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Dynamical Self-mass for Massive Quarks

arXiv:hep-ph/9206250v1 25 Jun 1992SFU-Preprint-92-6Dynamical Self-mass for Massive QuarksZheng Huang and K.S. ViswanathanDepartment of PhysicsSimon Fraser UniversityBurnaby, B.C.Canada V5A 1S6AbstractWe examine dynamical mass generation in QCD with large current mass quarks.A renormalization group analysis is performed to separate fermion self-mass into adynamical and a kinematical part.

It is shown that the energy scale of the Schwinger-Dyson (SD) equation and the effective gauge coupling are fixed by the current mass.The dynamical self-mass satisfies a homogeneous SD equation which has a trivial so-lution when the current mass exceeds a critical value. We therefore suggest that thequark condensate, as the function of the current mass, observes a local minimum aroundeΛQCD.November 15, 20181

1IntroductionDynamical chiral symmetry breaking in QCD has been extensively studied [1]. The standardtool to study this problem is the Schwinger-Dyson (SD) equation, i. e. the fermion gapequation.

Since one is usually interested in the chiral limit one studies the gap equation inthe limit of the vanishing current quark mass. In the presence of a quark whose current massis much larger than ΛQCD, the nature of solutions to SD equation may exhibit non-analyticbehavior in m. It has long been argued by Pagels [2] that if the current mass is large, it ispossible that m ̸= 0 does not belong to an analytic extension of m = 0.

In this article weexplore this scenario using the renormalization group equation (RGE) analysis.Recently, Langfeld, Alkofer and Reinhardt [3] have reported on a numerical study of theSchwinger-Dyson equation for massive quarks in the background field of a classical vacuumsolution. Besides the usual spontaneous chiral symmetry breaking solution for m = 0, theyfind that the total quark condensate as a function of the current mass exhibits a discontinuousdrop at mc ≃70MeV .

This is interesting because not much has been known about thebehavior of quark condensates for quarks of large current mass.However, in a recent conference report [4] we have discussed the problem of dynamicalmass generation in QCD with large current mass (m ≫ΛQCD). It was pointed out that usingthe renormalization group analysis one can separate unambiguously the quark self-mass (thepart of the self-energy which commutes with γ5) into a dynamical and a kinematical part.The dynamical part BD satisfies a homogeneous SD equation but with an effective couplingconstant ¯g2(t) ≃(lnmΛQCD )−1 and hence when m ≫ΛQCD it describes weak coupling regime.As a consequence, the SD equation for BD can be solved quite reliably in a single effectivegluon exchange and one finds that BD has only a trivial solution when m exceeds a criticalvalue mc which is of the order of 2.7ΛQCD.

The kinematical part BK, however, is a non-singular function of the current mass and satisfies an inhomogeneous equation and henceis proportional to m. The total fermion condensate can be written similarly as a sum of a2

dynamical and kinematical part which is expected to take a drop near current mass of theorder of mc1.The aim of this paper is to elaborate on our method which utilizes the renormalizationgroup techniques. The dynamical self-mass may be understood as arising from an effectiveinteraction with the gluons.

It would be of interest to explore if the dip in condensate hasany observable consequences [5].2The Renormalized SD EquationThe quark self-mass B(p2) satisfies the following SD equation (the gap equation) in themomentum spaceB(p2) = m + ig2C2(N)4TrZd4k(2π)4Dµν(p −k)γµS(k)Γν(p, k)(1)where S(k), Dµν(k) and Γν(p, k) are the complete quark, gluon propagators and the propervertex respectively.m is the current mass; and C2(N) =N2−12Nfor SU(N).The quarkpropagator S is related to B byS−1(k) ≠k+ ̸E(k) −B(k2)(2)where ̸E(k) is the wave function correction factor. Before trying to solve (1), we must rewriteit in terms of renormalized quantities and specify a renormalization prescription.

We takethe point of view that the renormalization constants in the renormalized SD equation may bedefined as in perturbation theory. We also require that after carrying out the renormalizationprescription we get a finite SD equation.

We adopt the dimensional regularization and massindependent renormalization scheme [6]. We define renormalized quantities as followsS(p; g, m, ξ; ǫ)=ZF(g(µ); ǫ)SR(p; g(µ), m(µ), ξ(µ); µ);1We have neglected the non-trivial topological gauge configurations.

