Extracting a model quark propagator’s spectral density

Zehao Zhu,1, ∗ Khepani Raya,1, † and Lei Chang1, ‡

1School of Physics, Nankai University, Tianjin 300071, China(Dated: May 11, 2020)

We propose a practical procedure to extrapolate the space-like quark propagator onto the complexplane, which follows the Schlessinger Point Method and the spectral representation of the propa-gator. As a feasible example, we employ quark propagators for different flavors, obtained from thesolutions of the corresponding Dyson-Schwinger equation. Thus, the analytical structure of thequark propagator is studied, capitalizing on the current-quark mass dependence of the observedfeatures.

PACS numbers:

I. INTRODUCTION

The strong-interaction part of the Standard Model,Quantum Chromodynamics (QCD), is characterised bytwo emergent phenomena: Dynamical Chiral SymmetryBreaking (DCSB) and confinement [1, 2]. DCSB is re-sponsible for the vast majority of the mass of the vis-ible universe and has a crucial impact on the observedhadron spectrum and properties; for example, it explainsboth the large mass of the proton and the unnaturallylight mass of the pion. Confinement entails that coloredstates, such as QCD’s fundamental degrees of freedom(quarks and gluons), cannot appear in the spectrum. Italso guarantees that condensates, typical order parame-ters of DCSB [3], are wholly contained within hadrons [4].Thus, DCSB and confinement might be, in fact, inti-matelly connected. Both phenomena can be potentiallyunderstood from QCD’s 2-point functions [5], namely,propagators. Studying the analytical properties of thepropagators could shed some light [6, 7] on their con-finement properties and the connection with DCSB andDSEs have been a cornerstone in handling such ender-vours [8, 9]. Thus, focusing on the matter sector, we ob-tain the quark propagator (in the space-like axis) throughits corresponding Dyson-Schwinger equation (DSE) [10–12], properly truncated and with interaction model in-puts [13]. Subsequently, the Schlessinger Point Method(SPM) [14, 15] is employed to extrapolate the propaga-tor into the complex plane, allowing us to study the cor-responding spectral function and its analytic structure.The article is organized as follows: in Section II we writethe DSE for the quark propagator, the truncation andmodel inputs. Section III describes the SPM and theanalytic continuation procedure. It is worth mentioningthat the algorithm is quite general and can be employedto study different inputs, such as lattice QCD propaga-tors. Section IV shows the numerical results and SectionV summarizes the obtained conclusions.

∗ 1710315@mail.nankai.edu.cn† khepani@nankai.edu.cn‡ leichang@nankai.edu.cn

II. GAP EQUATION

The DSE for the quark propagator, gap equation, isthe starting point for analyses of DCSB and confinementin the continuum, as well as the fundamental ingredientfor hadron physics studies based upon Bethe-Salpeter orFaddeev equations [16, 17]. The gap equation, in Eu-clidean space, reads

S−1f (p) = Z2(iγ · p+mbm

f ) + Σf (p) ,

Σf (p) =4

3Z1

∫ Λ

dq

g2Dµν(p− q)γµSf (q)Γfν (p, q) , (1)

where∫ Λ

dq=∫ Λ d4q

(2π)4 stands for a Poincare invariant reg-

ularised integration, with Λ regularisation scale. Therest of the pieces are defined as usual: Sf is the f -flavor quark propagator, Dµν is the gluon propagatorand Γν the fully-dressed quark-gluon vertex; Z1,2 arethe quark-gluon vertex and quark wavefunction renor-malisation constants, respectively; g is the Lagrangiancoupling constant and mbm

f is the bare-quark mass. The

latter is related with the renormalisation point (ζ) depen-

dent current-quark mass, mζf , via Slavnov-Taylor identi-

ties [18, 19]. Each Green function involved obeys its ownDSE, thus forming an infinite tower of coupled equations,which must be systematically truncated to extract theencoded physics [20, 21]. Regardless of the truncation,a general solution for the fully-dressed quark propagatorcan be expressed as follows

Sf (p) = Zf (p2)(iγ · p+Mf (p2))−1 , (2)

in analogy with its bare counterpart,

S(0)f (p) = (iγ · p+mbm

f )−1 .

