Confinement studies in QCD with Dyson-Schwinger equations
NNuclear and Particle Physics Proceedings 00 (2020) 1–5
Nuclear andParticle PhysicsProceedings
Confinement studies in QCD with Dyson-Schwinger equations
Marco Frasca a,1,1 a Via Erasmo Gattamelata, 3, 00176 Rome (Italy)
Abstract
We provide a study of quantum chromodynamics with the technique of Dyson-Schwinger equations in di ff erentialform. In this way, we are able to approach the non-perturbative limit and recover, with some approximations, the ’tHooft limit of the theory. Quark mass in the propagator term goes o ff -shell at low-energies signaling confinement. Acondition for such occurrence in the theory is provided. Keywords:
1. Introduction
Yang-Mills set of Dyson-Schwinger equations can besolved through a class of exact solutions of the 1-pointfunction, similarly to the φ theory [1]. Indeed, boththeories can map each other. Adding quarks to the La-grangian makes the theory no more amenable to an ex-act treatment. But, notwithstanding such a di ffi culty,full QCD can be treated with the identical approachthrough Dyson-Schwinger equations.The technique we use is due to Bender, Milton andSavage [2]. This method has the important advantagethat the di ff erential form of the Dyson-Schwinger equa-tions is retained. This is especially useful when, as inour case, we have exact solutions for the 1-point and2-point equations in the classical case.Therefore, starting from the results for Yang-Millstheory without quarks, the set of Dyson-Schwingerequations for quantum chromodynamics (QCD) be-comes amenable to a perturbative treatment in thestrong coupling limit provided the ’t Hooft limit N →∞ , Ng = constant , Ng (cid:29) ∗ Talk given at 23th International Conference in Quantum Chromo-dynamics (QCD 20, 35th anniversary), 27 - 30 October 2020, Mont-pellier - FR
Email address: [email protected] (Marco Frasca) approximation [3] when the condition for confinementis assumed to have the quark propagator o ff -shell dueto the behaviour of the mass function of quarks [4, 5].Therefore, no free quark is observable being not any-more a state of the theory. In this paper we will followthe derivation given in Ref.[3].
2. Bender-Milton-Savage method
The principal point in the Bender-Milton-Savage(BMS) technique is to derive the Dyson-Schwingerequations retaining their PDE form [2]. In this way,vertexes are never introduced and there is no need tomove to momenta space obtaining cumbersome integralexpressions.We consider the partition function of a given theory,e..g a scalar field theory to fix the ideas, given by Z [ j ] = (cid:90) [ D φ ] e iS ( φ ) + i (cid:82) d x j ( x ) φ ( x ) . (1)For the 1P-function, it is (cid:42) δ S δφ ( x ) (cid:43) = j ( x ) (2)being (cid:104) . . . (cid:105) = (cid:82) [ D φ ] . . . e iS ( φ ) + i (cid:82) d x j ( x ) φ ( x ) (cid:82) [ D φ ] e iS ( φ ) + i (cid:82) d x j ( x ) φ ( x ) (3) a r X i v : . [ h e p - ph ] N ov arco Frasca / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 Then, we set j =
0. We derive this equation again withrespect to j to get the equation for the 2P-function. Weassume the following definition of the nP-functions (cid:104) φ ( x ) φ ( x ) . . . φ ( x n ) (cid:105) = δ n ln( Z [ j ]) δ j ( x ) δ j ( x ) . . . δ j ( x n ) . (4)This will yield δ G k ( . . . ) δ j ( x ) = G k + ( . . . , x ) . (5)This procedure can be iterated to any desired ordergiving, in principle, all the hierarchy of the Dyson-Schwinger equations in PDE form [1]. This is advan-tageous when the solutions for 1P- and 2P-functions areknown in the classical case.