Inclusion of them may lead to acontribution from the gluon condensate. However, the essential feature of the behavior of the total quarkcondensate should not be substantially affected by this.3

Dµν(p; g, m, ξ; ǫ)=ZA(g(µ); ǫ)DµνR (p; g(µ), m(µ), ξ(µ); µ);Γµ(p; g, m, ξ; ǫ)=Z−1F Z−1/2AΓRµ (p; g(µ), m(µ), ξ(µ); µ)(3)B(p; g, m, ξ; ǫ)=Z−1F BR(p; g(µ), m(µ), ξ(µ); µ)whereg(ǫ)=Zg(g(µ); ǫ)g(µ);m(ǫ)=Zm(g(µ); ǫ)m(µ);ξ(ǫ)=Zξ(g(µ); ǫ)ξ(µ).The renormalized quantities are functions of g(µ) and ǫ. ξ in (3) is the gauge parameter.Substituting (3) into (1) we obtain the renormalized integral equation (in Euclidean 4 −ǫdimensions)BR(p; µ)=ZF(ǫ)[Zm(ǫ)m(µ) + Z2g(ǫ)Z1/2A (ǫ)g2(µ)C2(N)(4)· µǫ4 −ǫTrZd4−ǫk(2π)4−ǫγµDµνR (p −k; µ)SR(k; µ)ΓνR(p, k; µ)].In (4) we have written SR(p; g(µ), m(µ), ξ(µ); µ) as SR(p; µ) for short.Eq. (4) is in general not self-consistent since its right hand side contains divergent renor-malization constants as ǫ →0.

It has been suggested by Johnson, Baker and Willey [7] thata suitable choice of the gauge parameter ξ (Landau gauge) would lead to a finite ZF as ǫ →0.Subsequently, it would be possible to require that the divergences arising from the integralcancel those in Zm(µ, ǫ)m(µ) and make (4) finite. Thus, a consistent solution to (4) shouldsatisfy the condition of finiteness of ZF(µ, ǫ) as ǫ →0.

We show below using RGE analysisthat when the quark current mass is large, this condition can be approximately satisfied.4

3Renormalization Group AnalysisEquation (4) contains the exact propagators and vertex.If the current mass is small,BR(p; g, m, ξ, µ, ) may be expanded in powers of m(µ)BR(p; g, m, ξ; µ)=BR0 (p; g, 0, ξ; µ)|{z}BRD(5)+ m(µ)BR1 (p; g, 0, ξ; µ) + m(µ)2BR2 (p; g, 0, ξ; µ) + · · ·|{z}BRKwhere we call the first term B0 as the ‘dynamical’ part of BR and other terms as the‘kinematical’ part. A non-vanishing dynamical part signals the spontaneous chiral symmetrybreaking.

(5) is just the chiral perturbation expansion, which can only make sense when theseries is convergent or m is small (m ≪ΛQCD). We would like to develop a scheme applicablewhen m > ΛQCD, when the power series solution does not converge.

A convenient methodin this case is to apply RGE treatment. We may write (4) as a functional equationF(p; g(µ), m(µ), ξ(µ); µ) = 0.

(6)From (6) it follows that(µ ∂∂µ + β(g) ∂∂g + γmm ∂∂m + δ(g)ξ ∂∂ξ)F = 0. (7)In a mass-independent renormalization scheme, β(g), γ(g), and δ(g) depend only on g(µ).Introducing t byt = ln mPµ(8)with mP = m(µ = mP) and observing that F is homogeneous of order 1 in p, m and µ, wehave( ∂∂t + p ∂∂p + µ ∂∂µ −1)F(p; g, etm′, ξ; µ) = 0.

(9)5

In (9) m′(µ) = e−tm(µ). From (7) and (9), it follows that F satisfies the following RGE:(−∂∂t −p ∂∂p+β(g) ∂∂g + γmm′ ∂∂m′+δ(g)ξ ∂∂ξ + 1)F(p; g, etm′, ξ; µ) = 0.

