Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a

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1 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 1/30 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a D. Klabučar b, D. Horvatić b, D. Kekez c a Talk at Mini-workshop BLED 2014: QUARK MASSES AND HADRON SPECTRA, Bled, Slovenia, July 6-13, 2014 b Physics Department, University of Zagreb, Croatia c Rudjer Bošković Institute, Zagreb, Croatia

2 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 2/30 Dyson-Schwinger approach to quark-hadron physics = the bound state approach which is nopertubative, covariant and chirally well-behaved. e.g., GMOR relation: lim emq 0 M 2 q q/2 m q = qq /f 2 π a) direct contact with QCD through ab initio calculations b) phenomenological modeling of hadrons as quark bound states (used also here, for example) DS eq s: coupled system of integral equations for Green functions of QCD... but... equation for n-point function calls (n+1)-point function... cannot solve in full the growing tower of DS equations various degrees of truncations, approximations and modeling is unavoidable (more so in phenomenological modeling of hadrons, as here)

3 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 3/30 Phenomenologically most important DS equations: Gap eq. for propagators q of dynamically dressed quarkq = + λ a 2 γµ S q (l) Γ a ν(l, p) Homogeneous Bethe-Salpeter (BS) equation for a Meson q q bound state vertex Γ q q M = K M Γ q q Γ q q S q

4 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 4/30 Gap and BS equations in rainbow-ladder truncation S q (p) 1 = iγ p + m q d 4 l (2π) 4 g2 G eff µν(p l)γ µ S q (l)γ ν S q (p) = 1 ip A q (p 2 ) + B q (p 2 ) = ip A q(p 2 ) + B q (p 2 ) p 2 A q (p 2 ) 2 + B q (p 2 ) 2 = 1 A q (p 2 ) ip + m q (p 2 ) p 2 + m q (p 2 ) 2 Γ q q (p,p) = 4 3 d 4 l (2π) 4g2 G eff µν(p l)γ µ S q (l+ P 2 )Γ q q (l,p)s q(l P 2 )γ ν Euclidean space: {γ µ,γ ν }=2δ µν, γ µ=γ µ, a b = 4 i=1 a ib i P is the total momentum, M 2 = P 2 meson mass 2 G eff µν(k) an effective gluon propagator - modeled!

5 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 5/30 From the gap and BS equations... solutions of the gap equation the dressed quark mass function m q (p 2 ) = B q(p 2 ) A q (p 2 ) propagator solutions A q (p 2 ) and B q (p 2 ) pertain to confined quarks if m 2 q(p 2 ) p 2 for real p 2 The BS solutions Γ q q enable the calculation of the properties of q q bound states, such as the decay constants of pseudoscalar mesons: f PS P µ = 0 q λps 2 γ µ γ 5 q Φ PS (P) d 4 l f π P µ = N c tr s (2π) 4 γ 5γ µ S(l + P/2) Γ π (l; P) S(l P/2)

6 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 6/30 Some renormalization-group improved interactions Landau gauge gluon propagator : g 2 G eff µν(k) = G( k 2 )( g µν + k µk ν k 2 ), G(Q 2 ) 4π αeff s (Q 2 ) Q 2 = G UV (Q 2 ) + G IR (Q 2 ), Q 2 k 2. s (Q 2 ) G UV (Q 2 ) = 4π αperturbative Q 2 4π 2 d Q 2 ln(x 0 + Q2 ) {1+b Λ 2 QCD ln[ln(x 0 + Q2 Λ 2 QCD)] ln(x 0 + Q2 Λ 2 QCD ) }, but modeled non-perturbative part, e.g., Jain & Munczek: G IR (Q 2 ) = G non-pert (Q 2 ) = 4π 2 a Q 2 exp( µq 2 ) (similar : Maris, Roberts...) or, dressed propagator with dynamical gluon mass induced by dim. 2 gluon condensate A 2 (Kekez & Klabučar, PRD 71 (2005) ): 2 G(Q 2 ) = 4π αpert s (Q 2 ) Q 2 Q 2 Q 2 Q 2 M 2 gluon + c ghost Q 2 Q 2 + M gluon 2 + c. gluon Q 2

7 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 7/30 hese effective strong couplings α eff s (Q 2 ) Q 2 G(Q 2 )/4π Q 2 Blue = Munczek & Jain model. Red = K & K propagator with A 2 -induced dynamical gluon mass. Green = Alkofer. Magenta = Bloch. Turquoise dashed: Maris, Roberts & Tandy model. Important: integrated IR strength must be sufficient for DChSB!

