Spectral representation of nonperturbative quark propagators for - PowerPoint PPT Presentation

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Spectral representation of nonperturbative quark propagators – for microscopic calculations in strong matter NewCompStar Annual Conference 2017, Warsaw, Poland 27.–31. of March 2017. car (1) in collaboration with Dalibor Kekez (2) Dubravko Klabuˇ (1) Physics Department, Faculty of Science – PMF, University of Zagreb, Croatia (2) Rudjer Boˇ skovi´ c Institute, Zagreb, Croatia

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Overview Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis f π calculation Spectral representation Stieltjes transform 3R Quark Propagator Quark loops Electromagnetic form factor Transition form factor Quark Propagator with Branch Cut Form factors Conclusions

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Introduction • Many QFT studies (on lattice, large majority of Schwinger–Dyson studies, etc.) are not done in the physical, Minkowski spacetime, but in 4-dim Euclidean space. • ⇒ Situation with Wick rotation (relating Minkowski with Euclid) must be under control, but this is highly nontrivial in the nonperturbative case – most importantly, the nonperturbative QCD. • Do nonperturbative Green’s functions permit Wick rotation? • For solving Bethe-Salpeter equation and calculation of processes, extrapolation to complex momenta is necessary. ⇒ Knowledge of the analytic behavior in the whole complex plane is needed. • Very complicated matters ⇒ studies of Ansatz forms are instructive and can be helpful to ab initio studies of nonperturbative QCD Green’s functions. ... (and vice versa of course) ... • Among them, quark propagator is the “most unavoidable” one, especially for quark matter applications.

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra How Schwinger–Dyson approach generates quark propagators • Schwinger-Dyson (SD) approach: ranges from solving SD equations for Green’s functions of non-perturbative QCD ab initio , to higher degrees of phenomenological modeling, esp. in applications including T , µ > 0. e.g., [Alkofer, v.Smekal Phys. Rept. 353 (2001) 281], and [Roberts, Schmidt Prog.Part.Nucl.Phys. 45 (2000)S1] • SD approach to quark-hadron physics = nonpertubative, covariant bound state approach with strong connections with QCD. The “gap” Schwinger–Dyson equation “dressing” the quark propagator : d 4 k � A ( p 2 ) � p − B ( p 2 ) ≡ S − 1 ( p ) = p / − m − iC F g 2 (2 π ) 4 γ µ S ( k )Γ ν ( k , p ) G µν ( p − k ) . (1) -1 -1 = + A 2 ( q 2 ) q 2 − B 2 ( q 2 ) = Z ( − q 2 ) � q + M ( − q 2 ) A ( q 2 ) � q + B ( q 2 ) q 2 − M 2 ( − q 2 ) = − σ V ( − q 2 ) � q − σ S ( − q 2 ) S ( q ) = [ M ( x ) = dressed quark mass function, Z ( x ) = wave-function renormalization]

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Already just S ( p ) enables calculations of some observables; e.g., pion decay constant π ≈ − 2 B ( q 2 ) ... since close to the chiral limit , Γ BS γ 5 is a good approximation f π q − P 2 l − d P π + u ¯ ν l q + P 2 d 4 q − 2 B ( q 2 ) � � � � f π = i N c 1 � � P γ 5 S ( q + P S ( q − P (2 π ) 4 tr 2 ) γ 5 2 ) 2 M 2 f π π π cos 2 θ c (1 − m 2 Γ( π + → e + ν e ) = 1 4 π G 2 f 2 e ) 2 M π m 2 e . M 2 π

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Wick rotation • Even with much approximating & modeling, solving SD equations like Eq. (1), and related calculations (e.g., of f π ) with Green’s functions like S ( q ), are technically very hard to do in the physical, Minkowski space-time. • ⇒ additional simplification sought by transforming to 4-dim. Euclidean space by the Wick rotation to the imaginary time-component: q 0 → i q 0 : Im(q ) 0 I inf R I I eu I pe R II I res I pp Re(q ) 0 Unlike the perturbative case, propagator singularities can cause problems!