Here Z(p2) and M(p2) are dressing functions; the latteris known as the mass function. In the Rainbow approxi-mation [13, 22, 23], Eq. (1) is modified according to thereplacement:

g2Z1Dµν(p− q)Γfν (p, q)→ Z22D

fµν(p− q)γν , (3)

arX

iv:2

005.

0418

1v1

[nu

cl-t

h] 8

May

202

0

2

FIG. 1. The closed contour shown in this figure is usedto calculate the Cauchy integral. This contour keeps all thepoles outside and has a infinitesimal distance ε away from thebranch cut.

where Dfµν(k) := (δµν − kµkν/k2)Gf (k2) and Gf (k2) is

the effective coupling, expressed as [13, 24]:

Gf (s = k2) = GfIR(s) + GfUV(s) , (4)

GfIR(s) =8π2D2

f

ω4f

e−s/ω2f , (5)

GfUV(s) =8π2γmF(s)

ln[τ + (1 + s/ΛQCD)2]. (6)

The term GfIR(s) provides an infrarred enhancement,which is controlled by the product ωfD

2f . Conversely,

GfUV(s) is set to reproduce the one-loop renormalisation-group behavior of QCD in the gap equation. We have de-fined: sF(s) = (1−exp[−s/(4m4

t )]), γm = 12/(33−2Nf ),τ = e10 − 1, mt = 0.5 GeV, ΛQCD = 0.234 GeV andNf = 5. Key features of the gluon propagator, such asthe infrarred saturation and the connection with pertur-bation theory, are conveniently captured by the modeldefined in Eqs. (4) - (6). Thus, we solve Eq. (1) for dif-ferent current-quark masses, whose specific values andinteraction strength are given in Table I.

TABLE I. One-loop evolved current quark masses and gluonmodel parameters. Masses, ωf and Df are listed in GeV. Therenormalisation scale is set to ζ = 2 GeV.

Flavor mζf ωf D2

f Flavor mζf ωf D2

f

u/d 0.005 0.500 1.060 c 1.170 0.730 0.599

s 0.112 0.530 1.040 b 4.070 0.766 0.241

III. ANALYTIC CONTINUATION

By solving the gap equation, it is almost straightfor-ward to obtain the quark propagator in the p2 > 0 axis.The extension to the complex p2-plane can be numer-ically challenging since, among other issues, one mightencounter singularities [25–30]. Thus, we follow an ana-lytic continuation scheme, based upon the spectral repre-sentation [31–35] of the quark propagator and the SPM,to extrapolate our numerical solutions onto the complexplane.

Firstly, it is convenient to re-expresss the quark prop-agator, in Eq. (2), in terms of its vector (σV ) and scalar(σS) functions:

Sf (p) = −iγ · pσV (p2) + σS(p2) . (7)

Both σV,S(p2) are obtained on a large, discrete set ofpoints (p2

i > 0, i = 1, · · · , N). Following the SPM, weemploy a continued fraction representation such that

σ(p2) =σ(p2

1)

1+

a1(p2 − p21)

1+· · · aN (p2 − p2

N )

1

= σ(p21)

[1 +

a1(p2 − p21)

1 +a2(p2−p22)

1+···

]−1

, (8)

interpolates (extrapolates) these functions (the labelsV, S are implicit). The coefficients ai are recursively ob-tained, ensuring that ∀p2

i , the interpolated value of σ(p2i )

exactly reproduces the numerically obtained one, [14].Therefore, as discussed in Ref. [34], the SPM gives al-most exact reconstructions, of the continous function, ifthe number of input points is adequate.