8 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 8/30 Agreement with lattice and with perturbative QCD of realistic DS approaches to QCD (also incorporating pqcd at high p 2 ) Lattice data for M(p 2 ) compared with numeric sol ns of gap eq. for realistic DS model in Bhagwat+al, PRC 68, (2003). Dashed curve: sol n in chi.lim, m = 0. Solid crvs: sol ns for M(p 2 ) for current-quark masses m = 30 MeV, 55 MeV, and 110 MeV. Red dashed curve is the chiral-limit solution for M(p 2 ) from the MN model with G = GeV 2, and the solid green curve is the corresponding numerical sol n with m = 5 MeV.

9 Results for B(q 2 ) for the flavors u,d,s,c,b and chiral limit B(q 2 )[GeV] q 2 [GeV 2 ] Our (Kekez+al, Int.J.Mod.Phys. A14 (1999) 161) solutions for propagator function B(q 2 ) in chi. limit m = 0 (solid curve) and for various flavors with various masses m(λ) 0 (dotted curves). Roberts-Frank s Ansatz for u, d quarks is dashed. Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 9/30

10 Results for A(q 2 ) for the flavors u,d,s,c,b and chiral limit A(q 2 ) q 2 [GeV 2 ] Comparison of our chiral-limit solution (solid curve) for the propagator function A(q 2 ) with our massive solutions for various m(λ) 0 given by the dotted lines marked by flavors, and with Roberts-Frank Ansatz for u, d-quarks (dashed line). Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 10/30

11 Results for M(q 2 ) for the flavors u, d, s, c, b and chiral limit B(q 2 )/A(q 2 )[GeV] q 2 [GeV 2 ] Our constituent masses M(q 2 ) = B(q 2 )/A(q 2 ): solid curve is in chiral limit, while dotted ones (marked by pertinent flavors) denote our constituent quark mass functions for m(λ) 0 (Kekez+al, Int.J.Mod.Phys. A14 (1999) 161). Dashed curve is M(q 2 ) following from A(q 2 ) & B(q 2 ) Ansätze of Roberts & Frank+al. Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 11/30

12 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 12/30 Dyson-Schwinger estimate of g 2 0/g 2 24 in GBE RCQM In the chiral limit, f π gives the normalization of the pseudoscalar q q bound-state vertex Γ π, whereas its O(p 0 ) piece is proportional to the scalar propagator function B(q 2 ), generalizing the GT relation: Γ π (q;p 2 =M 2 π =0)= 2B(q2 ) m=0 f π γ 5. In some applications, this is a reasonable approximation also realistically away from the chiral limit, B(q 2 ) m=0 B(q 2 ) u,d,b(q 2 ) s. Following the GT analogy, i.e., assuming that the constant pseudoscalar couplings in GBE RCQM are approximations to low-energy magnitudes of the pseudoscalar q q vertices, yields g 2 0 /g2 24 > 1 due to B(q2 ) s /f s s > B(q 2 ) u,d /f π. Further, assuming non-anomalous nonet yields g0 2/g , in agreement with Plessas, Day & Choi fitting GBE RCQM to phenomenology.