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra General properties a quark propagator should have: • S ( q ) → S free ( q ) because of asymptotic freedom ⇒ σ V , S ( − q 2 ) → 0 for q 2 ∈ C and q 2 → ∞ • σ V , S ( − q 2 ) → 0 cannot be analytic over the whole complex plane • positivity violating spectral density ↔ confinement Try Ans¨ atze of the form (meromorphic parametrizations, like Alkofer&al. [1]): n p � � 1 � p + a j + ib j � p + a j − ib j � S ( p ) = r j p 2 − ( a j + ib j ) 2 + (2) p 2 − ( a j − ib j ) 2 Z 2 j =1 n p 2 r j ( x + a 2 j − b 2 j ) 1 Dressing f ′ nctns are thus � σ V ( x ) = j ) 2 + 4 a 2 ( x + a 2 j − b 2 j b 2 Z 2 j j =1 n p 2 r j a j ( x + a 2 j + b 2 1 j ) � ( e . g ., see Ref . [ 1 ]) and σ S ( x ) = j ) 2 + 4 a 2 ( x + a 2 j − b 2 j b 2 Z 2 j j =1 Constraints: n p n p r j = 1 � � r j a j = 0 2 j =1 j =1

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra 2CC quark propagator - with just 2 pairs of complex conj. poles Parameters (yielding 2CC propagator of Alkofer & al.): n p = 2, a 1 = 0 . 351, a 2 = − 0 . 903, b 1 = 0 . 08, b 2 = 0 . 463, r 1 = 0 . 360, r 2 = 0 . 140, Z 2 = 1. • Naive Wick rotation 0.3 impossible • In calculated quantities, 0.2 proliferation of poles from propagators: 0.1 • contour plot of the f π integrand - the complex 0 function q 0 �→ | trace ( q 0 , ξ ) | -0.1 for ξ ≡ | q | = 4 . 0 . -0.2 Dots are the poles of this function. -0.3 3.6 3.7 3.8 3.9 4 4.1 4.2

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Are quark propagators with CC poles really OK for nonperturbative QCD? 2CC propagator Ansatz by Alkofer&al. gives at ξ = 0 . 5, these integrals over q 0 : I pp = 0 . 184102 I pe = 0 . 184102 + 0 . 0397371 i I inf ≈ 0 I res = 0 . 0166534 i I eu = 0 . 0230837 i | I pp + I res + I eu − I pe | = 1 . 4 · 10 − 9 ⇒ Cauchy’s residue theorem is satisfied: f π = 0 . 071 GeV = Euclidean result (from I eu ), cannot be reproduced in Minkowski space (i.e., from I pp + I res ). * Quark propagators with CC poles have been ”popular” since they have no K¨ allen-Lehmann representation ⇒ corresponding states cannot appear in the physical particle spectrum ⇒ seem appropriate for confined QCD ... but, ∃ objections besides not allowing naive Wick rotation: possible problems for causality and unitarity of the theory. * Also, Beni´ c et al. , PRD 86 (2012) 074002, have shown that CC poles of the quark propagator cause thermodynamical instabilities at nonvanishing temperature and density.

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Quark propagators with singularities only on the real axis? Investigations (e.g. [1] Alkofer et al., Phys.Rev. D70 (2004) 014014 ) based on the available ab initio SD and lattice results indicate: gluon, but probably also quark, propagators have singularities (isolated poles or branch cuts) only for real momenta. ⇒ Use such Ans¨ atze ! This brings also a practical calculational advantage: No obstacles to Wick rotation ⇒ π decay constant f π calculated equivalently * in the Minkowski space: � ∞ � + ∞ N c � � P γ 5 S ( q + P 2 ) γ 5 S ( q − P � ξ 2 d ξ dq 0 B ( q 2 ) tr f 2 π = − i 2 ) 4 π 3 M 2 π 0 −∞ where q 2 = ( q 0 ) 2 − ξ 2 , ξ ≡ | q | , and q · P = M π q 0 , OR * in the Euclidean space: � ∞ � π 3 � � P γ 5 S ( q + P 2 ) γ 5 S ( q − P � d β sin 2 β B ( q 2 ) tr f 2 π = dx x 2 ) 8 π 3 M 2 π 0 0 √ x cos β . where q 2 = − x and q · P = − iM π

Introduction Wick rotation General properties of quark propagators CC quark propagator Quark propagators with singularities only on the real axis Spectra Example: f π calculation with a propagator singular only on the real axis 2 1.5 1 0.5 Im � q 0 � 0 � 0.5 � 1 � 1.5 � 2 � 1 0 1 2 3 Re � q 0 � Contour plot of the integrand and path of integration in the complex q 0 –plane. f π + f π ∼ 10 − 5 ... great ... but f π = 67 MeV too small ⇒ try out various ⇒ 2 f π − f π such models (Ans¨ atze)! But how to make such Ans¨ atze more generally? ...