:21) . Eq. (16) can be rewritten as a fixed point equation for the quark mass function m ( k ), m ( p ) = m + σp ∞ Z dk π I A ( k, p, µ ) m ( k ) p − I B ( k, p, µ ) m ( p ) k q k + m ( k ) . (18)Since the potential has an infinite constant independent of the mass, in thevariational equation we search for the extremum of the energy difference7 − E = − g π ∞ Z dp π (cid:26) p [ C ( p ) − C ( p )] + 2 p m [ S ( p ) − S ( p )] (19)+ σ ∞ Z dk π I A ( k, p, µ ) [ S ( k ) S ( p ) − S ( k ) S ( p )]+ I B ( k, p, µ ) [ C ( p ) C ( k ) − C ( k ) C ( p )] (cid:27) . where E is constant, and where we use the subindex when the mass m ( p )is substituted by the constant current mass m . These two Eqs. (18) and (19)constitute the main object of our study. We use both Eq. (18) and the minimization of Eq. (19) to find the dynamicalquark mass m ( k ), but first we must regulate correctly their divergences. Theinfrared divergences are present in the term pk/ [( p − k ) + µ ], infrared diver-gent in the limit of a vanishing regulator µ →
0, which is present bot in thefunctions I A ( k, p, µ ) and I B ( k, p, µ ). We must show that this infrared diver-gence is cancelled both in the fixed point Eq. (18) and in the energy densityEq. (19). In what concerns the fixed point Eq. (18), while the denominatordiverges quadratically in 1 / ( k − p ) , the numerator m ( k ) p − m ( p ) k is of order( k − p ) and thus the integrand diverges like 1 / ( p − k ) only, and it’s integral hasfinite principal value. In the energy density difference the infrared divergencealso cancels, since the numerator common to the infrared divergent terms, S ( p ) S ( k ) − S ( p ) S ( k ) + C ( p ) C ( k ) − C ( p ) C ( k )= − × [ ϕ ′ ( p ) − ϕ ′ ( p )]( k − p ) (20) −
12 [ ϕ ′ ( p ) ϕ ′′ ( p ) − ϕ ′ ( p ) ϕ ′′ ( p )]( k − p ) + o ( p − k ) is then of order ( k − p ) .In the ultraviolet part of the integrals, while each sub-term in the propagatorfunction integrands I A ( k, p, µ ) and I B ( k, p, µ ) is divergent, the actual sum isultraviolet convergent, since in the limit of large k we have,8 I A ( k, p,
0) = ∞ X n =1 n p n +1 k n ,k I B ( k, p, µ ) = ∞ X n =1 n n + 1 n + ! p n +1 k n , (21)and thus the integrals in k have ultraviolet integrable integrands decayinglike p /k . Also, the ultraviolet divergence in the kinetic terms of the energydensity 2 p C ( p ) + 2 p m S ( p ) cancels due to the difference with 2 p C ( p ) +2 p m S ( p ) if the mass difference m ( k ) − m vanishes sufficiently fast in theultraviolet.To address correctly both the infrared and ultraviolet divergences of the in-tegrals in k of Eqs. (18) and (19), we divide the integral in two sections, theinfrared one for 0 < k < p and the ultraviolet one for 2 p < k < ∞ . Inthe infrared region we compute the respective principal value, performing asymmetric sum centred in k = p , maintaining a very small regulator µ just tocancel automatically the contribution of k = p . In the ultraviolet region weuse the change of variable [33] of Adler and Davis k → x/ (1 − x ) with Jacobian1 / (1 − x ) and with integration between x = 2 p/ (1 + 2 p ) and 1. The changeof variable in the ultraviolet transforms, say an integral of rational functions1 / (1 + k ) n into the integral of polynomials (1 − x ) n − and thus it is adequateto the integral of rational functions as we have here. Our numerical integralsin k of a generic integrand I singular in p are thus computed in the form, ∞ Z dk I ( k ) = P p Z dk I ( k ) + Z p p dx (1 − x ) I (cid:18) x − x (cid:19) , (22)where each of the two numerical integrals can either be computed with arectangular, trapezoidal or Simpson sum or with the gauss quadrature method.I one would discretize the quark mass m ( p ) in a series of momenta p i , thenthe integrals of Eq. (22) loses accuracy. Finite differences would require manyinterpolations, both in the infrared end and in the ultraviolet end, since theprincipal value requires requires that k has many summation points smallerthan p and many other larger than p . Moreover the correct integration of theintegrand in the ultraviolet large k limit, where the integrand behaves like (cid:16) pk (cid:17) , also requires an integration extending beyond any value of p . To solvethis problem, we utilize a well defined ansatz for m ( k ), formally describe theparametrized as, m ( p ) = m ( p ; c , c , · · · , c n ) . (23)and this allows the numerical summation for the integrals in any point of the9ntegration domain.In what concerns the numerical convergence to the solution for m ( k ), thefixed point equation is relatively unstable, particularly in the infrared regionof p ≃