3. 1P and 2P functions of QCD
For our computations, we choose the Landau gaugethat permits to simplify the computations and decouplesthe ghost field.By applying the BMS tecnhinque to the QCD parti-tion function [3], one has for the 1P-functions ∂ G a ν ( x ) + g f abc ( ∂ µ G bc µν (0) + ∂ µ G b µ ( x ) G c ν ( x ) − ∂ ν G ν bc µ (0) − ∂ ν G b µ ( x ) G µ c ( x )) + g f abc ∂ µ G bc µν (0) + g f abc ∂ µ ( G b µ ( x ) G c ν ( x )) + g f abc f cde ( G µ bde µν (0 , + G bd µν (0) G µ e ( x ) + G eb νρ (0) G ρ d ( x ) + G de µν (0) G µ b ( x ) + G µ b ( x ) G d µ ( x ) G e ν ( x )) = g (cid:88) q , i γ ν T a S iiq (0) + g (cid:88) q , i ¯ q i ( x ) γ ν T a q i ( x ) , (6)and for the quarks( i /∂ − ˆ M q ) q i ( x ) + g T · / G ( x ) q i ( x ) = , (7)the mass function being given byˆ M iq = m q I − g T · / W iq ( x , x ) . (8)Here and in the following Greek indexes ( µ, ν, . . . ) arefor the space-time and Latin index ( a , b , . . . ) for thegauge group. It is not di ffi cult to see that, as expected, inthe equations for the correlation functions of lower or-der appear contributions from higher order correlationsfunctions. This is peculiar to the Dyson-Schwingerscheme. We will prove this harmless in the followingsections. We can obtain a reduced set of such equations by us-ing the selected solutions [3] G a ν ( x ) → η a ν φ ( x ) (9)being φ ( x ) a scalar field, and we introduce the η -symbols with the following properties η a µ η a µ = N − .η a µ η b µ = δ ab ,η a µ η a ν = (cid:16) g µν − δ µν (cid:17) / . (10)This yields the set of reduced 1P-function equations ∂ φ ( x ) + Ng ∆ (0) φ ( x ) + Ng φ ( x ) = N − g (cid:88) q , i η a ν γ ν T a S iiq (0) + g (cid:88) q , i ¯ q i ( x ) η a ν γ ν T a q i ( x ) ( i /∂ − ˆ M iq ) q i ( x ) + g T · /ηφ ( x ) q i ( x ) = . (11)We now give here equations for the 2P-functions. Wemake the choice for the gluon 2P-function G ab µν ( x − y ) = (cid:32) η µν − ∂ µ ∂ ν ∂ (cid:33) ∆ φ ( x − y ) (12)where η µν is the Minkowski metric, and ∆ φ ( x − y ) is thepropagator of the φ we introduced above to solve theequations to map them onto. Then, the set of remapped2P-functions is ∂ ∆ φ ( x − y ) + Ng ∆ φ (0) ∆ φ ( x − y ) + Ng φ ( x ) ∆ φ ( x − y ) = g (cid:88) q , i ¯ Q ia ν ( x − y ) γ ν T a q i ( x ) + g (cid:88) q , i ¯ q i ( x ) γ ν T a Q ia ν ( x − y ) + δ ( x − y ) ∂ P ad ( x − y ) = δ ad δ ( x − y )( i /∂ − ˆ M iq ) S i jq ( x − y ) + g T · /ηφ ( x ) S i jq ( x − y ) = δ i j δ ( x − y ) ∂ W aiq ν ( x − y ) + Ng ∆ φ (0) W aiq ν ( x − y ) + Ng φ ( x ) W aiq ν = g (cid:88) j ¯ q j ( x ) γ ν T a S jiq ( x − y )( i /∂ − ˆ M iq ) Q ia µ ( x − y ) + g T · /ηφ ( x ) Q ia µ ( x − y ) + gT a γ µ ∆ φ ( x − y ) q i ( x ) = . (13) arco Frasca / Nuclear and Particle Physics Proceedings 00 (2020) 1–5
