Universal Scale Factors: A Bridge Between Chiral Lagrangians and QCD Sum-Rules
NNuclear and Particle Physics Proceedings 00 (2020) 1–5
NuclearandParticlePhysicsProceed-ings
Universal Scale Factors: A Bridge Between Chiral Lagrangians andQCD Sum-Rules ∗ Amir H. Fariborz a , J. Ho b , T.G. Steele c,1 a Department of Mathematics / Physics, SUNY Polytechnic Institute, Utica, NY 13502, U.S.A. b Department of Physics, Dordt University, Sioux Center, Iowa, 51250, USA c Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada
Abstract
Chiral Lagrangian mesonic fields can be connected to QCD quark operators via matrix operators containing scale factors. Thesescale factor matrices are shown to be constrained by chiral symmetry, resulting in a universal scale factor for each Chiral Lagrangiannonet. QCD sum-rules, combined with mixing angles from Chiral Lagrangian analyses, are used to determine the scale factors forthe a isotriplet and K ∗ isodoublet scalar mesons. The resulting scale factors verify the universality property, providing a validationof the scale factor matrices connecting Chiral Lagrangian mesonic fields and quark operators. Keywords:
The description of light scalar mesons in terms ofquark and gluonic constituents provides some of thegreatest challenges in hadronic physics [1–3]. It is gen-erally expected that mixtures of two-quark, four-quark,and gluonic substructures is necessary to describe themultitude of scalar mesons below 2 GeV [4–12]. Thestrong mixing of quark and gluonic components is a cru-cial feature that emerges in many approaches, includingChiral Lagrangians [5, 8, 13], QCD sum-rules [14–20],and other methods [21, 22]. Another important aspectis the inverted hierarchy between two-quark and four-quark states (e.g., Chiral Lagrangians [8, 10, 23], bagmodel [24] and QCD sum-rules [25–27]).The focus of this proceedings is reviewing our workfrom Refs. [28–30] on the interconnections betweenChiral Lagrangian and QCD sum-rule methodologies asapplied to the scalar mesons. Each method is basedon a set of key principles: Chiral Lagrangian linear ∗ Talk given at 23rd International Conference in Quantum Chro-modynamics (QCD 20), 27–30 October 2020, Montpellier - FR
Email addresses: [email protected] (Amir H. Fariborz), [email protected] (J. Ho), [email protected] (T.G. Steele) Speaker, Corresponding author. models [7, 10–12] or non-linear models [4, 8, 23, 31–33] are founded on chiral symmetry and its breakdown,while QCD sum-rules are founded on QCD-hadron du-ality [34, 35]. Each method faces challenges in ap-plication to the scalar mesons: Chiral Lagrangians areoblivious to the underlying four-quark structures (e.g.,molecules, diquarks) while QCD sum-rules require pa-rameterization of the broad (and possibly overlapping)scalar states. In pursuing the interconnections betweenChiral Lagrangian and QCD sum-rule methods, we aremotivated by the pursuit of synergies leading to newinsights and approaches within each method. Further-more, we are motivated by the philosophy that featuresof the scalar sector that emerge from our analysis bridg-ing the two methodologies are inherently more robustthan features emerging from isolated analyses.In Ref. [28] we developed the general framework ofscale factor matrices providing the linkage between Chi-ral Lagrangian mesonic fields and quark-level compos-ite operators (two-quark and four-quark) used in QCDsum-rules to probe