Dynamical chiral symmetry breaking in SU(N_{c}) gauge theories with large number of fermion flavors
aa r X i v : . [ h e p - ph ] O c t Dynamical chiral symmetry breaking in SU ( N c ) gauge theories with large number offermion flavors O.Gromenko ∗ Department of Physics Kiev Shevchenko National University,pr.Gluskova 2, Kiev 03022, UkraineDepartment of Physics Clarkson University8 clarkson ave. Potsdam, NY 13699-5822 USA (Dated: December 11, 2018)In this paper we examine a phase transition in SU ( N c ) gauge theories governed by the existence ofan infrared fixed point of the renormalization group β function. The nonlinear integral Schwinger-Dyson equation for a mass function of massless fermions is solved numerically using the exactexpression of the running coupling in two-loop approximation for an SU (3) gauge theory. Basedon the obtained solution of the Schwinger-Dyson equation, the value of the chiral condensate, h ¯ qq i ,and the decay constant, f π , of bound states (mesons) are calculated for several values of fermionflavors N f . We show that this kind of phase transition is a transition of finite order. PACS numbers: 11.30.Rd, 11.30.Qc, 12.38.Aw
I. INTRODUCTION
Gauge field theories with a large number of masslessfermions are becoming an attractive topic for theoreticalresearch. A recent discovery of a phase transition in suchtheories [1], [2] has led to additional interest. Depend-ing on the number of fermions, there are two possiblephases. The first phase is a phase with broken chiral sym-metry and confinement which occurs when the numberof fermions is less than some critical value ( N f < N crf ),where N crf is a critical value of the number of fermions.The second phase, occurs when ( N f > N crf ), is a phasewith strict chiral symmetry and absence of confinement.This type of phase transition occurs for example in min-imal supersymmetrical QCD [3]. The dynamics of thesetwo phases is well studied in the approximation discussedbellow.The reason for the phase transition is the existenceof an infrared fixed point (IFP), with coupling constant α ∗ = α (0), in two loop approximation for the renormal-ization group β function. Also the mathematical aspectof the phase transition becomes clear from the analysisof the integral Schwinger-Dyson equation (SDE) for themass function. Assigning to the running coupling theconstant value at the IFP, α ( Q ) ≡ α ∗ , the SDE turnsout to be an equation for eigenvalues and has only trivialsolution with sufficiently small α ∗ . In contrast, if α ∗ islarger than a critical value α c = π/ (3 C ( F )), then theSDE has nontrivial solutions. The critical value N crf isthen determined from the relation α ∗ = α ∗ ( N f , N c ), atconstant N c . However the situation changes when we usethe running coupling. The kernel of the integral SDE,like the running coupling, is a function of the numberof fermions and colors. And it is unclear what should ∗ Electronic address: Oleksandr.Gromenko @ cern.ch be considered as a critical value of α ∗ . The most obvi-ous solution would be the introduction of a certain in-tegral characteristic for the SDE (as has been done forthe symmetric kernel in the theory of integral equations).We could then consider number of fermions as the freeparameter, which would determine the value of α ∗ .If a chiral symmetry is broken then there exist bosondegrees of freedom, which of course arise as Goldstonebosons. We examine the chiral phase transition by study-ing order parameters like the quark condensate, h ¯ qq i , andthe decay constant, f π , of bound states. Examining thesequantities near the point of phase transition may shedlight on the nature of the transition. In this paper westudy numerical solution of the SDE for a mass functionusing an exact expression for the running coupling. Theexistence of both trivial and nontrivial solutions of theSDE confirms that the phase transition takes place in an SU (3) gauge theory.In section II we discuss the equation for the runningcoupling in two loop approximation. In section III webriefly go over the conservation of Ward-Takahashi iden-tities and set up the SDE in the local gauge. Section IVis devoted to the numerical calculations and discussion. II. SOME PROPERTIES OF THE GAUGETHEORIES WITH THE INFRARED FIXEDPOINT