Next we change the space-like p2 to a complex value,p2 → p2 = x + iy, (x, y ∈ R). An important featureof the SPM is its capability to identify singularities andbranch points [36, 37]. However, it is seen that if p2

remains real and positive in the training set, the SPMmost likely will not introduce any singular structure be-sides scattered poles. This is due to the simple shapes ofσv,s on the space-like axis: finite, continuous and mono-tonically decreasing functions. Thus, it is found advanta-

geous to deal with a√p2-grid instead [36]. The training

set is simply mapped as:

{p2i , σ(p2

i )} → {√p2i , σ(p2

i )} . (9)

Clearly, this mapping introduces a branch point atp2 = 0, and its corresponding branch cut along its neg-ative real axis. Besides, other singular structures, in theform of complex conjugate poles (CCP), could also ap-pear disperse on the complex plane [25, 26]. The closedcontour, which leaves all the poles and branch cut out-side, is sketched in Fig. 1. At this point, one can ap-peal to complex analysis theorems to represent σ(p2) ina convenient way. An arbitrary point inside the chosencontour can be written as a Cauchy integral. Moreover,

3

from the residue theorem [38, 39], the quark propagatorcan be usefully re-expressed as follows:

σ(p2) =1

π

∫ ∞0

ρ(ω)

p2 + ωdω +

∑i

(Ri

p2 − qi+

R∗ip2 − q∗i

),

ρ(ω) = Im[σ(−ω − iε)] , (10)

where ε is a positive infinitesimal real number; Ri and qitake complex values, which are respectively identified asthe residues and poles. The integral part is a standarddispersion relation for the propagator, such that ρ(ω)corresponds to its spectral function; the fraction partaccounts for the posible presence of complex conjugatepoles.

In principle, ρ(ω) can be computed straightforwardlyfrom the SPM on the numerical data. However, thereare many components that can impact the stability ofthe extrapolations. We have identified key factors whichalter the outcome: the number and distributions of pointsand, potentially, the mapping of Eq. (9). Thus, in orderto get a more accurate and stable result, we proceed asfollows:

• Redistribute the big set of Np = m · n points inton subsets of m points.

• Select one point from each subset, randomly, toform a new small set (of n points).

• Implement SPM on the latter and obtain σ(p2).

This strategy makes it easy to control the number ofpoints, while also covering most of the domain of the ini-tial, much bigger, set. The SPM is performed severaltimes, keeping the mean value as the final result and en-suring a small error is produced. Besides, the particularvalues of m and n are properly fixed by the requirementthat the produced spectral functions are similar in shapeand the location of the poles is stable. The latter is dis-cussed below.

Another important issue is the fact that ρ(ω) comesfrom the integral along the branch cut, but the branchcut might seem artificial, in the sense that it could ap-

pear only due to the choice of a√p2-grid. This is a

standard procedure [36], however. The presence of thebranch cut serves a crucial purpose: it allows to write thepropagator as in Eq. (10), enabling the access to its polestructure. An effective method to calculate the positionsand residues of the poles has been already introduced inRefs. [40, 41], but it is not practical enough to recon-struct the propagator. For instance, we have seen thata single pair of CCP is sufficient to accurately determinethe quark propagator. This implies that Eq. (10) can beconveniently reduced to

σ(p2)− 1

π

∫ ∞0

ρ(ω)

p2 + ωdω =

(R

p2 − q+

R∗

p2 − q∗

),(11)

such that, with ρ(ω) already determined, q and R (polesand residues) can be straightforwardly obtained followingstandard minimization procedures.

IV. NUMERICAL RESULTS

From the solutions of the gap equation, Eq. (1), fordifferent quark flavors (u/d, s, c, b), one gets the space-like quark propagators. Then the corresponding spectralfunctions, poles and residues, are identified following thealgorithm described in the previous section. The SPMis performed 50 times for each quark flavor; the outputsare averaged to produce a final result and error estimates.The particular values of m and n, for each case, are spec-ified in Table II.

TABLE II. Chosen (m,n) values for each quark flavor; vectorand scalar parts separately.