hadronic properties. Chiral sym-metry was shown to constrain the scale factor matricesto contain one universal scale parameter for each Chi-ral Lagrangian scalar nonet, and the scale factors were1 a r X i v : . [ h e p - ph ] D ec Nuclear and Particle Physics Proceedings 00 (2020) 1–5 successfully extracted for the a isotriplet sub-system,thereby demonstrating the feasibility of the approach tolink Chiral Lagrangian and QCD sum-rules [28]. De-pendence of these extracted scale factors on the non-perturbative gluon condensate QCD sum-rule input pa-rameter as studied in [29], allowing an estimate of the-oretical uncertainties. In Ref. [30] the crucial univer-sality property was established by demonstrating thatthe scale factors extracted from the K ∗ isodoublet and a isoptriplet sub-systems were in excellent numericalagreement. Other key features establishing the validityof the scale factors linking Chiral Lagrangians and QCDsum-rules included agreement between vacuum expec-tation values in both approaches as related by the scalefactors [28, 29].We begin by reviewing the chiral symmetry con-straints on the scale-factor matrices connecting Chi-ral Lagrangian fields and QCD sum-rule operators. Inthe generalized linear sigma model notation of [10, 12]there are two chiral nonets M and M (cid:48) with identical chi-ral transformation properties but with di ff erent U A (1)transformations M → U L M U † R , M → e i ν MM (cid:48) → U L M (cid:48) U † R , M (cid:48) → e − i ν M (cid:48) , (1)and hence M is associated with two-quark ( ¯ qq ) struc-tures and M (cid:48) with four-quark ( ¯ qq ¯ qq ) structures. Themesonic nonets contain the bare (un-mixed) chiral La-grangian mesonic scalar and pseudoscalar fields M = S + i φ , M (cid:48) = S (cid:48) + i φ (cid:48) (2) S = S a + κ + a − S κ κ − ¯ κ S , S (cid:48) = S (cid:48) a (cid:48) + κ (cid:48) + a (cid:48)− S (cid:48) κ (cid:48) κ (cid:48)− ¯ κ (cid:48) S (cid:48) (3)and similarly for the pseudoscalar components in φ and φ (cid:48) . QCD operators that satisfy the chiral transformationproperties (1) for M are( M QCD ) ba = ( ¯ q R ) b ( q L ) a ⇒ (cid:0) S QCD (cid:1) ba = q a ( x ) ¯ q b ( x ) (4)where a and b are flavor indices and each can take valuesof 1 to 3, and there is no loss of generality in the chosennormalization of the QCD operator. Similarly, M (cid:48) QCD can be mapped to four-quark composite operators, andthe specific forms chosen among the many choices willbe outlined below.The relation between the QCD operators and mesonicfields occurs via scale factor matrices I M and I M (cid:48) [28–30] M = I M M QCD , M (cid:48) = I M (cid:48) M (cid:48) QCD , (5) which have the chiral transformation following proper-ties governed by (1):[ U R , I M ] = [ U L , I M ] = , (6)[ U R , I M (cid:48) ] = [ U L , I M (cid:48) ] = , (7)and hence the scale factor matrices are multiples of theidentity matrix [28–30] I M = − m q Λ × , I M (cid:48) = Λ (cid:48) × , (8)where the (constant) scale factor parameters Λ and Λ (cid:48) have dimensions of energy and the quark mass factor m q = ( m u + m d ) / Λ and Λ (cid:48) must be identical for all mem-bers of the chiral nonets. Chiral symmetry thus leadsto the remarkable result that only two universal scalefactors are needed to relate Chiral Lagrangian mesonicfields to QCD operators as expressed in (5) and (8). Forthe remainder of this proceedings, the methodology fordetermining