(10)From (10) it follows thatF(p; g, etm′, ξ; µ) = etF(e−tp; ¯g(t), ¯m′(t), ¯ξ(t); µ)(11)where the effective parameters ¯g(t), ¯m′(t), and ¯ξ(t) are defined byt =¯g(t)Zgdxβ(x),(12)¯m′(t) = m′(µ) exp¯g(t)Zgdxγm(x)β(x) ,(13)and¯ξ(t) = ξ(µ) exp¯g(t)Zgdx δ(x)β(x). (14)It should be noted that even though t is a function of µ, the effective coupling ¯g(t) dependson mP only.An explicit calculation of (12)-(14) gives the following results in one-loopapproximation in the minimum subtraction schemeα(mP) ≡¯g2(t)4π=2πβ0 lnmPΛQCD,(15)¯m′(t) = e−tm(µ) exp¯g(t)Zgdxγm(x)β(x) = e−t ¯m(t) = e−tm(µ = mP) = µ,(16)and¯ξ(t) = 1 −1ξ0(lnmPΛQCD )dξ ,(17)6

where β0 = 33−2nf3, dξ =39−4nf2(33−2nf ) [8] and ξ0 is a constant of integration. It is interesting toobserve that ¯m′(t) is just the scale parameter µ.

This arises because we have defined t bythe on-shell current mass mP .The physical consequence of the RGE analysis follows from equation (11). Instead ofsolving the SD equation (1) with a large current mass m directly by approximation proceduresfor Γν and Dµν, we can solve, equivalently an effective SD equation govern by a much smallermass µ (= ¯m′(t)) and running coupling and gauge parameters.

A suitable approximation tothese are given in (15)-(17). It is seen from (17) that when mP ≫ΛQCD, ¯ξ(t) ≃1 (Landaugauge) and ¯g(t) →0 as mP →∞.

Thus a large current mass justifies the single gluonexchange approximation for the effective SD equation and ensures the compatibility of thecondition ZF(¯g, ¯ξ, ǫ) →1 + O(¯g2) as ¯ξ →1, whereas, it would have been hard to justify suchan approximation in (1).As a consequence of these results, the renormalized Green’s functions DµνR and ΓµR maybe approximated byDµνR (p; ¯g, µ, ¯ξ; µ)=δµν −pµpν/p2p2+ O(¯g2);ΓµR(p; ¯g, µ, ¯ξ; µ)=γµ + O(¯g2). (18)Substituting (18) into the effective SD equation as given in (11), we arrive at the RGEimproved effective SD equation to O(¯g2)B(p2) = Zm(¯g; ǫ)µ + 3¯g2C2(N)Z d4−ǫk(2π)4(2πµ)ǫ(p −k)2B(k2)k2 + B2(k2) + O(¯g4),(19)where we have used the notation B(p2) ≡BR(p; ¯g, µ, ¯ξ; µ).

Finally, from a solution to (19),we can reconstruct BR(p; g(µ), m(µ), ξ(µ); µ) defined in Eq. (4) from the following equation,which is easily derivedBR(p; g(µ), etm(µ), ξ(µ); µ)(20)= et exp[−¯g(t)ZgdxγF(x)β(x) ]BR(e−tp; ¯g(t), µ, ¯ξ(t); µ).7

Now in (19) the inhomogeneous term contains a mass of the order of ΛQCD, we can thususe a chiral perturbation theory to expand BR(p; ¯g(t), m0, ¯ξ(t); µ) as power series in m0 (inour case, of course, m0 = µ. We use a different symbol to make this decomposition moreobvious)BR(p; ¯g(t), m0, ¯ξ(t); µ)=BRdyn(p; ¯g(t), 0, ¯ξ(t); µ)+m0BRkin(p; ¯g(t), m0, ¯ξ(t); µ)(21)where we have called the term independent of m0 as the dynamical self-mass and have lumpedthe rest as the kinematical part.

In this way of splitting, it is obvious that the kinematicalpart is a regular function of the m0 and it goes to zero as m0 →0. But what is crucial isthat both Bdyn and Bkin depend on mP in a non-analytical way through ¯g(t).Substituting (21) into (20) we obtain the following structure for the self-mass for a heavyquark (taking γF = 0)BR(p; g(µ), m(µ), ξ(µ); µ) =etBdyn(e−tp; ¯g(t))+m(µ) exp(−Z ¯g(t)gdxγm(x)β(x) )Bkin(e−tp; ¯g(t), m0).

(22)It is to be emphasized again that the separation of the self-mass into a dynamical and akinematical part is not the usual power series expansion in current mass. In fact, substituting(21) into (19) we derive an effective homogeneous SD equation for BdynBdyn(p; ¯g(t)) = 3¯g2(t)C2(N)Zd4k(2π)41(p −k)2Bdyn(k2)k2 + B2dyn(k2) + O(¯g4)(23)while Bkin satisfiesBkin(p; ¯g(t)) = 1 + 3¯g2C2(N)Zd4k(2π)41(p −k)2Bkin(k2)(k2 + B2dyn(k2))2(k2 −3B2dyn).

