the pion form factor in the covariant spectator theoryalfred stadler, university of Évora the pion...

The Pion Form Factor in the Covariant Spectator Theory

Alfred StadlerUniversity of Évora, and Centro de Física Teórica de Partículas (CFTP) Lisbon

in collaboration with

Teresa Peña, Elmar Biernat (CFTP Lisbon)Franz Gross (JLab)

Electron-Nucleus Scattering XIII, June 23-27, 2014, Marciana Marina, Isola d’Elba

published in PRD89, 016005 (2014) and PRD89, 016006 (2014)

Elba XIII, 2014Alfred Stadler, University of Évora The Pion Form Factor in the Covariant Spectator Theory

New challenges in meson physics

Motivation

F. Jegerlehner, A. Nyffeler / Physics Reports 477 (2009) 1–110 49

(a) [L.D.]. (b) [L.D.]. (c) [S.D.].

Fig. 34. Hadronic light-by-light scattering diagrams in a low energy effective model description. Diagrams (a) and (b) represent the long distance [L.D.]contributions at momenta p ⇤, diagram (c) involving a quark loop which yields the leading short distance [S.D.] part at momenta p � ⇤ with⇤ ⇠ 1 to 2 GeV an UV cut-off. Internal photon lines are dressed by ⇢–� mixing.

Table 6Orders with respect to 1/Nc and chiral expansion of typical leading contributions shown in Fig. 34.

Diagram 1/Nc expansion p expansion Type

Fig. 34(a) Nc p6 ⇡0, ⌘, ⌘0 exchangeFig. 34(a) Nc p8 a1, ⇢,! exchangeFig. 34(b) 1 p4 Meson loops (⇡±, K±)Fig. 34(c) Nc p8 Quark loops

Kinoshita (HK 1998) [246] (see also Kinoshita, Nizic and Okamoto (KNO 1985) [79]). Although the details of the calculationsare quite different, which results in a different splitting of various contributions, the results are in good agreement andessentially given by the ⇡0-pole contribution, which was taken with the wrong sign, however. In order to eliminate thecut-off dependence in separating L.D. and S.D. physics, more recently it became favorable to use quark–hadron duality, asit holds in the large Nc limit of QCD [252,253], for modeling of the hadronic amplitudes [245]. The infinite series of narrowvector states known to show up in the large Nc limit is then approximated by a suitable lowest meson dominance (LMD)ansatz [255], assumed to be saturated by known low lying physical states of appropriate quantum numbers. This approachwas adopted in a reanalysis by Knecht and Nyffeler (KN 2001) [17,250], in which they discovered a sign mistake in thedominant ⇡0, ⌘, ⌘0 exchange contribution (see also [256,257]), which changed the central value by +167 ⇥ 10�11, a 2.8� shift, and which reduced a larger discrepancy between theory and experiment. More recently Melnikov and Vainshtein(MV 2004) [258] found additional problems in previous calculations, this time in the short distance constraints (QCD/OPE)used in matching the high energy behavior of the effective models used for the ⇡0, ⌘, ⌘0 exchange contribution. Mostevaluations have adopted the pion–pole approximation which, however, violates four-momentum conservation at theexternal⇡0� ⇤� vertex, if used too naively, as pointed out in Refs. [258,44,46]. In the followingwewill attempt an evaluationwhich avoids such manifest inconsistencies. Maybe some of the confusion in the recent literature was caused by the factthat the distinction between off-shell and on-shell (pion–pole) form factors was not made properly.

Let us start now with a setup of what one actually has to calculate. We will closely follow Ref. [17] in the following.The hadronic light-by-light scattering contribution to the electromagnetic vertex is represented by the diagram Fig. 31.According to the diagram, a complete discussion of the hadronic light-by-light contributions involves the full rank–fourhadronic vacuum polarization tensor

⇧µ⌫�⇢(q1, q2, q3) =Z

d4x1d4x2d4x3 ei (q1x1+q2x2+q3x3) h0 | T {jµ(x1)j⌫(x2)j�(x3)j⇢(0)} | 0 i. (145)

The external photon momentum k is incoming, the qi’s of the virtual photons are outgoing from the hadronic ‘‘blob’’.Here jµ(x) ⌘ ( Q�µ )(x) ( = (u, d, s), Q = diag(2, �1, �1)/3 the charge matrix) denotes the light quark part ofthe electromagnetic current. Since jµ(x) is conserved, the tensor ⇧µ⌫�⇢(q1, q2, q3) satisfies the Ward–Takahashi identities{qµ

1 ; q⌫2; q�3; k⇢}⇧µ⌫�⇢(q1, q2, q3) = 0 , with k = (q1 + q2 + q3) which implies

⇧µ⌫�⇢(q1, q2, k � q1 � q2) = �k� (@/@k⇢)⇧µ⌫�� (q1, q2, k � q1 � q2), (146)and thus tells us that the object of interest is linear in k when we go to the static limit kµ ! 0 in which the anomalousmagnetic moment is defined. As a consequence the electromagnetic vertex amplitude takes the form ⇧⇢(p 0, p) =k�⇧⇢� (p 0, p) and the hadronic light-by-light contribution to the muon anomalous magnetic moment is given by (seealso [158])