13 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 13/30 Illustrate this with B(q 2 ) u,d /f π and B(q 2 ) s /f s s from the separable model we used in Horvatić & al, Phys.Rev.D76 (2007) [arxiv: ] as separable model good fits, + easier to calculate, especially at T > 0: Calculations simplify with the separable Ansatz for G eff µν: G eff µν(p q) δ µν G(p 2,q 2,p q) G(p 2,q 2,p q) = D 0 f 0 (p 2 )f 0 (q 2 ) + D 1 f 1 (p 2 )(p q)f 1 (q 2 ) two strength parameters D 0, D 1, and corresponding form factors f i (p 2 ). In the separable model, gap equation yields B f (p 2 ) = em f ˆAf (p 2 ) 1 p 2 = 8 Z 3 Z d 4 q (2π) 4 G(p2, q 2, p q) d 4 q (2π) 4 G(p2, q 2, p q) B f (q 2 ) q 2 A 2 f (q2 ) + B 2 f (q2 ) (p q)a f (q 2 ) q 2 A 2 f (q2 ) + B 2 f (q2 ). This gives B f (p 2 ) = m f + b f f 0 (p 2 ) and A f (p 2 ) = 1 + a f f 1 (p 2 ), reducing to nonlinear equations for constants b f and a f.

14 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 14/30 A simple choice for interaction form factors of the separable model (no perturbative part, but omitting it is not important at low energies): f 0 (p 2 ) = exp( p 2 /Λ 2 0) f 1 (p 2 ) = [1 + exp( p 2 0/Λ 2 1)]/[1 + exp((p 2 p 2 0))/Λ 2 1] gives good description of pseudoscalar properties if the interaction is strong enough for realistic DChSB, when m u,d (p 2 small) the typical constituent quark mass scale M ρ /2 M N /3. f 0,1 p f 1 p f 0 p p 2 GeV 2

15 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 15/30 Nonperturbative dynamical propagator dressing Dynamical Chiral Symmetry Breaking (DChSB) gives B u,d and B s yielding g0 2/g similar to GBE RCQM

16 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 16/30 DChSB = nonperturb. generation of large quark masses even in the chiral limit ( m f 0), where the octet pseudoscalar mesons are Goldstone bosons of DChSB!

17 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 17/30 Good DS results for PSeudoscalar mesons π,k and unphysical" s s: Separable model parameter values reproducing experimental data: m u,d = 5.5 MeV, Λ 0 = 758 MeV, Λ 1 = 961 MeV, p 0 = 600 MeV, D 0 Λ 2 0 = 219, D 1 Λ 4 1 = 40 fixed by fitting M π, f π, M ρ, g ρπ+ π, g ρe + e pertinent predictions a u,d = 0.672, b u,d = 660 MeV, i.e., m u,d (p 2 ), ūu m s = 115 MeV (fixed by fitting M K predictions a s = 0.657, b s = 998 MeV, i.e., m s (p 2 ), ss, M s s, f K, f s s ) Summary of results (all in GeV) for q = u, d, s and pseudoscalar mesons without the influence of gluon anomaly: PS M PS M exp PS f PS f exp PS m q (0) q q 1/3 0 π ± K ± s s Using these f π = f uū = f d d and f s s with B u,d and B s shown before, yields g 2 0/g similar to GBE RCQM of Plessas et al.

18 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 18/30 Summary Dyson-Schwinger approach to the pseudoscalar meson nonet provides an explanation for couplings in GBE RCQM DChSB leads to nonperturbatively dressed quarks. Their propagator funct ns A(q 2 ) and B(q 2 ) yield dressed masses M(q 2 ) explaining the notion of constituent quarks & their various relationships with current quarks for all flavors in spite of very different current masses. Thanks to B(q 2 ) s /f s s > B(q 2 ) u,d /f π, Dyson-Schwinger approach explains qualitatively the value g 2 0/g 2 24 > 1 in GBE RCQM. More specifically, assuming non-anomalous pseudoscalar nonet, we get g0/g , in agreement with Plessas, Day & Choi fitting GBE RCQM to phenomenology. Therefore, we propose to try GBE RCQM where: 1.) octet would have π, K but η NS = (uū + d d)/ 2 [degenerate with π!] instead of η 8, and 2.) instead of the singlet η 0, the unphysical" η S = η s s = s s. [Note that off-shell particles need not be mass eigenstates anyway!]