Let’s start from the Lagrangian of the SU ( N c ) gaugefield theories. It appears as follows: L = N f X k =1 ¯ ψ k ( i b D ) ψ k − F aµν F aµν , (II.1)where ψ k is a four component spinor of flavor k , D µ = ∂ µ − igA µa T a , T a are the generators of the gauge groupand g is the coupling constant. This Lagrangian is clearlyinvariant under the global symmetry group SU ( N f ) L × SU ( N f ) R × U (1) L,R because all fermions are massless.Nonetheless, this symmetry may be broken to diagonalsubgroup SU ( N f ) L + R × U (1) L,R .The next step is an analysis of the equation for therunning coupling in two loop approximation (the firsttwo coefficients are independent of the renormalizationscheme, the higher-order coefficients are scheme depen-dent). It takes the form: dαd ln( Q /ν ) = − bα − cα − ..., (II.2)where α = g / π and according to Ref.[4] coefficients b and c look as follows: b = 112 π (11 C ( A ) − T f N f ) , (II.3) c = 116 π (cid:18) C ( A ) − C ( A ) T f N f − C ( F ) T f N f (cid:19) , (II.4)The theory is asymptotically free if b >
0. The IFP existsif c <
0, which for SU (3) takes place when N f >
8. Therunning coupling at the IFP takes the value α ∗ = − b/c .The fixed point coupling α ∗ can be made sufficientlysmall to perform a calculation by perturbation theory.Certainly we assume that 0 < g . N f . For an SU (3) gauge theory this variation interval is10 . N f < / α ( Q ) + 1 α ∗ ln (cid:18) b + cα ( Q ) (cid:19) = b ln Q Λ , (II.5)where we have introduced the scaleΛ = ν (cid:18) bα s ( ν ) + c (cid:19) c/b exp (cid:18) − bα ( ν ) (cid:19) , (II.6)which has the same physical sense as the dimensionalΛ QCD parameter in ordinary QCD. For further calcula-tions we consider Λ to be independent of N f , for a fixedvalue of N c . This is a good approximation when we aretaking into consideration the small variation interval of N f .Equation (II.5) is a transcendental equation for α ( Q ).It can be solved analytically using complex Lambert W k ( z ) function [5]. Lambert’s function satisfies thetranscendental equation W k ( z ) exp( W k ( z )) = z , where k = 0 , ± , ± , ... . We note that there are only two pos-sible real solutions k = 0 , − z > − /e . The Lambert W function has simpleasymptotics: W ( z → ∼ z − z , W − diverges when z →
0, and W k ( z → ∞ ) ∼ ln z + 2 πik − ln(ln z + 2 πik ). Then we have: α ( Q ) = α ∗ (cid:18) W i (cid:18) − ( Q / Λ ) bα ∗ ce (cid:19)(cid:19) − , (II.7) i = (cid:18) − , c > , c < (cid:19) . The case i = − c < α ( Q →
0) = α ∗ (cid:18) Q / Λ ) bα ∗ ce (cid:19) , (II.8) α ( Q → ∞ ) = 1 b ln( Q / Λ ) (cid:18) bα ∗ ln ln( Q / Λ )ln( Q / Λ ) (cid:19) . (II.9)It is important to note that as the number of fermionsdecreases, the value of asymptote (II.8) increases, andthe value of asymptote (II.9) decreases. III. SCHWINGER-DYSON EQUATION FORTHE MASS FUNCTION