FM(0) = 148mµ

Tr�

(6p + mµ)[� ⇢, � � ](6p + mµ)⇧⇢� (p, p)

. (147)

The required vertex tensor amplitude is determined by

⇧⇢� (p 0, p) = �ie6Z

d4q1(2⇡)4

d4q2(2⇡)4

1q21 q

22 (q1 + q2 � k)2

1(p 0 � q1)2 � m2

µ

1(p � q1 � q2)2 � m2

µ

⇥ � µ(6p 0� 6q1 + mµ)� ⌫(6p� 6q1� 6q2 + mµ)� �@

@k⇢⇧µ⌫�� (q1, q2, k � q1 � q2), (148)

F. Jegerlehner, A. Nyffeler / Physics Reports 477 (2009) 1–110 47

Fig. 31. Assignment of momenta for the calculation of the hadronic contribution of the light-by-light scattering to the muon electromagnetic vertex.

Fig. 32. The invariant � � mass spectrum obtained with the Crystal Ball detector [242]. The three spikes seen represent the � � ! pseudoscalar (PS)! � � excitations: PS = ⇡0, ⌘, ⌘0 .

Fig. 33. Hadronic light-by-light scattering is dominated by ⇡0-exchange in the odd parity channel, pion loops etc. at long distances (L.D.) and quark loopsincluding hard gluonic corrections at short distances (S.D.). The photons in the effective theory couple to hadrons via � � ⇢0 mixing.

As a contribution to the anomalous magnetic moment three of the four photons in Fig. 31 are virtual and to be integratedover all four-momentum space, such that a direct experimental input for the non-perturbative dressed four-photoncorrelator is not available. In this case one has to resort to the low energy effective descriptions of QCD like chiral perturbationtheory (CHPT) extended to include vector-mesons. Note that early evaluations assumed that the main contribution tohadronic light-by-light scattering comes frommomentum regions around themuonmass. Itwas later observed in Refs. [243,244] that the higher momentum region, around 500–1000 MeV, also gives important contributions. Therefore, hadronicresonances beyond the Goldstone bosons of CHPT need to be considered as well. The Resonance Lagrangian Approach (RLA)realizes vector-meson dominance model (VMD) ideas in accord with the low energy structure of QCD [88]. Other effectivetheories are the extendedNambu–Jona–Lasinio (ENJL)model [244] (see also [245]) or the very similar hidden local symmetry(HLS) model [243,246]; approaches more or less accepted as a framework for the evaluation of the hadronic LbL effects. Theamazing fact is that the interactions involved in the hadronic LbL scattering process are the parity conserving QED and QCDinteractions while the process is dominated by the parity odd pseudoscalar meson-exchanges. This means that the effective⇡0� � interaction vertex exhibits the parity violating �5 coupling, which of course in � � ! ⇡0 ! � � must appear twice

‣ Upcoming intense experimental activity to explore meson structure GlueX (Jlab), PANDA (GSI)‣ Search for exotic mesons (hybrids, glueballs) ‣ Need to understand “conventional” -mesons in more detail‣ Pion transition form factors

Hadronic contributions to light-by-light scattering Source of uncertainty in prediction of anomalous magnetic moment of the muon Important in search for physics beyond the Standard Model

qq


‣ Construct a model to describe all -type mesons‣ Covariant framework (CST) - light quarks require relativistic treatment

Work in Minkowski space (physical momenta — but have to confront singularities) Improve on previous work by Gross, Milana, Savkli‣ Quark self-energy from interaction kernel (consistent quark mass function)‣ Chiral symmetry: massless pion in chiral limit of vanishing bare quark mass (NJL)‣ Calculate meson spectrum, bound-state vertex functions and form factors‣ Learn about confining interaction (scalar vs. vector, etc.)

qq

qq

A unified model for all mesonsqq

Our objectives

Much important work was done on meson structure

‣ Cornell-type static potential models (Isgur and Godfrey, Spence and Vary, etc.) But: nonrelativistic (or “relativized”); structure of constituent quark and relation to existence of zero-mass pion in chiral limit not addressed‣ Dyson-Schwinger approach (C. Roberts et al.)

But: Euclidean space; only Lorentz vector confining interaction‣ Lattice QCD (also Euclidean space), EFT, Bethe-Salpeter, Light-front, Point-form, ...


Covariant two-body bound-state equation

Start from the Bethe-Salpeter (BS) equation for the bound-state vertex function Γ

P = k1 � k2

k = (k1 + k2)/2

total momentumrelative momentum

vertex function�(k, P ) �(k1, k2)or

V(p, k) kernel

�BS(p, P ) = i

Zd4k

(2⇡)4V(p, k;P )S1(k1)�BS(k, P )S2(k2)

One-body equation for dressed propagatorquark interact with itself: self energy Σ(p) = A(p) + /pB(p) ⇒ Dyson (mass gap)

equation for dressed quark propagator S(p) = 1m0+Σ(p2)−/p−iϵ

≡ Z(p2)M(p2)−/p−iϵ

and bare propagator S0(p) = 1m0−/p−iϵ

Dyson PR 75, 1949

S S0 S0 SΣ

= +

V

dressed quark mass function M = A+m0

1−B generated dynamically from theinteractions

constituent quark massm = M(m2) ⇔spontaneous dynamicalchiral symmetry breaking!