the scale factors will be presented, and theuniversality property will be established for the scalefactors extracted from the K ∗ isodoublet and a isop-triplet sectors [30].As an illustration of the methodology, we beginwith the K ∗ isodoublet system and relate the physi-cal K ∗ (700) and K ∗ (1430) chiral Lagrangian fields toQCD operators J QCD κ via a combination of a Chiral La-grangian rotation matrix L κ and the scale factor matrix I κ [28–30]: K = (cid:32) K ∗ (700) K ∗ (1430) (cid:33) = L − κ (cid:32) S ( S (cid:48) ) (cid:33) = L − κ I κ J QCD κ (9) L − κ = (cid:32) cos θ κ − sin θ κ sin θ κ cos θ κ (cid:33) , I κ = (cid:32) − m q Λ Λ (cid:48) (cid:33) , (10) J QCD κ = (cid:32) J κ J κ (cid:33) , J κ = ¯ ds (11) J κ = sin( φ ) u T α C γ µ γ s β (cid:16) ¯ d α γ µ γ C ¯ u T β − α ↔ β (cid:17) + cos( φ ) d T α C γ µ s β (cid:16) ¯ d α γ µ C ¯ u T β + α ↔ β (cid:17) (12)where C is the charge conjugation operator and cot φ = / √ a (980)2 Nuclear and Particle Physics Proceedings 00 (2020) 1–5 and a (1450) isotriplet system is A = (cid:32) a (980) a (1450) (cid:33) = L − a S − S √ S (cid:48) − S (cid:48) √ = L − a I a J QCD a (13) L − a = (cid:32) cos θ a − sin θ a sin θ a cos θ a (cid:33) , I a = I κ = (cid:32) − m q Λ Λ (cid:48) (cid:33) (14) J QCD a = (cid:32) J a J a (cid:33) , J a = (cid:16) ¯ uu − ¯ dd (cid:17) / √ J a = sin φ √ d T α C γ µ γ s β (cid:16) ¯ d α γ µ γ C ¯ s T β − α ↔ β (cid:17) + cos φ √ d T α C γ µ s β (cid:16) ¯ d α γ µ C ¯ s T β + α ↔ β (cid:17) − u ↔ d . (16)Because the physical a and K ∗ chiral Lagrangian fieldsoccur in Eqs. (9) and (13) and result in a diagonalhadronic correlation function, their associated QCD op-erators define projected physical currents J Ps that leadto a diagonal QCD physical correlation function matrix Π P ( Q ) J Ps = L − s I s J QCD s , s = { κ , a } (17) Π P ( Q ) = (cid:101) T s Π QCD ( Q ) T s , T s = I s L s (18) Π QCD mn (cid:16) Q = − q (cid:17) = (cid:90) d x e iq · x (cid:104) | T (cid:104) J QCD m ( x ) J QCD n (0) † (cid:105) | (cid:105) (19)where (cid:101) T denotes the transpose of the matrix T . Be-cause the physical QCD correlator matrix Π P ( Q ) is di-agonal, the o ff -diagonal correlator term Π P ( Q ) mustsatisfy Π QCD12 = − (cid:101) T κ Π QCD11 T κ + (cid:101) T κ Π QCD22 T κ (cid:101) T κ T κ + (cid:101) T κ T κ , (20)which will be used as one of our QCD theoretical in-puts. Similarly, the projected physical correlator can beexpressed as a diagonal hadronic correlation function Π H ( Q ) containing the resonance properties and a con-tinuum contribution Π P ( Q ) = Π H ( Q ) , H = { K , A } (21) Π H i j (cid:16) Q = − q (cid:17) = (cid:90) d x e iq · x (cid:104) | T (cid:104) H i ( x ) H j (0) (cid:105) | (cid:105) = δ i j m si + Q − im si Γ si + continuum . (22)Imposing QCD-hadron duality, the hadronic and QCDcontributions to the projected physical correlation func-tions are equated Π H ( Q ) = Π P ( Q ) = (cid:101) T s Π QCD ( Q ) T s , (23) and by applying an appropriate transform (e.g., Boreltransform [34, 35]) to both sides of (23) any desiredQCD sum-rule can be obtained. To ensure that the ro-tation matrices L s properly disentangle the mixture ofChiral Lagrangian mesonic fields and projected phys-ical QCD correlator, Gaussian sum-rules will be usedbecause they can probe multiple states with similar sen-sitivity [36, 37]; by contrast Laplace sum-rules will sup-press heavier states and