(24)Both (23) and (24) have been studied in the context of chiral symmetry breaking [9].8

4Solutions and DiscussionsThe effective SD equations for the heavy quark differ from the those for the light quark wherethe chiral symmetry breaking is of concern. In the latter case the gauge coupling constant isa function of the momentum transfer.

In the infrared range the coupling becomes arbitrarilylarge and a non-trivial solution exists for light quarks. In (23) and (24) the coupling constantis fixed and allows us to seek for a solution in the wider range (not just the asymptotic solutionin the limit p2 →∞).

However, a non-trival solution to (23) should not be referred to asthe signal of chiral phase transition.Eq. (23) is equivalent to the following differential equationx2d2Bdyn(x)dx2+ 2xdBdyn(x)dx+ λ xBdyn(x)x + B2dyn(x) = 0(25)together with the boundary conditions (x = p2)xdBdyn(x)dx+ Bdyn(x)x=∆= 0;x2dBdyn(x)dxx=δ= 0(26)where ∆→∞and δ →0 and λ = 3¯g2C2(N)/16π2.

We are interested in finding not somuch in existence of non-trivial solution to (25) but in determining when the trivial solutionBdyn(x) = 0 is in fact a stable solution. To examine the stability of the trivial solution, letus linearize (25) about Bdyn = ε(x).

We findx2ε(x) + 2xε(x) + λxε(x) = 0(27)withxε(x) + ε(x)|x=∆= 0;x2ε(x)x=δ = 0(28)whose general solution isε(x) = c1xρ+ + c2xρ−(29)whereρ± = −12 ± 12√1 −4λ. (30)9

c1 and c2 are to be determined by the boundary conditions. If c1 and c2 are non-zero, thenthe solution ε(x) →∞as x →0 and thus the trivial solution is not stable.

Thus stabilityrequires c1 = c2 = 0. Substituting (29) into the b.c.

(28), we find the following condition forthe trivial solution to be stable:1 +√1 −4λ1 −√1 −4λ = ( δ∆)√1−4λ2. (31)If λ ≤1/4, a solution to (31) is not possible since the left hand side of (31) is finite while theright hand side tends to zero in the limit δ →0 and ∆→∞.

If λ > 0,√1 −4λ is purelyimaginary, then (31) is a transcendental equation for λ [10]. For fixed ∆and δ, it has aninfinite set of solutions for λ which becomes dense over the whole domain λ > 1/4 as ∆/δbecomes large.

Hence the critical point is λc = 1/4 or αc = π/4. For λ ≤λc we have a stabletrivial solution Bdyn(x) = 0.

The critical current mass ismc = ΛQCD exp 2πβ0αc≃eΛQCD. (32)Implicit in the analysis above is the assumption that there are no other stable solutions.The solution to (24) for Bkin can be derived by the iteration process while Bdyn = 0 mustbe substituted into (24).

To the first order one has Bkin = 1 + O(¯g2(t)) and the kinematicalpart of total self-mass reads from (22)BRK(p; g(µ), m(µ), ξ(µ); µ) ∼= m(µ)(b ln mµ )−c/b(1 + O(¯g2))(mP ≫ΛQCD)(33)where c = 3C2(N)/8π2 and b = β0/8π2.We have concerned ourselves with the self-mass for the heavy quark. Having assumed thechiral phase transition for light quarks, we may draw some conclusions for the self-mass of themassive quark as the function of its current mass.

In the range of m ≃0, the BRD dominates.As m becomes large, the kinematical part grows almost linearly and the dynamical partchanges slowly. At mP = mc, the dynamical part drops to zero and the total self-mass willexhibit a local minimum.

As m gets even larger, the self-mass is completely governed by10

the kinematical part (except for the possible contributions from the gluon condensate) andeventually blows up as shown in (33) when m →∞. The quark condensate defined as⟨¯qq⟩=−Zd4k(2π)4TrSF(k)=−Zd4k(2π)4TrBRKk2 −(BRK + BRD)2 −Zd4k(2π)4TrBRDk2 −(BRK + BRD)2=⟨¯qq⟩D + ⟨¯qq⟩K(34)is expected to observe the same drop when BRD = 0.11

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