m0DCSB←→ m

Fig.: Bowman et al, PRD 71, 2005

Elmar Biernat (CFTP/IST) Quarks and mesons in CST May 22, 2014 10 / 23

The dressed propagator satisfies the Dyson equation

S0(p) =1

m0 � /p� i✏

S(p) =1

m0 � /p+ ⌃(p)� i✏=

Z(p2)

M(p2)� /p� i✏

bare propagator

bare quark mass

⌃(p) = A(p2) + /pB(p2) Z(p2) =1

1�B(p2)M(p2) =

m0 +A(p2)

1�B(p2)self-energy wave function renormalization mass function

dressed quark propagator

dressed mass satisfiesm = M(m2)

p1 k1

V �P

p1

�P

p2 k2

=

p2


Kernel truncation

In the BS equation the kernel is effectively iterated to all orders

But the complete kernel is a sum of an infinite number of irreducible diagrams → has to be truncated (most often: ladder approximation)

Two-body equation for scattering amplitudeinhomogeneous Bethe-Salpeter equation (BSE) for quark-antiquark scatteringamplitude M with interaction kernel VBethe, Salpeter, PR 84, 1951

= +M V

q

qMV

= + + + . . .

S

S

if kernel V =!

(all 2-particle irreducible diagrams)

V = + + + + . . .

⇒ exact result for scattering amplitude Min practice impossible ⇒ need approximations


The kernel contains all two-body irreducible diagrams

Problems with the ladder truncation

‣ No one-body limit(missing crossed ladders)

‣ Not best suited to describe bound states (crossed-ladder contributions are significant)

Two-body equation for scattering amplitudeinhomogeneous Bethe-Salpeter equation (BSE) for quark-antiquark scatteringamplitude M with interaction kernel VBethe, Salpeter, PR 84, 1951

= +M V

q

qMV

= + + + . . .

S

S

if kernel V =!

(all 2-particle irreducible diagrams)

V = + + + + . . .

⇒ exact result for scattering amplitude Min practice impossible ⇒ need approximations


see Nieuwenhuis and Tjon, PRL77, 814 (1996)


If bound-state mass is small:both poles are close together (both important)

From Bethe-Salpeter to CST

3

Charge conjugation, denoted by the operator C, trans-forms quarks into antiquarks and vice versa, accom-plished by taking the transpose of the vertex functionand changing p1 $ �p2. The amplitude is invariant un-der charge conjugation if it remains unchanged up to aphase ⌘, with ⌘2 = 1. The required condition is therefore

C�1 �TBS(�p, P ) C = ⌘ �BS(p, P ) , (8)

where we have used that p1 = p+ 12P $ �p2 = �p+ 1

2Pimplies p $ �p. Performing this operation on Eq. (1),and using C�1 �µT C = ��µ and the charge conjugationinvariant conditions

C�1VT (p, k;P )C = V(�p,�k;P )

C�1ST (k)C = S(�k) (9)

gives

C�1�TBS(�p, P )C = i

Z

d4k

(2⇡)4V(p,�k;P )S(�k2)

h

C�1�TBS(k, P )C

i

S(�k1)

= i

Z

d4k

(2⇡)4V(p, k;P )S(k1)

h

C�1�TBS(�k, P )C

i

S(k2) , (10)

which shows that C�1�TBS(�p, P )C satisfies the same

equation as �BS(p, P ) (and hence the two are equal upto a phase), provided conditions for the propagators andkernel, Eqs. (7) and (9), are satisfied. We will alwayschoose kernels that satisfy condition (9).

Note that a crucial step in the derivation was our abil-ity to change the four-dimensional integration variablek ! �k. This condition must be preserved when wespecialize to the Covariant Spectator Theory (CST).

B. Charge conjugation invariant CST equations

Next we introduce a charge conjugation invariant formof the bound-state CST equations. For cases when wewant the correct limit as P ! 0 these are the “four-channel” equations previously discussed [3].

To motivate the structure of these equations, beginwith the BS equation (1) and consider the k0 integration.The dressed propagator of quark i with dressed mass mand renormalization constant Z0 can be written

S(ki) ' Z0(m+ /ki)

m2 � k2i � i✏(11)

near its poles at ki0 = ±Eki , where Eki ⌘ p

m2 + k2i .

Figure 2 shows the positions of the four propagator polesin the complex k0 plane in the bound-state rest frame(note that k0 is the zero component of the relative mo-

mentum k, not of the individual particle momenta ki).In the rest frame, the total momentum is Pr = (µ,0),the quark and antiquark three-momenta ki are equal tothe relative three-momentum k, and therefore Eki = Ek,with Ek ⌘ p

m2 + k2. However, in the following we willcontinue working in an arbitrary frame with total mo-mentum P in order to emphasize the manifest covarianceof our framework.

To perform the k0 integration we can close the contourin the lower or upper half plane. In the CST frameworkonly poles of propagators are included, whereas the polesof the kernel are moved to higher order kernels, and ne-glected. As one can see in Fig. 2, in either half plane the

�Ek � µ2 �Ek +

µ2

Ek � µ2 Ek +

µ2

Im k0

Re k0

FIG. 2. (color online) The positive-energy poles (colored crosses

with positive Ek) and negative-energy (white crosses with negative

Ek) poles of the propagators of quark 1 (red with �µ/2) and quark

2 (cyan with +µ/2) in the complex k0-plane in the bound-state rest

frame.

respective two poles are separated by the bound-statemass µ. If µ is large, the pole closer to the origin dom-inates the integral, and the more distant pole can beneglected. However, in the limit P ! 0 the two polesmove close together and the contributions of both mustbe taken into account.First we close the k0 contour in the lower half plane.