0, when we are searching for the vacuum groundstate. Notice that themass gap equation had not only one, but an infinite tower of solutions [38]. Theexcited solutions dominate the fixed point iteration, converging to the largereigenvalue of the iterated matrix, because they minimize the denominator √ k + m . Previously in the literature, the fixed point method was providedwith extra stability with two different methods, Adler and Davis used a cubicequation and relaxation, to select the best solution [33]. Bicudo and Nefedievquasi-linearized the fixed point equation and selected the desired eigenvalue,corresponding either to the stable vacuum or to excited, false vacua [38]. Thusthey were able to find both the stable vacuum and the excited false vacua.But these works have not yet determined in detail m (0), since this demands avery large numerical precision, and since most previous authors have focusedin computing the function S ( p ) which in the infrared region is S (0) = 1regardless of the actual finite value of m (0). Importantly, the present techniqueof minimizing the energy density directly tends to the right solution, which isthe groundstate vacuum.Interestingly, the variation of the ansatz parameters c , c , · · · , c n of the energydensity of the vacuum E = E ( c , c , · · · , c n ) utilized in minimization codeswith gradient method, utilizes the fixed point equation. Computing the partialderivatives of the energy density we get, ∂ E ( c i ) ∂c i = Z dp δ E δϕ ∂ϕ∂c i = − g π ∞ Z dp π ( − p ) R ( p ; c i ) pp + m ( p ; c i ) ∂m ( p ; c i ) ∂c i , (24)where R ( p ; c i ) is the right hand side of the mass gap Eq. (8), R ( p ; c i ) = + S ( p ; c i ) B ( p ; c i ) − C ( p ; c i ) A ( p ; c i ) , (25)utilized in the fixed point Eq. (18). m ( p )To guide our choice of ansatze m ( p ; c , c , · · · , c n ), we first notice that the seriesexpansion for I A and I B in Eq. (21), also apply when k ↔ p . When replacedin the integral of the fixed point equation (18) , the series suggest that a series10xpansion of m ( p ) should only have even terms, i. e. m ( p ) should be a functionof p . m ( p ) should also be a finite function since our integrals are finite.In what concerns the asymptotic ultraviolet tail of the integral of the fixedpoint equation (18), there are two different limits we can address. In the caseof a large current quark mass m , m k + m interpolates between 1 in the infraredregion of the integral and m k in the ultraviolet region of the integral. Usingthese approximation, the components of the integral are analytical, ∞ Z dk π p I A ( k, p, µ ) = p µ , (26) ∞ Z dk π k I B ( k, p, µ ) = p µ , ∞ Z dk π pk I A ( k, p, µ ) = p arctan pµ π µ = p µ − pπ + o ( µ ) , ∞ Z dk π I B ( k, p, µ ) = − pµ + ( p + µ ) arctan pµ π µ = p µ − pπ + o ( µ ) , and thus adding the respective components we find that in the infareddominated approximation the integrals cancel, while in the ultraviolet domi-nated approximation the integral with m k produces an ultraviolet behaviourof m ( p ) − m → m σπ p . Thus in the case of large m we expect that m ( p ) − m decays in the ultraviolet like 1 /p .In the case where m ≃
0, assuming then that in the large p limit the dy-namical quark mass vanishes sufficiently fast, the fixed point equation leadsto, m ( p ) → σp ∞ Z dk π q k + m ( k ) k p m ( k ) p → σp ∞ Z dk π k m ( k ) q k + m ( k ) (27)and, providing m ( p ) decays faster than 1 /p for a finite integral, this decaysin the ultraviolet like 1 /p . 11he simplest possible ansatz for m ( p ) − m we may have, function of p ,and encompassing both the behaviour in 1 /p for a large current quark mass m and the behaviour in 1 /p for a small current quark mass is the rationalfunction, A ( p ) = 1 c + c p + c p . (28)This ansatz is a Pad´e approximant, and to check whether our simple ansatz issufficient, it is convenient to check that the next Pad´e approximant, a moreflexible rational ansatz with two more parameters, A ( p ) = 1 + n p d + d p + d p + d p (29)leads to the same result. In both ansatze of Eq. (28) and Eq. (29) we assumethat the parameters are positive. While Eq. (28) is a decreasing function,Eq. (29) may have a different behaviour at the origin, either with an initialincrease, or