4. ’t Hooft limit ’t Hooft limit corresponds to solve the theory when[6, 7] N → ∞ , Ng = constant , Ng (cid:29) . (14)We are assuming a SU(N) gauge group and so, N is thenumber of colors. Therefore, we are able to evaluate ourset of Dyson-Schwinger equations in this limit. We needa perturbation series for a coupling formally running toinfinity as the one proposed in Ref.[8]. To this aim, were-scale x → (cid:112) Ng x . So, e.g., the equation for thegluon field will become ∂ φ ( x (cid:48) ) + ∆ φ (0) φ ( x (cid:48) ) + φ ( x (cid:48) ) = (15)1 (cid:112) Ng √ N ( N − (cid:88) q , i η · γ · T S iiq (0) + (cid:88) q , i ¯ q i ( x (cid:48) ) η · γ · T q i ( x (cid:48) ) . Then, taking formally the ’t Hooft limit, it yields the1P-equations at the leading order ∂ φ ( x ) + Ng ∆ φ (0) φ ( x ) + Ng φ ( x ) = , ( i /∂ − ˆ M iq ) ˆ q i ( x ) = . (16)From this equations we see that the e ff ect of the interac-tions is on the masses. We can solve this set of equationsas φ ( x ) = (cid:115) µ m + (cid:112) m + Ng µ × sn ( p · x + χ, κ ) , (17)being sn a Jacobi elliptical function, µ and χ arbitraryintegration constants and m = Ng ∆ φ (0). Then, κ = − m + (cid:112) m + Ng µ − m − (cid:112) m + Ng µ . (18)This is true provided that the following dispersion rela-tion holds p = m + Ng µ m + (cid:112) m + Ng µ . (19)In the same limit we get the set of 2P-equations ∂ ∆ φ ( x , y ) + Ng ∆ φ (0) ∆ ( x − y ) + Ng φ ( x ) ∆ φ ( x − y ) = g (cid:88) q , i ¯ Q ia ν ( x , y ) γ ν T a ˆ q i ( x ) + g (cid:88) q , i ¯ˆ q i ( x ) γ ν T a Q ia ν ( x , y ) + δ ( x − y ) ∂ P ad ( x − y ) = δ ad δ ( x − y )( i /∂ − ˆ M iq ) ˆ S i jq ( x − y ) = δ i j δ ( x − y ) ∂ W aiq ν ( x , y ) + Ng ∆ φ (0) W aiq ν ( x , y ) + Ng φ ( x ) W aiq ν ( x , y ) = g (cid:88) j ¯ˆ q j ( x ) γ ν T a ˆ S ji ( x − y )( i /∂ − ˆ M iq ) ˆ Q ia µ ( x , y ) + gT a γ µ ∆ φ ( x − y ) ˆ q i ( x ) = . (20)In order to solve this set of equations, we consider ∂ ∆ ( x − y ) + [ m + Ng φ ( x )] ∆ ( x − y ) = δ ( x − y ) (21)that admits the following solution in momenta space [1,8] ∆ ( p ) = M ˆ Z ( µ, m , Ng ) 2 π K ( κ ) × ∞ (cid:88) n = ( − n e − ( n + ) π K (cid:48) ( κ ) K ( κ ) − e − (2 n + K (cid:48) ( κ ) K ( κ ) π × (2 n + p − m n + i (cid:15) (22)being M = (cid:115) m + Ng µ m + (cid:112) m + Ng µ , (23)and ˆ Z ( µ, m , Ng ) a given constant. This gives rise to agap equation for the mass shift m on the theory spectrum m n [9]. Given the gluon propagator, one getsˆ S i jq ( x , y ) = δ i j ( i /∂ − ˆ M iq ) − δ ( x − y )ˆ Q ia µ ( x , y ) = − g (cid:90) d y (cid:48) (cid:88) j ˆ S i jq ( x − y (cid:48) ) T a γ µ ∆ ( y (cid:48) , y ) ˆ q j ( y (cid:48) ) W aiq ν ( x , y ) = g (cid:90) d y (cid:48) ∆ ( x − y (cid:48) ) (cid:88) j ¯ˆ q j ( y (cid:48) ) γ ν T a ˆ S jiq ( y (cid:48) − y ) ∆ ( x , y ) = ∆ ( x − y ) + g (cid:90) d y (cid:48) ∆ ( x − y (cid:48) ) (cid:88) q , i ¯ˆ Q ia ν ( y (cid:48) , y ) γ ν T a ˆ q i ( y (cid:48) ) + (cid:88) q , i ¯ˆ q i ( y (cid:48) ) γ ν T a ˆ Q ia ν ( y (cid:48) , y ) . (24)Finally, the quark propagator can be obtained by thisset of equations as ( i /∂ − ˆ M iq ) ˆ q i ( x ) = i /∂ − ˆ M iq ) ˆ S i jq ( x − y ) = δ i j δ ( x − y ) , (25) arco Frasca / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 given the mass matrixˆ M iq = m q I − g (cid:90) d y (cid:48) ∆ ( x − y (cid:48) ) T a γ ν × (cid:88) k ¯ˆ q k ( y (cid:48) ) γ ν T a ˆ S kiq ( y (cid:48) − x ) . (26)This can be solved by iteration starting from the freequark propagator. When the on-shell condition fails, wewill have quark confinement.