Introducing the on-shell momenta ki = (Eki ,ki) and us-ing the form (11) for the dressed propagators permits thetwo propagator pole contributions to the right hand sideof (1) to be written

�(p, P ) = �Z0

Z

k1

V(p, k1 � 12P ;P )(m+ /k1)

⇥�(k1 � 12P, P )S(k1 � P )

�Z0

Z

k2

V(p, k2 + 12P ;P )S(k2+ + P )

⇥�(k2 +12P, P )(m+ /k2) , (12)

Integration over relative energy k0:‣ Keep only pole contributions from propagators‣Move kernel poles to higher-order kernels

(reorganization of the BS series)‣ Cancellations between ladder and crossed ladder

diagrams can occur‣ Reduction to 3D loop integrations, but covariant‣Works very well in few-nucleon systems

Symmetrize pole contributions from both half planes:resulting equation is symmetric under charge conjugation

Covariant Spectator Theory (CST)

CST verticesBS vertex (approx.)

µ

Mini-review: A.S., F. Gross, Few-Body Syst. 49, 91 (2010)

= + + +12—{ }


Four-channel CST equationClosed set of equations when external legs are systematically placed onshell

=

=

=

=

+ + +

+ + +

+ + +

+ + +

2×

2×

2×

2×

Approximations for special cases ‣ Heavy-light quark systems: 2 channels‣ Large bound-state mass: 1 channel

Smooth nonrelativistic limit: Schrödinger equation

Solutions: bound state masses μ and corresponding vertices Γ

Smooth one-body limit: Dirac equation


Confining potential in momentum space

Linear confinement

confinement cannot be obtained from finite number of gluon exchanges ⇒non-perturbative treatment of QCD necessary: e.g. lattice simulations of QCD

phenomenological ‘Cornell’ qq potentialEichten et al PRD 17, 1978,0 and 21, 1980; Richardson PLB 82, 1979

V (r) = −αs

r + σr + C

! good description of quarkonia (ccand bb mesons)

value σ = 0.85 GeV/fm at r ∼ 2 fm:∃ enough energy to produce light qqpair

light mesons require relativistictreatmente.g. “relativized” quark modelsGodfrey, Isgur PRD 32, 1985

! good description of meson spectrumnot covariantno off-shell propagation of quarks

Allton et al, UKQCD Collab., PRD 65, 2002


Allton et al, UKQCD Collab., PRD 65, 054502 (2002)

Static QCD potential from the latticePhenomenological kernelqq

Inspired by Cornell potential:

NR linear potential in momentum space: Fourier transform of screened potential

�r = lim✏!0

�@2

@✏2e�✏r

rUsually:

But simpler: �r = lim✏!0

��

✏

�e�✏r � 1

�⌘ VA(r)� VA(0)

VA(q) = �8⇡�

q4Gross, Milana, PRD 43, 2401 (1991)Savkli, Gross, PRC 63, 035208 (2001)

highly singular automatic subtraction

only a Cauchy principal value singularity remains

V (r) = �r � C � ↵s

r

hVL�i(p) =Z

d3k

(2⇡)3VL(p� k)�(k) = �8⇡�

Zd3k

(2⇡)3�(k)� �(p)

(p� k)4

VL(q) = VA(q)� (2⇡)3�(q)

Zd3q0

(2⇡)3VA(q

0)FT:

with


Covariant confining kernel in CST

Covariant generalization: q2 ! �q2

This leads to a kernel that acts like

k = (Ek,k)

pR = (EpR ,pR)

pR = pR(p0,p) value of k at which kernel becomes singular

kp

any regular function

initial state:either quark or antiquark onshell

hVL�i(p) =Z

d3k

(2⇡)3m

EkVL(p, k)�(k) = �8⇡�

Zd3k

(2⇡)3m

Ek

�(k)� �(pR)

(p� k)4

V nrL (r = 0) = 0corresponds to

Properties:Subtraction regularizes kernel to Cauchy principal valueNonrelativistic limit → linear potential Satisfies the condition

But does it still confine?

Yes: the vertex function vanishes if both quarks are onshell!= 0

Zd3k

(2⇡)3m

Ek!