with a steeper decrease, an thus it has room in it’s parameterspace to verify if the ansatz of Eq. (28) is close to the correct solution of themass gap equation.We also check numerically that ansatze with steeper infrared behaviours, in-cluding in the denominator terms like a c k or a c − /k would not improvethe solution since the best solution would have c = c − = 0. A better ansatzthan A ( p ) is A ( p ), however the improvement of the solution is very small,almost invisible to the naked eye in graphics. The partial redundancy betweenthe numerator and denominator parameters of A ( p ) already slows the con-vergence to the minimum of the energy density E . Thus an ansatz with moreparameters than A ( p ) is not necessary. The only ansatze we adopt here arethe ones of the rational functions A ( p ) and A ( p ). m >> √ σ limit We now compute the first iteration of the fixed point method starting with m ( k ) = m . In the simple case case of a constant mass m ( k ) = m , wecan compute the integral in the fixed point equation with a large precision.Defining the mass difference D , D ( p ) = m ( p ) − m , (30)12e compute D ( p ) in the case a constant mass m is used in the integrand, D ( p ) = m σp ∞ Z dk π I A ( k, p, µ ) p − I B ( k, p, µ ) k q k + m . (31)We get an accurate result for the integral D ( p ), with a numerical integrationdecomposed according to Eq. (22).This provides a good quark mass solution to m ( p ) = m + D ( p ) whenever thecurrent quark mass is large, i. e. when m >> D ( p ). In that case the integral D ( p ) only needs to be computed once, since this one-loop approximation isalready excellent.Moreover we can rescale in m , and then with a single computation we get m ( p ) for for any constant mass m . Denoting ˜ k = k/m and so forth we get, D ( p ) = σm ˜ p ∞ Z d ˜ k π I A (˜ k, ˜ p, ˜ µ ) ˜ p − I B (˜ k, ˜ p, ˜ µ ) ˜ k q ˜ k + 1= σm F ( pm ) , F ( p ) = 1 p ∞ Z dk π I A ( k, p, µ ) p − I B ( k, p, µ ) k √ k + 1 . (32)Thus we only need to compute the dimensionless function F ( p ) and this willproduce the dynamical quark mass for any current quark mass m larger thanthe typical scale √ σ of our problem, m ( p ) ≃ m + σm F ( pm ) . (33)The solution of the integration, obtained with the simplest numerical rect-angular sum in both integrals, but needing 10 integration points at least, isrepresented in Fig. 1.To be able to perform an accurate integration both in the infrared and theultraviolet, it is necessary to have an analytical function. We obtain an analyt-ical function by fitting the function F ( p ) , and we get an excellent fit alreadywith the very simple ansatz A ( p ) of Eq. (28) for the parameter set, c = 4 . , c = 3 . , c = 0 . , (34)with almost no graphically visible difference in Fig. 1 from the ansatz A ( p )with two more parameters of Eq. (29) for the parameter set,13 Fig. 1. Dimensionless mass difference function F (˜ p ) computed in the limit whenthe current quark mass m is large. The dots show the numerical integral of Eq.(32) and the two almost overlapping solid lines show our fits with the two differentrational ansatze of Eq. (28) and Eq. (29). d = 4 . , d = 4 . , d = 0 . , d = 0 . , n = 0 . , (35)and both fits confirm the 1 /p decay of the mass in the ultraviolet, for largecurrent quark masses.Now we can include the large quark limit scaling in m , and also apply to thequark mass difference D ( p ) the ansatz A ( p ) of Eq. (28), and in this case theansatz parameters should be respectively, c = 4 . m , c = 3 . m − , c = 0 . m − . (36)We further compute the vacuum energy density difference for a large currentquark mass m . Although the mass difference D ( p ) in Eq. (32) decreases like m − in the large current quark mass m limit, a dimensional analysis of E −E shows that possibly it does not vanish. Since the energy difference is finite andindependent of the infrared cutoff µ , then our only dimensionfull parametersare the string tension σ (scaling like a mass square) and the current quarkmass m (scaling like a mass). Because the vacuum energy density per volumescales like a mass to the fourth power, a large mass expansion may have termsof the form m , m σ , σ , σ m − , · · · However the first term in this seriesclearly vanishes in the vacuum energy difference
E − E . Then the question iswhat is the first non-vanishing term in this expansion.14 E−E g c c c . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . .
536 0 . . A ( p ) for m ( p ) − m obtained with ourminimization code. All results are in dimensionless units of σ = 0 .
19 GeV = 1. To answer this question, we expand the vacuum energy density in a σm series,similar to the variation in Eq. (24), but now based in the expansion of thequark dynamical mass, m ( p ) = m + σm m (cid:18) pm (cid:19) + σ m m (cid:18) pm (cid:19) + · · · (37)and then we get E = − g π ∞ Z dp π (cid:26) p (cid:20) pC ( p ) + m S ( p ) + pδC ( p ) + m δS ( p ) δm ( m ( p ) − m )+ 12 pδ C ( p ) + m δ S ( p ) δm ( m ( p ) − m ) + · · · (cid:21) (38)+ σ ∞ Z dk π I A ( k, p, µ ) (cid:20) S ( p ) S ( k ) + 2 δS ( p ) δm S ( k ) ( m ( p ) − m ) + · · · (cid:21) + I B ( k, p, µ ) (cid:20) C ( p ) C ( k ) + 2 δC ( p ) δm C ( k ) ( m ( p ) − m ) + · · · (cid:21)(cid:27) . The zeroth order term vanishes when we perform the difference of the vacuumenergy densities
E − E . Then the first order term in the kinetic energy density15 E−E g d d d d n A ( p ) for m ( p ) − m obtained with ourminimization code. All results are in dimensionless units of σ = 0 .
19 GeV = 1. also vanishes since the kinetic energy density is minimized by m ( p ) = m .Thus the leading term is of second order in the kinetic energy density, and offirst order in the potential energy density, and both are proportional to σ .Using the intermediate steps, pδ C ( p ) + m δ S ( p ) δm = − p ( p + m ) / , (39) σ ∞ Z dk π (cid:20) δS ( p ) δm I A ( k, p, µ ) S ( k ) + I B ( k, p, µ ) C ( k ) (cid:21) δC ( p ) δm = 2 p ( p + m ) / σm D (cid:18) pm (cid:19) , and changing variable to the dimensionless ˜ p , we get, E − E = σ − g π ∞ Z d ˜ p π ˜ p [ F (˜ p )] (˜ p + 1) + o σ m ! . (40)Finally using the ansatz A ( p ) of Eq. (28), with the parameter set of Eq. (34)this results in 16 æ æ æ æ æ æ æ æ æ æ æ - - - - - D Ε H a L æ æ æ æ æ æ æ æ æ æ æ æ - - H b L æ æ æ æ æ æ æ æ æ æ æ æ æ - - H c L æ æ æ æ æ æ æ æ æ æ æ æ æ - H d L Fig. 2. Plots of our numerical solution with the ansatz A ( p ), as a function of thecurrent quark mass m : (a) the vacuum energy density shift E − E , (b) parameter c , (c) parameter c , (d) Parameter c . The dots show our numerical solution, thesolid line is the large m limit obtained with Eqs. (36) and (41), and the verticaldot-dashed lines represent the current masses of the quarks u , d , s , c , b , t . E − E ≃ − − . × − σ g . (41) Utilizing our ansatze of Eq. (28) and Eq. (29), we may now compute withgreat accuracy the integrals of Eq. (19) which now are a function of the ansatzparameters, and apply a standard minimizing code to determine the optimalparameters. We use 1000 × A ( p ), due to the partial redundancy of the parameters. Then wealso minimize the energy starting from different randomly generated initialvalues for the parameters. In Tables 1 and 2 we only show the digits which arestable, in the sense that they do not depend on the initial values. Notice thatthe energy density obtained with the two different ansatze differ only by afew per mil and that the ansatz of Eq. (29) already exhibits some redundancein the parameters. This shows that the ansatz of Eq. (28) is already quiteaccurate for the parametrization of the quark mass m ( p ).17 æ æ æ æ æ æ æ æ æ æ æ æ - - H L - m0 H a L æ æ æ æ æ æ æ æ æ æ æ æ æ - H L H b L Fig. 3. Plots of the mass gap as a function of the current quark mass m : (a) themass gap difference m (0) − m , measuring the amount of generated dynamical mass,(b) the mass gap m (0) . The dots show our numerical solution, the solid line is theleading order obtained when m → ∞ , and the vertical dot-dashed lines representthe current masses of the quarks u , d , s , c , b , t . Notice that the dynamical massgeneration has a maximum for finite quark masses close to the strange quark mass.All results are in dimensionless units of σ = 0 .