5. Non-local NJL approximation
From eq.(26) we can define the the self-energy Σ ( x , x ) = g (cid:90) d y (cid:48) ∆ ( x − y (cid:48) ) T a γ ν × (cid:88) k ¯ˆ q k ( y (cid:48) ) γ ν T a ˆ S kiq ( y (cid:48) − x ) (27)From this, we can introduce a non-local-Nambu-Jona-Lasinio model (nlNJL) [3]( i /∂ − m q + Σ iNJL ( x , x )) ˆ S i jq ( x − y ) = δ i j δ ( x − y ) (28)being now the quark self-energy computed at the firstiteration by the free quark propagator giving Σ iNJL ( x , x ) = g (cid:90) d y (cid:48) ∆ ( x − y (cid:48) ) T a γ ν × (cid:88) j ¯ˆ q j ( y (cid:48) ) γ ν T a ˆ S ji q ( y (cid:48) − x ) (29)or, in momentum space, Σ iNJL ( p ) = g (cid:90) d p (2 π ) ∆ ( p ) T a γ ν × (cid:88) j ¯ˆ q j ( p ) γ ν T a ˆ S ji q ( p − p ) . (30)This nlNJL-model gives rise, as usual, to a mass gapequation for quarks implying the formation of a con-densate. This generally grants a pole in the quark prop-agator. Failing to find such a pole means that the quarkpropagator has no physical mass states and the quarksare confined [4, 5]. Then, at very low energies M q = m q − Tr Σ iNJL (0) (31)where the trace is over flavors, colors and spinor in-dexes. This yields M q = m q + N f ( N − Ng × (32) (cid:90) d p (2 π ) ∆ ( p ) M q p + M q We consider the gluon propagator neglecting the massshift, as this is generally small as shown in [9], ∆ ( p ) = ∞ (cid:88) n = B n p + m n . (33)Therefore, we have to compute M q = m q + N f ( N − Ng (cid:90) d p (2 π ) × (34) ∞ (cid:88) n = B n p + m n M q p + M q . The corresponding integral can be evaluated exactlywhen a cut-o ff Λ is used, as usual for Nambu-Jona-Lasinio models that are generally expected not to berenormalizable [10]. This yields M q = m q + N f ( N − Ng π ∞ (cid:88) n = B n M q m n − M q ) × m n ln (cid:32) + Λ m n (cid:33) − M q ln + Λ M q . (35)This equation is amenable to a numerical treatment pro-vided that M q ≥ m q and M q (cid:28) Λ , Λ is the nlNJL-modelcut-o ff . The ultraviolet cut-o ff represents, at least, theboundary of the region where asymptotic freedom startsto set in (generally taken at Λ ≈ x = m / Λ and y = M q / Λ having set m n = (2 n + m . The mass gap m can beassumed to be that of the σ meson or f(500) that we fixto m = .
417 GeV [11]. Then, the function to study is y = m q Λ + κα s ∞ (cid:88) n = B n y (2 n + x − y × (cid:34) (2 n + x ln (cid:32) + n + x (cid:33) − y ln (cid:32) + y (cid:33)(cid:35) . (36)From this, we derive the mass function µ ( α s , y ) = y − m q Λ − κα s ∞ (cid:88) n = B n y (2 n + x − y × (37) (cid:34) (2 n + x ln (cid:32) + n + x (cid:33) − y ln (cid:32) + y (cid:33)(cid:35) . that we plotted in fig.1 for its zeros. arco Frasca / Nuclear and Particle Physics Proceedings 00 (2020) 1–5 -8 -6 -4 -2 0 2 4 6 8 M q / -10-8-6-4-20246810 M q = s =0.18 s =0.56 s =1.32 s =2.08 s =2.84 Figure 1: Zeros of the quark mass function µ given by the red curve forthe 3D figure (above) and the corresponding profiles with the physicallimit in red (below). Zero curve exists then, there is a range of physicalparameters where the chiral symmetry is broken. At en-ergies higher than 1 GeV, asymptotic freedom starts toset in and quarks retain their bare masses. At increas-ing α s , the e ff ective quark mass starts to be nonphys-ical overcoming the cut-o ff . So, we have no solutionsand the quark propagator has no physical poles for afree quark. A confinement condition can be straight-forwardly obtained by taking M q = Λ , the limit of thephysical region. This yields α s = min q = u , d , s − m q Λ κξ . (38)being κ = N f N ( N − / π , ξ = ξ ( m / Λ ) afunction justdepending on the mass gap and the UV cut-o ff obtain-able by eq.(36), and α s = g / π .
6. Conclusions
We have derived the set of Dyson-Schwinger equa-tions, till to 2P-functions, for QCD with the Bender-Milton-Savage technique. Then, we were able to solvethem in the ’t Hooft limit. We recognized that thelow-energy limit is given by a nonlocal-Nambu-Jona-Lasinio approximation. Consequently, in the low-energy limit, we were able to show that the model isconfining. A confinement condition was also obtainedobtained.
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