Z

k

hVLi =Z

kVL(p, k) = 0

More details: Savkli, Gross, PRC 63, 035208 (2001)

Shorthand


Spontaneous chiral symmetry breaking - pion equation

= + + +12—{ }

Pion bound-state equation for finite mass μ

For μ→0 this becomes (in the rest frame)

= +12— 1

2—lim

µ!0

P = (µ,0) P = 0 P = 0

�5G0 �5G0�5G(p2)

For a kernel with Lorentz scalar and vector structure V(p, k) = VS(p, k)11 ⌦ 12 +

1

4VV (p, k)gµ⌫�

µ1 ⌦ �⌫

2

one gets a condition for the existence of a massless pion solution

1 =Z0

4m(1�B0)

Z

k

VV (p, k) + VV (p,�k)� VS(p, k)� VS(p,�k)

�

Z0 = Z(m2)

B0 = B(m2)


Spontaneous chiral symmetry breaking

Dressed quark propagator S(p) =1

m0 � /p+ ⌃(p)� i✏= Z(p2)

M(p2) + /p

M2(p2)� p2 � i✏

⌃(p) = A(p2) + /pB(p2) Z(p2) =1

1�B(p2)M(p2) =

m0 +A(p2)

1�B(p2)self-energy wave function renormalization mass function

The kernel determines the quark self-energyqq

+= 12— + 1

2—

m0 � /p+ ⌃(p) m0 � /p

Dressed quark propagator is

with

the self-energy Σ is computed from the kernel:

The gap equation gives the constraint

compare with the zero mass pion condition

Spontaneous chiral symmetry breaking in the CST (2)

Σ( p ) = A( p

2) + p B( p

2)

S( p ) =1

m0− p + Σ( p )

=

Z( p2) M ( p

2) + p⎡

⎣⎤⎦

M2( p

2) − p

2 − iε

M ( p

2) =

m0+ A

1 − B; Z( p

2) =

1

1 − B

M (m2) = m

These are the

same if:

• m0 = 0

•

VS( p, k) = 0

k

∫

m

0− p + Σ( p )

m

0− p

Σ( p )

1

2

1

2+ +=( )-1


with





Σ( p ) = A( p

2) + p B( p

2)

S( p ) =1

m0− p + Σ( p )

=

Z( p2) M ( p

2) + p⎡

⎣⎤⎦

M2( p

2) − p

2 − iε

M ( p

2) =

m0+ A

1 − B; Z( p

2) =

1

1 − B

M (m2) = m

These are the

same if:

• m0 = 0

•

VS( p, k) = 0

k

∫

m

0− p + Σ( p )

m

0− p

Σ( p )

1

2

1

2+ +=

-1)(

⌃(p)


with





Σ( p ) = A( p

2) + p B( p

2)

S( p ) =1

m0− p + Σ( p )

=

Z( p2) M ( p

2) + p⎡

⎣⎤⎦

M2( p

2) − p

2 − iε

M ( p

2) =

m0+ A

1 − B; Z( p

2) =

1

1 − B

M (m2) = m

These are the

same if:

• m0 = 0

•

VS( p, k) = 0

k

∫

m

0− p + Σ( p )

m

0− p

Σ( p )

1

2

1

2+ +=


with





Σ( p ) = A( p

2) + p B( p

2)

S( p ) =1

m0− p + Σ( p )

=

Z( p2) M ( p

2) + p⎡

⎣⎤⎦

M2( p

2) − p

2 − iε

M ( p

2) =

m0+ A

1 − B; Z( p

2) =

1

1 − B

M (m2) = m

These are the

same if:

• m0 = 0

•

VS( p, k) = 0

k

∫

m

0− p + Σ( p )

m

0− p

Σ( p )

1

2

1

2+ +=

CST-Dyson equation:

The dressed quark mass is the solution of the gap equation M(m2) = m

in the equation for A at

this yields another constraint

A(p20) =Z0

4

Z

k

hVS(p, k) + VS(p,�k) + VV (p, k) + VV (p,�k)

i

B(p20) =Z0

4p0

Z

k

Ek

m

VS(p, k)� VS(p,�k)� 1

2VV (p, k) +

1

2VV (p,�k)

�

p = (p0,0)In rest frame, :

M(m2) = m = m0 +A0 +mB0We can use

Z0 = Z(m2)

A0 = A(m2)

B0 = B(m2)

p0 = m


Spontaneous chiral symmetry breaking

1 =Z0

4m(1�B0)

Z

k

VV (p, k) + VV (p,�k)� VS(p, k)� VS(p,�k)

�

1 =1 +m0/A0

4m(1�B0)Z0

Z

k

hVV (p, k) + VV (p,�k) + VS(p, k) + VS(p,�k)

i

The two equations become identical for m0 = 0

provided thatZ

k

hVS(p, k) + VS(p,�k)

i= 0

constraint from gap equation for A

zero-mass pion condition

the Lorentz scalar interaction is zeroor it satisfies the constraint

The linear confining kernel satisfiesZ

kVL(p, k) = 0

Compare: {When

The confining interaction does not contribute to

in the chiral limit (m0 → 0),

then a zero-mass pion solution exists (Goldstone boson)

if a finite quark mass m can be generated

the confining kernel can have a scalar component!