19 GeV = 1. Unlike the fixed point method, converging quite fast (with a single iteration)for large current quark masses m >> √ σ , the variational method convergesfaster for small current quark masses. Thus both methods are complementary.In Tables 1 and 2 and in Fig. 2 we show the results of our minimization fortwo ansatze and for different current quark masses spanning over five orders ofmagnitude. In Figs. 3 and 4 we illustrate the mass difference at the momentumorigin m (0) − m , the mass gap m (0) and the running mass m ( p ) for differentcurrent masses m .Now that we have an excellent and simple ansatz for the running quark mass,we may compute the regularized quark condensate and the quark dispersionrelation. The quark condensate h ¯ ψψ i is another possible order parameter, to becompared with the other order parameter we computed, i. e. the quark massat vanishing momentum m (0). The quark condensate is computed from theone-loop quark propagator functions in Eq. (10), and it is ultraviolet divergentfor finite quark masses. Thus we regularize the quark condensate, subtractingthe quark condensate for a current quark, h ¯ ψψ i − h ¯ ψψ i = − gπ ∞ Z k dk m ( k ) q k + m ( k ) − m √ k + m (42)The one quark dispersion relation E ( p ), defined in Eq. (9), is relevant for theboundstate equation of mesons or of baryons. For instance, in the instanta-neous Salpeter equation, a hamiltonian H = E q + E ¯ q + V q ¯ q can be definedfor mesons (actually the hamiltonian is a matrix [37] including negative and18 .0 0.5 1.0 1.5 2.0 2.5 3.0 p D H p L Fig. 4. The quark mass function difference D ( p ) = m ( p ) − m measuringthe extent of dynamical mass generation, is represented with increasing num-ber of dashes per curve for five different current quark masses m with values10 − , − , − / , , / , in dimensionless units of σ = 0 .
19 GeV = 1. positive energy components). The dispersion relation E ( p ) is infrared diver-gent due to the infinite constant of quark-antiquark potential detailed in Eq.(12). In momentum space, this leads to an infinite Dirac delta in the integralpresent in E ( p ). We can regularize the numerical integral, subtracting a termto cancel the integrand when k = p , a term that we add back analytically, E ( p ) = pC ( p ) + m S ( p ) + σp ∞ Z dk π (cid:26) I A ( k, p, µ ) [ S ( k ) − S ( p )] S ( p ) (43)+ I B ( k, p, µ ) [ C ( p ) − C ( p )] C ( k ) + h I A ( k, p, µ ) S ( p ) + I B ( k, p, µ ) C ( p ) i(cid:27) , in particular the integral of I A S + I B C is analytical thanks to Eq. (26),and in the limit of a vanishing infrared regulator µ → σp ∞ Z dk π I A ( k, p, µ ) S ( p ) + I B ( k, p, µ ) C ( p ) → σ µ − σπ pp − m ( p ) . (44)Importantly, this divergence is physically irrelevant since, in the hamiltonianof any colour singlet hadron, the sum of the divergences of the quark and of theantiquark energies cancel with the infrared divergence of the quark-antiquarkpotential detailed in Eq. (12). Finally for the purpose of future computations19 (cid:16) −h ¯ ψψ i + h ¯ ψψ i g (cid:17) e e e e . . . . .
65 451 . . . . . .
23 448 . . . . . .
94 436 . . . . . .
304 399 . . . . . .
717 317 . . . . . .
176 173 . . . . . .