Since the initial Lagrangian has only masslessfermions, we must require conservation of the vector andaxial vector Ward-Takahashi (WT) identities. It is essen-tial for the conservation of the axial WT identity that therunning coupling must depend on the same momentumas the gluon propagator [8]. The use of simple Landaugauge and other local covariant gauges violates the vec-tor WT identity when the ladder approximation with abare vertex is studied, and they are not suitable for thispurpose. This problem can be solved by using a nonlocalgauge which depends on the momentum: D µν = − i (cid:18) g µν − η ( p ) p µ p ν p (cid:19) p . (III.1)This nonlocal gauge was proposed by T.Kugo andM.Mitchard [9], where η ( p ): η ( p ) = 2 p α Z p dy ( yα ( y ) − y α ′ ( y )) . (III.2)It is clear that function (III.2) coincides with the Lan-dau gauge at small and large momenta, i.e., η (0) =1 , η ( ∞ ) = 1.The nonlocal Kugo gauge allows us to write the SDEfor the dressing fermion propagator S ( p ) = i/ ( A ( p )ˆ p − B ( p )). In the ladder approximation it is given by: iS ( p ) − = ˆ p − iC ( F ) Z d k (2 π ) g (( p − k ) ) D µν ( p − k ) iγ µ S ( k ) iγ ν , (III.3)where we use a bare vertex igγ µ T a . The another advan-tage is that in the Kugo gauge the fermion wave functionrenormalisation constant A ( p ) is equal to one. Using (III.3) we may retrieve the SDE for the dynamical massfunction in Euclidean momentum space: B ( p E ) = C ( F ) Z d k E π α (( p − k ) E ) 4 − η (( p − k ) E )( p − k ) E B ( k E ) k E + B ( k E ) . (III.4)However, the obtained equation is rather complicatedand can not be solved analytically without certain as-sumptions and approximations. One of these approxi-mations is to use the constant value at the IFP for therunning coupling, i.e., α ( Q ) = α ∗ . It may be used inthe region of momentum where the running coupling isslowly changing. Or in other words for α ∗ → α c fromabove and N f → N c from below. Expression (III.2) inthis approximation is equivalent to the Landau gauge andtherefore (III.4) becomes: B ( p ) = 3 C ( F ) α ∗ Z d k π p − k ) B ( k ) k + B ( k ) . (III.5) Equation (III.5) can be integrated over angular part: Z d k p − k ) = π p Z k dk p + π ∞ Z p k dk k . (III.6)In general this type of angular integration can not beperformed exactly in the more complicated cases. Sub-stituting (III.6) into (III.5), we obtain the final integralequation for the mass function: B ( p ) = 3 C ( F ) α ∗ π p Z dk k p B ( k ) k + B ( k ) + Λ ∗ Z p dk B ( k ) k + B ( k ) , (III.7)were we have introduced the cutoff parameter Λ ∗ . By us-ing simple differentiation with respect to p , this integralequation converts to a differential equation: p B ′′ ( p ) + 2 B ′ ( p ) + 3 α ∗ C ( F )4 π B ( p ) p + B ( p ) = 0 , (III.8)with two boundary conditions: dp B ( p ) dp | p =Λ ∗ = 0 , lim p → p dB ( p ) dp = 0 . (III.9)The solution of this equation can be found in terms of ahypergeometric function which has the following asymp-totic form at small momentum: B (0) ≈ Λ ∗ exp − C p α ∗ /α c − ! , (III.10) The origin of the critical constant α c now becomes fairlyclear. For recent reviews of the SDE and their applicationsee for example Refs. [6], [7]. IV. NUMERICAL CALCULATIONS ANDRESULTS
In this section we discuss the method which was usedto solve the SDE (III.4). First of all we note that it is im-possible to perform the angular integration analyticallyin this case. To solve the nonlinear SDE, we use a sim-ple quadrature method. All calculations were performedusing Mathematica software and consisted of the follow-ing steps. We set up a square lattice ( p i , k j ) where both p i and k j pass the number of discrete values from lowerboundary q to upper boundary Λ ∗ and then carry out FIG. 1: Numerical solution of the Schwinger-Dyson equation.Here B/ Λ is dimensionless dynamical mass functionTABLE I: Numerical values of the physical quantities ob-tained from the numerical solution of the SDE N f B (0) / Λ −h ¯ qq i / Λ · − f π / Λ10 0.0951 5.3224 0.0279110.3 0.0540 2.2542 0.0184510.5 0.0336 1.0331 0.012610.7 0.0186 0.3686 0.007711 0.0055 0.0473 0.0028 numerical integration using these lattice sites. Next, wereplace the integral in (III.4) by quadrature sum of therectangles. As a result we obtain a system of nonlinearequations, where unknown variables serve as the valuesof the