Flavor Vector Scalar Flavor Vector Scalar

u/d (42, 24) (16, 60) c (28, 36) (32, 8)

s (10, 24) (16, 64) b (6, 38) (42, 24)

The spectral densities are displayed in Figs. 2 and 5;the latter shows the results separately, for each quarkflavor, and also includes the quark propagator dressingfunctions. As it is clear from the figures, the spectral den-sities are not positive definite. Moreover, for both scalarand vector parts, spectral densities consistently exhibitpeaks (a minimum). The position of the minimum movestowards ω →∞ as the current quark mass increases. Forthe scalar part, the absolute value of this minimum in-creases with the quark mass, while the opposite patternis observed for the vector part. A natural, kindred fea-ture is also observed with the quark propagator dressingfunctions: at infrared momenta, the vector part of the uquark is much larger that its scalar counterpart; this iscompletely reversed for masses above mcr ' mc. The po-sition of the poles (with the associated uncertainty) as afunction of the quark’s mass in shown Fig. 3. Their cen-tral values and the corresponding residues are capturedin Table III. Clearly, the poles move deeper into the com-plex plain as the current mass grows, i.e. it takes largerabsolute values of both real and imaginary parts. Thisfeature is consistent with realistic DSE studies [42, 43].

The non-positivity of the spectral functions (in partic-ular ρv) is often related to confinement [6, 8]. In thisconnection, we can also study the so-called spaced aver-aged (SA) Schwinger function [3, 44], which, at ~p = 0:

∆(τ) :=

∫d3x

∫d4p

(2π)4ei(tp4+~x·~p)

=1

π

∫ ∞0

dp4 cos(tp4)σ(p24) , (12)

where σ is any of the scalar functions of the propaga-tor. The SA Schwinger function for a real, m massive,scalar particle will decay exponentially, ∆(τ) ∼ e−mτ ,since the propagator is simply σ(p2) = 1/(p2 + m2) andthe mass shell can be reached in the real axis. On the

4

u

s

c

b

0.05 0.10 0.50 1 5 10 50

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

/GeV2

S(

)/GeV

-1

u

s

c

b

0.05 0.10 0.50 1 5 10 50

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

/GeV2

V(

)/GeV

-2

FIG. 2. [Upper panel] Spectral density associated with thescalar part of the propagator, σs(p

2). [Lower panel] Theanalogous for the vector part. The dark-gray and light-grayshaded areas represent the σ and 2σ confidence intervals, re-spectively. Consistently, the observed peaks are shifted to-wards ω →∞ as the quark mass increases.

other hand, if one has a propagator described by a pair ofCCPs instead, one should expect an oscillatory behavior,∆(τ) ∼ e−aτ cos(b t + δ) (here, a is the real part of theCCP mass). In this case, the propagator could be asso-ciated with either a short-lived excitation that decays, ora confined, fundamental particle [8]. Figure 4 displaysthe SA Schwinger functions for the scalar parts of differ-ent quark flavors. The presence of the peaks in the log-arithmic scale reveals a negative sign in the Schwingerfunction, thus another proof of violation of positivity.Naturally, the position of the peaks follow an oppositepattern, with respect to ρ(ω), i.e., peaks move towardsτ → 0 with increasing quark mass.

u s c

b Mean

0.5 1 5 10 50

-4

-2

0

2

4

Re p2/GeV2

Imp2/GeV2

FIG. 3. Position of the poles (q) in the complex plane, withextrapolation uncertainty. From left to right: u, s, c and bquark propagators.

u

s

c

0 2 4 6 8 10

0.001

0.010

0.100

1

τ GeV-1

|Δs(τ)|

FIG. 4. [Upper panel] Space averaged Schwinger functionsof the scalar part of the propagators: u, s and c quarks. Thepresence of the peaks, which moves towards τ → 0 as thequark mass increases, reveals a change of sign in ∆s(τ). Thefirst peak of the b quark propagator lies around τ ∼ 1, but theSchwinger function becomes numerically unstable for largervalues of τ .

V. CONCLUSIONS AND SUMMARY

Starting with space-like quark propagators, we de-scribed a viable procedure to get access to their complexstructure. The DSE inputs have been employed merelyas an illustration and the proposed method is quite gen-eral. This is based upon the SPM, which interpolatesthe space-like quark propagator and extrapolates it intothe complex plane. Then, a proper choice of integrationcontour, allows us to rewrite the propagator in terms of

5

TABLE III. Poles (q) and residues (R) for different quarkflavors. The mass units are expressed in appropriate powersof GeV.