19 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 19/30 Appendix on η-η For easier understanding how we estimated g0 2/g2 24, and what exactly are η NS = (uū + d d)/ 2 and η S = η s s = s s which we propose to try in GBE RCQM, we add this Appendix with additional slides explaining our treatment of the η-η complex based on the references D. Klabučar and D. Kekez, Phys. Rev. D 58 (1998) [hep-ph/ ]. D. Kekez, D. Klabučar and M. D. Scadron, J. Phys. G 26 (2000) 1335 [hep-ph/ ]. D. Kekez and D. Klabucar, Phys. Rev. D 73 (2006) [hep-ph/ ].

20 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 20/30 Anomaly and mixing in η-η complex Dyson-Schwinger approach yields mass 2 eigenvalues M 2 u d = M2 π +,M 2 u s = M 2 K,..., ˆM2 NA = diag(m 2 uū,m 2 d d,m2 s s) u d = π +, u s = K +,... but uū, d d and s s do not correspond to any physical particles (at T = 0 at least!), although in the isospin limit (adopted from now on) M uū = M d d = M u d = M π. I is a good quantum number! recouple into "more physical" I 3 = 0 octet-singlet basis I = 1 π 0 = 1 2 ( uū d d ), I = 0 η 8 = 1 6 ( uū + d d 2 s s ), I = 0 η 0 = 1 3 ( uū + d d + s s ).

21 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 21/30 Anomaly and mixing in η-η complex the non-anomalous (chiral-limit-vanishing!) part of the mass-squared matrix of π 0 and η s is (in π 0 -η 8 -η 0 basis) ˆM 2 NA = 0 M 2 π M 2 88 M M 2 08 M C A M 2 88 η 8 ˆM 2 NA η 8 = 2 3 (M2 s s M2 π), M 2 80 η 8 ˆM 2 NA η 0 = M 2 08 = 2 3 (M2 π M 2 s s) M 2 00 η 0 ˆM 2 NA η 0 = 2 3 (1 2 M2 s s + M 2 π), Not enough! In order to avoid the U A (1) problem, one must break the U A (1) symmetry (as it is destroyed by the gluon anomaly) at least at the level of the masses.

22 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 22/30 Gluon anomaly is not accessible to ladder approximation! P f f f f P Diamond graph: an example of a transition q q q q (q, q = u, d, s[...]), contributing to the anomalous masses in the η-η complex, but not included in the interaction kernel in the ladder approximation.

23 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 23/30 Anomaly and mixing in η-η complex All masses in ˆM NA 2 are calculated in the ladder approx., which cannot include the gluon anomaly contributions. Large N c : the gluon anomaly suppressed as 1/N c! Include its effect just at the level of masses: break the U A (1) symmetry and avoid the U A (1) problem by shifting the η 0 (squared) mass by anomalous contribution 3β. complete mass matrix is then ˆM 2 = ˆM 2 NA + ˆM 2 A where ˆM 2 A = β 1 C A does not vanish in the chiral limit. 3β = M 2 η 0 = the anomalous mass 2 of η 0 [in SU(3) limit incl. ChLim] is related to the YM topological susceptibility. Fixed by phenomenology or (here) taken from the lattice.

24 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 24/30 Anomaly and mixing in η-η complex we can also rewrite ˆM 2 A in the q q basis uū, d d, s s ˆM 2 A = β C A flavor breaking ˆM 2 A = β X 1 1 X X X X 2 1 C A We introduced the effects of the flavor breaking on the anomaly-induced transitions q q q q (q,q = u,d,s). s s transition suppression estimated by X f π /f s s. Then, ˆM2 A in the octet-singlet basis is modified to ˆM 2 A = β (1 2 X)2 2 3 (2 X X2 ) 3 (2 X X2 ) (2 + X) C A In the isospin limit, one can always restrict to 2 2 submatrix of etas (I =0), as π 0 (I =1) is decoupled then.