unknown mass function at the lattice sites: B ( p i ) = C ( F ) π k j =Λ ∗ X k j = q k j e K ( p i , k j ) B ( k j ) k j + B ( k j ) ∆ , (IV.1)where ∆ is the lattice step.Since the unknown function is smooth, it is possibleto use a quadrature formula of rectangles. Also, thesame results are reproduced when we use a more pre-cise quadrature trapezoid formula. The newly obtainedsystem of nonlinear equations was solved by an iterationmethod. We have calculated the mass function for a se-ries of values of N f near N crf .In Fig.1 we illustrate the dependence B/ Λ on p/ Λ forvarious numbers of fermions. The behavior of the massfunction is as expected: it differs from zero at smallmomentum and then go smoothly to zero at large mo-mentum. Also the quantity B/ Λ is obviously dimension-less and cannot depend on the transmutation parame-ter Λ. The mass function B increases when the numberof fermions decreases. There is a similar tendency in(III.10). The cuttoff Λ ∗ is approximately equal to 0 . α ( Q ) ∼ α ∗ is useful and it is interesting to compare B (0) with (III.10). We willdo this below.For better understanding of the physical nature of thephase transition, it is also useful to calculate other phys-ical quantities. One of them is the value of the vacuumcondensate, which can be easily found: h ¯ qq i = − lim x → +0 trS ( x,
0) = − N c N F π Z Λ ∗ dp E p E B ( p E ) p E + B ( p E ) , (IV.2)and the value of meson decay constant (it is described bywell known Pagels-Stokar formula) f π = N c N F π Z Λ ∗ dp p B ( p )( p + B ( p )) (cid:18) B ( p ) − p B ( p ) dp (cid:19) (IV.3)These values have also been calculated using quadratureformulas and are illustrated in Table I.Let us analyze the quantities B (0), h ¯ qq i , f π in more de-tail. These quantities are continuous functions near thecritical point. This fact confirms that the phase tran-sition is a transition of second or higher order, possiblyof infinite order. For this reason, the quark condensate,decay constant and B (0) may be fitted by polynomialfunctions near the critical point. B (0) ∼ (cid:0) N crf − N f (cid:1) α , (IV.4) h ¯ qq i ∼ (cid:0) N crf − N f (cid:1) β , (IV.5) f ∼ (cid:0) N crf − N f (cid:1) ρ . (IV.6)If we find that the critical exponents α , β , ρ are smallreal numbers, it will confirm that the phase transition isof finite order. Note, that Eg. (III.10) describes a phasetransition of infinite order as oppose to finite. The leastsquares fitting of the curves gives: B (0) : α = 2 . , N crf = 11 .
43; (IV.7) h ¯ qq i : β = 2 . , N crf = 11 .
21; (IV.8) f : ρ = 1 . , N crf = 11 . . (IV.9)We indeed find that the critical exponents are smallpositive numbers. The value of the critical number offermions is described by the well known formula: N crf = N c (cid:18) N c − N c − (cid:19) , obtained from the condition α ∗ and α c [1]. In case N c =3, the critical number of fermions is N crf = 11 . V. CONCLUDING REMARKS
In this paper we have shown numerically that the SDEfor a fermion propagator with an exact expression for therunning coupling has nontrivial solution B ( p ). Based onthe obtained solution, the value of chiral condensate andthe decay constant of pseudoscalar bosons were calcu-lated. These physical quantities are continuous functionsnear the critical point. Detailed analysis of B (0) nearthe critical point shows that the phase transition is atransition of finite order. Acknowledgments
Author would acknowledge V.P.Gusynin for manyhelpful discussions and useful notations. We also thank S.I.Vilchinsky for his support and the warm hospitalityat the Kiev Taras Shevchenko University. Also we thankDaniel T. Robb for careful reading and correction gram-mar mistakes. [1] T. Appelquist, A. Ratnaweera, J. Terning andL. C. R. Wijewardhana, Phys. Rev.
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