Flavor q R [σv] R [σs]

u/d −0.302± 0.364i 0.586∓ 0.542i −0.013∓ 0.480i

s −0.646± 0.660i 0.702∓ 0.311i 0.060∓ 0.719i

c −2.325± 1.145i 0.577∓ 0.712i 1.098∓ 0.157i

b −32.942± 4.260i 0.674± 0.498i 5.110∓ 3.287i

a Cauchy integral, Eq. (10); thus defining the propaga-tor’s dressing functions in terms of spectral densities andgranting us access to its pole structure. Remarkably, it isseen that a single pair of CCPs is sufficient to accuratelyrepresent the quark propagator. Among other things,this could expedite the computation of the form factors,which typically require two pairs of CCPs [43, 45–47] togive a precise result. It is observed that the spectral den-sities are not positive definite and present peaks, whichare shifted towards the ultraviolet region as the mass

gets larger. Similarly, the position of the poles movesfurther into the complex plane with increasing currentquark mass. The Schwinger functions exhibit the cor-responding analogous features. Such consistency is en-couraging. An immediate goal is to study whether ornot the features observed are still valid for truncationsbeyond the Rainbow approximation. This work is un-derway. Together with the practicity of our approachand the reduced error estimate, we believe that the al-gorithm discussed herein is also suitable to study otherGreen functions. Moreover, the SPM has been provenuseful in connecting Euclidean and Minkowskian quanti-ties [48–50], a key goal in modern hadron physics [51, 52].The present approach to obtain the analytical represen-tation of the quark propagator would take potential rolesin the calculation of the hadron spectrum [26, 53], espe-cially for the meson excited states [28–30].

VI. ACKNOWLEDGEMENTS

We are thankful to Daniele Binosi for sharing his codeto implement the SPM. KR wants to acknowledge Al-fredo Raya for his valuable comments.

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7

u-quark scalar part u-quark vector part u-quark propagator

0 1 2 3 4 5

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

/GeV2

S(

)/GeV

-1

0 1 2 3 4 5

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

/GeV2

V(

)/GeV

-2

Original σV

Reconstructed σV

Original σS

Reconstructed σS

10-6 0.001 1 1000

0.0

0.5

1.0

1.5

2.0

2.5

3.0

p2/GeV2

σS(p2)(GeV

-1)/σV(p2)(GeV

-2)

s-quark scalar part s-quark vector part s-quark propagator

0 1 2 3 4 5

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

/GeV2

S(

)/GeV

-1

0 1 2 3 4 5

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

/GeV2

V(

)/GeV

-2

Original σV

Reconstructed σV

Original σS

Reconstructed σS

10-6 0.001 1 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

p2/GeV2

σS(p2)(GeV

-1)/σV(p2)(GeV

-2)

c-quark scalar part c-quark vector part c-quark propagator

0 2 4 6 8

-0.8

-0.6

-0.4

-0.2

0.0

/GeV2

S(

)/GeV

-1

0 2 4 6 8

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

/GeV2

V(

)/GeV

-2

Original σV

Reconstructed σV

Original σS

Reconstructed σS

10-6 0.001 1 1000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

p2/GeV2

σS(p2)(GeV

-1)/σV(p2)(GeV

-2)

b-quark scalar part b-quark vector part b-quark propagator

0 10 20 30 40

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

/GeV2

S(

)/GeV

-1

0 10 20 30 40 50

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

/GeV2

V(

)/GeV

-2 Original σV

Reconstructed σV

Original σS

Reconstructed σS

10-6 0.001 1 1000

0.00

0.05

0.10

0.15

0.20

p2/GeV2

σS(p2)(GeV

-1)/σV(p2)(GeV

-2)

FIG. 5. The spectral and dressing functions. [Left Panel] Scalar part, ρs(ω). [Middle Panel] Vector part, ρv(ω). [RightPanel] Quark propagator dressing functions: σs(p

2), σv(p2). From top to bottom: u, s, c and b quarks. The depicted spectralfunctions are the mean result after 20 times SPM and analytic continuation. The dark-gray and light-gray bands represent σand 2σ confidence intervals, respectively.