A(p2) pion equation{ (m0 = 0)

Also: if m0 > 0 the condition forA(p2)guarantees that no zero-mass pion solution exists→ the real pion must have a finite mass


NJL mechanism in diagrammatic form

!ðpÞ ¼ Z0

4m

Zk

!m½VSðp; kÞ þ VSðp;&kÞ þ VVðp; kÞ

þ VVðp;&kÞ' þ 6k½VSðp; kÞ & VSðp;&kÞ

& 1

2VVðp; kÞ þ

1

2VVðp;&kÞ'

": (42)

For simplicity, evaluate this in the rest frame (p ¼ 0),

where the integral over d3k ensures that ^6k ! !0Ek, andextract the self-energy functions A and B:

Aðp20Þ ¼

Z0

4

Zk½VSðp; kÞ þ VSðp;&kÞ þ VVðp; kÞ

þ VVðp;&kÞ';

Bðp20Þ ¼

Z0

4p0

Zk

Ek

m

#VSðp; kÞ & VSðp;&kÞ & 1

2VVðp; kÞ

þ 1

2VVðp;&kÞ

$: (43)

Note that, since p ¼ 0, A and B are functions of p2 ¼ p20,

as required by Lorentz covariance.At p ¼ p (so that p0 ¼ m), the constraint in Eq. (30)

gives

mð1& B0Þ ¼ m0 þ A0: (44)

Using this reduces the equation for A at p0 ¼ m to theconstraint

1 ¼ 1þm0=A0

4mð1& B0ÞZ0


þ VVðp;&kÞ': (45)

Comparing Eqs. (39) and (45) shows that they are identicalif m0 ¼ 0 and if the integral over the scalar interactionvanishes:

Zk½VSðp; kÞ þ VSðp;&kÞ' ¼ 0: (46)

This condition is satisfied by the models discussed in thispaper.Unless m0 ¼ 0, constraint (39) will not be satisfied,

insuring that there is no pion bound state with zero mass.This means that the same constraint that makes it possiblethat m ! 0 [Eq. (45)] also ensures that there exists a pionwith zero mass [Eq. (39)] in the chiral limit. These con-sistency conditions link the spontaneous generation of adressed quark mass in the chiral limit, and hence thespontaneous breaking of chiral symmetry, to the existenceof a massless Goldstone pion.The equivalence between the zero-mass pion equation

and the self-energy equation in the chiral limit can alsoeasily be demonstrated in terms of Feynman diagrams, asshown in Fig. 6: The scalar self-energy Aðp2Þ becomesequal to the scalar part of the inverse dressed propagator inthe chiral limit. Multiplying with a !5 and attaching twooff-shell quark lines of momentum p and one pion line ofzero momentum yields a ‘‘spacelike’’ Yukawa vertex.

FIG. 6 (color online). The equivalence between the BS bound-state vertex function for a zero-mass pion and the scalar part (s.p.) ofthe CST self-energy in the chiral limit.

CONFINEMENT, QUARK MASS FUNCTIONS, AND . . . PHYSICAL REVIEW D 89, 016005 (2014)

016005-9

!ðpÞ ¼ Z0

4m

Zk



& 1

2VVðp; kÞ þ

1

2VVðp;&kÞ'

": (42)



Aðp20Þ ¼

Z0

4


þ VVðp;&kÞ';

Bðp20Þ ¼

Z0

4p0

Zk

Ek

m


2VVðp; kÞ

þ 1

2VVðp;&kÞ

$: (43)



gives

mð1& B0Þ ¼ m0 þ A0: (44)


1 ¼ 1þm0=A0

4mð1& B0ÞZ0










016005-9

!ðpÞ ¼ Z0

4m

Zk



& 1

2VVðp; kÞ þ

1

2VVðp;&kÞ'

": (42)



Aðp20Þ ¼

Z0

4


þ VVðp;&kÞ';

Bðp20Þ ¼

Z0

4p0

Zk

Ek

m


2VVðp; kÞ

þ 1

2VVðp;&kÞ

$: (43)



gives

mð1& B0Þ ¼ m0 þ A0: (44)


1 ¼ 1þm0=A0

4mð1& B0ÞZ0










016005-9

1-body CST-Dyson equation

2-body pion equation

In the chiral limit:

multiply by γ5and attach quark lines and a pion line with P=0

topologically equivalent

scalar part


Lorentz structure of the kernelWe construct a simple exploratory model:

VL(p, k) =⇥�11 ⌦ 12 � (1� �)�µ

1 ⌦ �µ2

⇤VL(p, k)

Mixed scalar-vector linear confining kernel (becomes linear potential in nonrel. limit)

Pure vector “constant” (in r-space) interaction

VC(p, k) =1

4gµ⌫�

µ1 ⌦ �⌫

2 VC(p, k)

VC(p, k;P ) = 2C h(p21)h(p22)h(k

21)h(k

22)(2⇡)

3Ek

m�3(p� k)with

12

�k2

k1p1

�p2

h(p21)

�p2

p1h(p21)

�k2

k1

FIG. 8. (Color online) The strong quark form factors can be viewed as gluon loop corrections to the quark vertices (shownhere at the two-loop level).

C. Strong quark form factors

The kernels given in Eqs. (49) and (50) include (strong)phenomenological form factors [36–38, 46, 51, 52]. Theirpurpose is twofold: they can be regarded as describingsome gluonic corrections to the vertices that would other-wise be overlooked, and they provide convergence in loopintegrals. The form factors are expressed as products ofindividual quark form factors h(p2), one for each quarkline associated with momentum p.

The idea that these form factors serve as an e↵ectivedescription of the infinite sum of overlapping gluon loopcorrections to the interaction vertices is illustrated inFig. 8 (shown only for two-loops). This raises the possi-bility that they could be calculated from first principlesat a later date.