obscure incomplete diagonal-ization. The resulting relation between the hadronic andprojected QCD Gaussian sum-rules for the a ( s = a )and K ∗ ( s = κ ) channels is G H ( ˆ s , τ ) = (cid:32) G H G H (cid:33) = (cid:101) T s G QCD ( ˆ s , τ, s ) T s , (24) G H ( ˆ s , τ ) = √ πτ ∞ (cid:90) s th dt exp (cid:34) − ( ˆ s − t ) τ (cid:35) ρ H ( t ) , (25) ρ H ( t ) = π Im Π H ( t ) + θ ( t − s ) 1 π Im Π QCD ( t ) , (26)where the individual continuum thresholds s (1)0 and s (2)0 for each diagonal entry in the 2 × s . Renormalization group behaviouridentifies the Gaussian width τ as the renormalizationscale and hence τ has a lower bound from QCD, andwe choose τ = consistent with the central valueused in Refs. [19, 37]. However, the Gaussian kernelpeak ˆ s is unconstrained by QCD and can be varied as aQCD sum-rule parameter analogous to the Borel scalein Laplace sum-rules.Because the constraint Eq. (20) is linear in the QCDcorrelation function, it also applies to the QCD Gaus-sian sum-rule matrix entries. Imposing the constraint(20) then provides the Gaussian sum-rules for the diag-onal terms in (24) for each channel ( s = { a , κ } ) [28–30] G H ( ˆ s , τ ) = aAG QCD11 (cid:16) ˆ s , τ, s (1)0 (cid:17) − bBG QCD22 (cid:16) ˆ s , τ, s (1)0 (cid:17) (27) G H ( ˆ s , τ ) = − aBG QCD11 (cid:16) ˆ s , τ, s (2)0 (cid:17) + bAG QCD22 (cid:16) ˆ s , τ, s (2)0 (cid:17) A = cos θ s cos θ s − sin θ s , B = sin θ s cos θ s − sin θ s (28) a = m q Λ , b = Λ (cid:48) ) (29)where G H and G H respectively represent K ∗ (700) and K ∗ (1430) contributions for the s = κ isodoublet channeland the a (980) and a (1450) contributions for the s = a isotriplet channel. The constraint (20) has been exam-3 Nuclear and Particle Physics Proceedings 00 (2020) 1–5 ined in [29], and agrees with the order-of-magnitude es-timate for the o ff -diagonal Gaussian sum-rule G QCD12 .Although the constraint (20) has been used in obtain-ing (27), the solutions for the diagonal elements retain ageneral property of (18). After applying the Borel trans-form (adapted for Gaussian sum-rules) to (18), takingthe trace, and noting that L s ˜ L s = so that T s (cid:101) T s = I s leads to Tr (cid:104) G P (cid:105) = Tr (cid:104) G H (cid:105) = Tr (cid:104) G QCD I s (cid:105) (30) G H ( ˆ s , τ ) + G H ( ˆ s , τ ) = aG QCD11 (cid:16) ˆ s , τ, s (1)0 (cid:17) + bG QCD22 (cid:16) ˆ s , τ, s (1)0 (cid:17) . (31)The solutions Eq. (27) that emerge from applying thediagonalization constraint (20) also satisfy the generalproperty (31) providing a valuable consistency check onour analysis methodology. However, it is remarkablethat Eqs. (30) and (31) are independent of the mixingangle matrix L s , and future analyses based on (30) mayyield valuable insights.Eq. (27) can now be solved for the scale factors Λ , Λ (cid:48) and the continuum thresholds are optimized to min-imize the ˆ s dependence of the scale parameters. Forour detailed analysis, the results for the QCD correla-tion functions are given in Refs. [19, 27, 37–39] andthe methods of [37] can then be used to form the Gaus-sian sum-rules. For the QCD input parameters we usePDG values [1] (quark masses, and α s ) and the follow-ing QCD condensate [35, 40–42] and instanton liquidmodel parameters [43, 44] (cid:104) α s G (cid:105) = (0 . ± .
02) GeV , (32) (cid:104) q σ Gq (cid:105)(cid:104) ¯ qq (cid:105) = (cid:104) s σ Gs (cid:105)(cid:104) ¯ ss (cid:105) = (0 . ± .