25 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 25/30 Anomaly and mixing in η-η complex nonstrange (NS) strange (S) basis 1 η NS = ( uū + d d ) = 1 r 2 η η 0, r 2 η S = s s = 3 η η 0. 3 the η η matrix in this basis is ˆM 2 = M2 η NS Mη 2 NS η S Mη 2 S η NS Mη 2 S 1 A = M2 uū + 2β 2βX 2βX M 2 s s + βx 2 1 A φ m2 η 0 0 m 2 η 1 A NS S mixing relations η = cos φ η NS sin φ η S, η = sin φ η NS + cos φ η S. θ = φ arctan 2

26 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 26/30 Anomaly and mixing in η-η complex Let lowercase m M s denote the empirical mass of meson M. From our calculated, model mass matrix in NS S basis, we form its empirical counterpart ˆm 2 exp by i) obvious substitutions M uū M π m π, M s s m s s ii) by noting that m s s, the empirical" mass of the unphysical s s pseudoscalar bound state, is given in terms of masses of physical particles as m 2 s s 2m 2 K m2 π. Then, ˆm 2 exp = [ m 2 ] π + 2β 2βX 2βX 2m 2 K m 2 π + βx 2 φ exp [ m 2 η 0 0 m 2 η ].

27 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 27/30 Finally, fix anomalous contribution to η-η : thetraceoftheempirical ˆm 2 exp demands the 1 st equality in β(2+x 2 ) = m 2 η+m 2 η 2m2 K = 2N f f 2 π χ YM (2 nd equality = WV relation) requiring that the experimental trace (m 2 η + m 2 η ) exp 1.22 GeV 2 be reproduced by the theoretical ˆM2, yields β fit = 1 2+X 2 [(m 2 η + m 2 η ) exp (M 2 uū + M 2 s s)] Or, get β from lattice χ YM! Then no free parameters! then, can calculate the N S-S mixing angle φ tan 2φ = 2 M2 η S η NS M 2 η S M 2 η NS = 2 2βX M 2 η S M 2 η NS and M 2 η NS = M 2 uū+2β = M 2 π +2β, M 2 η S = M 2 s s+βx 2 = M 2 s s+β f2 π f 2 s s

28 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 28/30 Anomaly and mixing in η-η complex The diagonalization of the N S-S mass matrix then finally gives us the calculated η and η masses: M 2 η = cos 2 φ M 2 η NS 2βX sin2φ + sin 2 φ M 2 η S M 2 η = sin2 φ M 2 η NS + 2βX sin2φ + cos 2 φ M 2 η S Equivalently, from the secular determinant, Mη 2 = 1 [ ] Mη 2 2 NS + Mη 2 S (M 2 ηns M 2 ηs ) 2 + 8β 2 X 2 = 1 [ ] Mπ 2 + Ms s 2 + β(2+x 2 ) (M 2 2π+2β M 2s s βx 2 ) 2 + 8β 2 X 2 Mη 2 = 1 [ ] M 2 η 2 NS + Mη 2 S + (M 2 ηns M 2 ηs ) 2 + 8β 2 X 2 = 1 [ ] Mπ 2 + Ms s 2 + β(2+x 2 ) + (M 2 2π+2β M 2s s βx 2 ) 2 + 8β 2 X 2

29 Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 29/30 Separable model results on η and η mesons (at T = 0) β fit β latt. Exp. θ M η M η X β masses are in units of MeV, 3β in units of GeV 2 and the mixing angles are dimensionless. β latt. was obtained from χ YM (T = 0) = (175.7 MeV) 4 X = f π /f s s as well as the whole ˆM 2 NA (consisting of M π and M s s ) are calculated model quantities.

30 For three DS models: summary of T = 0 results from WV from Ref. J-M&WV A 2 &WV separab&wv orig. Shore Experiment M π (138.0) isospin average M K (495.7) isospin average M s s f π ± 0.3 f K ± 1.0 f s s M η ± 0.12 M η ± 0.14 φ o o o o θ o o o 16.5 o θ o 5.12 o 6.80 o 12.3 o θ o o o 20.1 o f f f 0 η f 0 η f 8 η f 8 η Partial overview of Dyson-Schwinger approach to QCD and some applications to structure of hadrons a p. 30/30