The explicit form of h(p2) will be specified later. Ourquark form factors di↵er from those used in previousapproaches by their normalization at the on-shell pointp2 = m2. They will be normalized to 1 only in the chi-ral limit, h(m2

�) = 1, and therefore h(m2) 6= 1. Withthis normalization all free parameters of the quark massfunction are fixed in the chiral limit and the constituentquark mass m for finite m0 is then uniquely determinedfrom the mass constraint (30).

The use of factorized form factors [where the form fac-tor at a vertex, H(p, p0), is separated into a product ofseparate form factors, h(p2)h(p02)] is an advantage whencalculating electromagnetic current matrix elements inthe presence of strong form factors. If they are movedfrom the vertices to the quark propagators (leaving thevertices bare) as illustrated in Fig. 9, and reinterpretedas an additional correction to the quark propagator, thena dressed quark current can be constructed that includesthem and also satisfies the Ward-Takahashi identity [51].This is discussed in Ref. II.

VI. QUARK MASS FUNCTION

In this section we apply the interaction kernel definedin the previous section to calculate the CST self-energygiven in Eq. (43), and consequently the dynamical quarkmass function M(p2) from Eq. (29). Our Minkowski-

space results are then compared with LQCD calculationsperformed in Euclidean space. This comparison requiresthe computation of our mass function at negative valuesof p2.

A. Self-energy from the linear confining potential

The discussion on the decoupling of confinement fromchiral symmetry breaking revealed that the linear con-fining potential does not contribute to the dynamicalquark mass generation because the corresponding con-tribution to the scalar self-energy vanishes, as expressedin Eq. (54). For the remaining vector part of the self-energy we insert the confining kernel (48) into Eq. (43)and get

BL(p20) =

(�� 2)

2p0h2(m2)h2(p20)

⇥Z

d3k Z0

(2⇡)3⇥

VL(p, k;P )� VL(p,�k;P )⇤

= �(�� 2)2⇡�(p20 +m2)

p40h2(m2)h2(p20)

⇥Z

d3k

(2⇡)3Z0

E2k � E2

pR

, (57)

where the form factors h(p2) discussed in Sec.VC havebeen introduced at each vertex. Equation (57) shows thatBL is a function of p20, as it should be. Note that thiswould not be the case if we had not included the negative-energy propagator-pole contribution from the upper halfcomplex k0-plane. Furthermore, (57) displays the simpledependence of BL on the adjustable mixing parameter�. For the particular choice � = 2, BL is zero and theconfining potential does not contribute at all to the quarkself-energy, i.e.,

⌃L(p0) = 0 for � = 2 . (58)

This choice corresponds to a 2 : 1 ratio between scalarand vector coupling in the Lorentz structure of the con-fining kernel.The case � = 2 is appealing because of its simplicity

as it avoids the computation of the UV-divergent inte-gral in (57), and therefore we will explore this case in

Quark form factors h provide regularization (one factor for each quark line)

Can be viewed as vertex corrections

In the chiral limit, VL does not contribute to A (or the pion vertex)For one gets also � = 2

AL(p2) = 0

BL(p2) = 0 ⌃L(p

2) = 0

no contribution from the confining interaction to the self-energy


Quark mass function

The constant vector interaction contributes only to A(p2)

M(p2) = A(p2) +m0 = C h2(m2)h2(p2) +m0 C = m� + c1m0 +O(m20)with

in chiral limitM�(p2) = m�h

2(p2)m0 ! 0

Quark mass functionEB, Gross, Pena, Stadler PRD 89 2014

scalar-pseudoscalar linear confining potential does not contribute to self energy

constant interaction only contributes to A

⇒ mass function M(p2) = C (6+2ξ)8 h2(m2)h2(p2) +m0

parameters C = mχ + c1m0, Λ = 2.04 GeV and mχ = 0.308 GeV fixed by fit tolattice QCD data (at negative p2) in chiral limit

Lattice QCD data from Bowman et al PRD 71, 2005 extrapolated to chiral limit

!

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!!!!

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!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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!10 !5 00.0

0.1

0.2

0.3

0.4

p 2 ! GeV2 "

M!p2"!GeV"


Fit to (Euclidean) LQCD data p2 < 0fit for

⇤ = 2.042 GeV m� = 0.308 GeV

(to 50 points with p2 > -1.94 GeV2)

Lattice data don’t converge to m0 for large -p2 (finite lattice spacing effect)

h(p2) =

⇤2 �m2

�

⇤2 � p2

!2


Quark mass function for finite m0

Quark mass functions for finite m0

EB, Gross, Pena, Stadler PRD 89 2014; Lattice QCD data: Bowman et al PRD 71, 2005

m0 = 0.016 GeV, m = 0.363 GeV

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!10 !5 00.0

0.1

0.2

0.3

0.4

p 2 ! GeV2 "

M!p2"!GeV"

m0 = 0.032 GeV, m = 0.403 GeV

!

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!!!!!!!

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!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!10 !5 00.0

0.1

0.2

0.3

0.4

p 2 ! GeV2 "

M!p2"!GeV"

m0 = 0.047 GeV, m = 0.434 GeV

!!

!!!

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!!!!!

!!!!!

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!

!!!!!!!!!!!!!

!10 !5 00.0

0.1

0.2

0.3

0.4

p 2 ! GeV2 "

M!p2"!GeV"

m0 = 0.063 GeV, m = 0.462 GeV

!

!

!!!