484 49 . . . . . .
582 3 . . . . . . . . . . . . . . . .
999 2 . . . . . .
113 2 . . . σ = 0 .
19 GeV = 1. of the hadron spectra, it is convenient to write the dispersion relation as asum of the analytical infrared term of Eq. (44), plus the free quark dispersionrelation dominating the ultraviolet, and plus a finite and compact term e E ( p ),an integral that we compute numerically, E ( p ) = σ µ − σπ pp − m ( p ) + pC ( p ) + m S ( p ) + ˜ E ( p ) . (45)The numerical integral e E ( p ) decays like 1 /p in the chiral limit of small currentquark masses and decays like 1 /p in the case of large current quark masses. e E ( p ) is negative and we can conveniently fit it with the rational function, orPad´e approximant with odd powers of p only, e E ( p ) ≃ − e + e p + e p + e p . (46)We show the best fitting parameters e , e , e , e in Table 3. With our excellentfits of the dynamical quark mass m ( p ) and of the quark dispersion relation E ( p ) we are well equipped to address further problems, such as the breakingof chiral symmetry or the hadronic excited spectra at finite temperature T .20 æ æ æ æ æ æ æ æ æ æ æ - - < qq > H a L p - - E H p L - σ (cid:144)H L Fig. 5. (a) We show a log log plot of minus the regularized quark condensate −h ¯ ψψ i + h ¯ ψψ i as a function of the current quark mass m , with vertical dot–dashed lines representing the current masses of the quarks u , d , s , c , b , t . (b) Werepresent the regularized quark dispersion relation E ( p ) − σ µ with increasing num-ber of dashes per curve for five different current quark masses m with respectivevalues 10 − , − , − / , , / . The quark condensate has an inflection pointfor finite quark masses close to the strange quark mass, and for large masses it riseslinearly with m . All results are in dimensionless units of σ = 0 .
19 GeV = 1. While the chiral limit of m << √ σ was already well known in the literature,we find unanticipated effects for finite m ≃ √ σ and for heavy m >> √ σ current quark masses. We study in detail the large m limit, performing an one-loop expansion in the dimensionless number σ/m . We also develop a newtechnical approach to solve the mass gap equation, utilizing the variationalprinciple to increase the precision of our mass solution m ( p ) in the infraredlimit of p →
0, relevant to compute the mass gap m (0), an order parameterfor the chiral phase transition. We also show that the dynamical generatedconstituent quark mass m ( p ) can be quite well fitted by our inverse evenquartic polynomial ansatz, a Pad´e approximant with parameters c , c and c depending only on the current mass m .Our surprising results are that the dynamical mass generation has finite effectspersistent beyond the chiral limit. At m ≃ √ σ , in particular for massessimilar to the strange quark s mass, the quark mass generation m (0) − m is maximum, as shown in Fig. 4, while one would naively expect the quarkmass generation to be maximum in the chiral limit close to the up u or down d quark masses. A second order parameter, the regularized quark condensate h ¯ ψψ i − h ¯ ψψ i , monotonously grows in absolute value with m and shows aninflexion point for masses similar to the strange quark s mass, as depicted inFig. 5. In the limit of heavy quark masses m >> √ σ of the charm c , bottom b and top t quarks, although the mass gap difference m (0) − m vanishes like21 /m , it occurs that the energy density difference E − E is maximum. In theheavy quark limit, the energy density difference converges to a constant limitwhen m → ∞ , and we show in Fig. 2 that it is one order of magnitude largerthan in the chiral limit. This may be relevant for cosmology, contributing tothe dark energy.The numerical variational technique developed here, together with the detailedsolutions for the running quark mass m ( p ) and for the quark dispersion relation E ( p ) as a function of the current quark mass m and of the string tension σ ,are necessary tools for the continuation of our program to study the QCDphase diagram and the hadron spectrum at finite temperature T and density µ , when the string tension σ becomes quite small, possibly smaller than thelight current quark masses. acknowledgements I am very grateful to Gast˜ao Krein on the variational method, and to MarleneNahrgang, to Pedro Sacramento and to Jan Pawlowski for discussions on theQCD phase diagram motivating this paper. I acknowledge the financial sup-port of the FCT grants CFTP, CERN/ FP/ 109327/ 2009 and CERN/ FP/109307/ 2009.
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