1) GeV (33) (cid:104) ¯ qq (cid:105) = − (0 . ± . , (cid:104) ¯ ss (cid:105) = (0 . ± . (cid:104) ¯ qq (cid:105) (34) n c = . × − GeV , ρ = /
600 MeV , (35) m ∗ q =
170 MeV , m ∗ s =
220 MeV . (36)The instanton parameters ρ and n c have an estimateduncertainty of 15% and the quark zero-mode e ff ectivemasses m ∗ have a correlated uncertainty with ρ and thequark condensate [45]. The m s / m q = . θ κ = . θ a = . m s / m q = .
3. Forthe physical mass and width of the K ∗ states we use thefollowing values [1]: m κ =
824 MeV and Γ κ =
478 MeV for the K ∗ (700), m K = Γ K =
270 MeVfor the K ∗ (1430), and for the a (980 , m a = { . , . } GeV and Γ a = { . , . } GeV.The optimized values for the scale factors separatelydetermined for the a and K ∗ channels are shown inFig. 1 as a function of the Gaussian sum-rule param-eter ˆ s and their final determinations are given in Ta-ble 1 [30]. Theoretical uncertainties in the scale fac-tors arising from the gluon condensate (the dominantnon-perturbative input parameter) have been studied inRef. [29]. The scale factor results of Fig. 1 and Ta-ble 1 demonstrate the following three important proper-ties that establish the integrity of our proposed connec-tion of Chiral Lagrangian mesonic fields to QCD oper-ators via scalar factor matrices: • Refs. [28, 29] have examined the relation betweenthe Chiral Lagrangian vacuum expectation valuesand scale factors given by (cid:104) S (cid:105) = − m q (cid:104) ¯ uu (cid:105) / Λ and (cid:104) S (cid:48) (cid:105) ≈ . (cid:104) ¯ dd (cid:105)(cid:104) ¯ ss (cid:105) / Λ (cid:48) (n.b., vacuum sat-uration e ff ects are embedded in the 1.31 numericalfactor), and conclude that any numerical disagree-ments can be attributed to the known deviationsfrom the vacuum saturation hypothesis. • The scale factors have minimal energy ( ˆ s ) depen-dence indicating that no significant additional dy-namics is necessary to supplement our fundamen-tal relation (5) relating Chiral Lagrangian fieldsand QCD operators. • The scale factors are nearly identical for the K ∗ and a subsystems, establishing universality of thescale factors as required by chiral symmetry.Our future work will continue to seek new insights onthe scalar mesons by building upon the universal scalefactor matrices that we have firmly established in theisodoublet K ∗ and isotriplet a sectors. Of particu-lar interest is the challenging case of the isosinglet f system, which in addition to quark-antiquark and four-quark chiral nonets involve the inclusion of scalar andpseudoscalar glueballs. This will further test the uni-versality requirement and allow a more careful exami-nation of the substructure of f states. Informed by thescale factor matrix relations between mesonic fields andQCD operators, we will also pursue new methodologiesthat exploit this synergistic connection. Acknowledgments
This research was generously supported in part by theSUNY Polytechnic Institute Research Seed Grant Pro-gram. TGS is grateful for the hospitality of AHF and4
Nuclear and Particle Physics Proceedings 00 (2020) 1–5 s ( GeV ) Scale Factor ( MeV ) ΛΛ ' Figure 1. The scale factors Λ (lower pair of curves) and Λ (cid:48) (upperpair of curves) are shown as a function of ˆ s for optimized continuumthresholds in Table 1. Solid curves are for the K ∗ channel and dashedcurves are for the a channel. Channel s (1)0 s (2)0 Λ Λ (cid:48) K ∗ a Table 1. Values for the optimized scale factors Λ , Λ (cid:48) and continuumthresholds s (1)0 , s (2)0 for the a and K ∗ channels. All quantities are inappropriate powers of GeV. SUNY Polytechnic Institute while this work was initi-ated. TGS and JH are grateful for research funding fromthe Natural Sciences and Engineering Research Coun-cil of Canada (NSERC), and AHF is grateful for a 2019Seed Grant, and the support of the College of Arts andSciences, SUNY Polytechnic Institute. We thank Re-search Computing at the University of Saskatchewan forcomputational resources.
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