!

!!!!!!!

!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!

!10 !5 00.0

0.1

0.2

0.3

0.4

p 2 ! GeV2 "

M!p2"!GeV"


C = m� + c1m0 +O(m20) is fit (other parameters fixed in the chiral limit)c1


Pion form factor

Quark and pion currents EB, Gross, Pena, Stadler PRD 89 2014

Electromagnetic pion current in relativistic impulse approximation:

q = P+ − P−

P−P+

S(p+) S(p−)

Λ(−k)

Γ Γ

jµ

+

q = P+ − P−

P−P+

S(−p+) S(−p−)

Λ(k)

Γ Γ

jµ

h(p2) h(p′2)jµ

S(p) S(p′)h2S(p) h′2S(p′)

h−1jµh′−1

≡

(reduced) off-shell quark current (quark-photon vertex)jµR=f(γµ + κ iσµνqν

2m ) + δ′Λ′γµ + δγµΛ+ gΛ′γµΛ

with Λ =M(p)−/p

2M(p) ; f, δ(′), g form factors

satisfies vector Ward-Takahashi identity ⇒ pion current conserved !

differs in chiral limit by transverse component from Ball-Chiu currentBall, Chiu PRD 22, 1980


The pion current in CST: triangle diagrams with 6 propagator poles (each diagram)RIA (Relativistic Impulse Approximation): keep only the spectator poles

Analysis of pole structure: RIA is a good approximation for ‣ large Q2 (any μ)

‣ large μ (any Q2)

Needed ingredients:pion vertex functiondressed quark current

For small Q2 and small μ, the remaining poles contribute significantly→ need to calculate the CIA (Complete Impulse Approximation)


A simple pion vertex functionIn the chiral limit, the self-energy function A(p2) and the vertex Γ(p) satisfy the same equation → we already have a pion vertex!

�BS(p1, p2) =G1(p21, p

22)�

5 +G+(p21, p

22)(/p1�

5 + �5/p2)

+G�(p21, p

22)(/p1�

5 � �5/p2) +G3(p

21, p

22)/p1�

5/p2

General BS vertex structure for pseudoscalar bound states�(p, p) = G(p2)�5

in chiral limit (and rest frame)

For real pion, assume the chiral limit structure dominates

Pion vertex functionEB, Gross, Pena, Stadler PRD 89 2014

p

−p

P = 0

P

=

=

+

+

−p

p + P

12

12

12

12

⇒ CST pion vertex function near chiral limit Γ(p, P ) ∼ γ5h(p2)


near the chiral limit �(p, p) = G0h(p2)�5


Dressed quark currentUse the prescription by Gross & Riska to derive a conserved quark current

Quark and pion currents EB, Gross, Pena, Stadler PRD 89 2014

Electromagnetic pion current in relativistic impulse approximation:

q = P+ − P−

P−P+

S(p+) S(p−)

Λ(−k)

Γ Γ

jµ

+

q = P+ − P−

P−P+

S(−p+) S(−p−)

Λ(k)

Γ Γ

jµ

h(p2) h(p′2)jµ

S(p) S(p′)h2S(p) h′2S(p′)

h−1jµh′−1

≡

(reduced) off-shell quark current (quark-photon vertex)jµR=f(γµ + κ iσµνqν

2m ) + δ′Λ′γµ + δγµΛ+ gΛ′γµΛ

with Λ =M(p)−/p

2M(p) ; f, δ(′), g form factors

satisfies vector Ward-Takahashi identity ⇒ pion current conserved !

differs in chiral limit by transverse component from Ball-Chiu currentBall, Chiu PRD 22, 1980


Vertex form factors are absorbed into modified propagatorsImpose Ward-Takahashi identity on reduced current

jµR(p0, p) = h�1(p02)jµ(p0, p)h�1(p2)

S(p) = h2(p2)S(p)

qµjRµ(p0, p) = eS�1(p)� eS�1(p0)

jµR = f

�µ +

i�µ⌫q⌫2m

�+ �0⇤0�µ + ��µ⇤+ g⇤0�µ⇤Structure of the current:

off-shell form factors determined by WTI in terms of h(p2)⇤ =

M(p2)� /p

2M(p2)

Differs from Ball-Chiu current (used by Tandy and Maris) by a transverse pieceWTI cannot determine the current uniquely → need a dynamical calculation


Results for the pion form factor

Pion form factor EB, Gross, Pena, Stadler PRD 89 2014

µ = 0.42 GeV, µ = 0.28 GeV, µ = 0.14 GeV

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q 2 ! GeV2 "

Λi2F Π!Q2,Μ

i"

0 0.25 0.5

1

1.5

2

Q 2 ! GeV2 "

Λi2F Π!Q2,Μ

i"

ρ pole

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

Q 2 ! GeV2 "

Q2F Π!Q2"

Fπ independent of h form factors !

scaling relation

Fπ(Q2,λµ)

Q2≫µ2

≃ λ2Fπ(Q2, µ)

correct monopole behaviour

Fπ(Q2)

Q2≫µ2

∼ 1Q2+ν2

RIA fails for small pion masses µdata: Amendolia et al 1986; Brown et al 1973; Bebek et al 1974; 1976; 1978; Huber et al 2008

scalingElmar Biernat (CFTP/IST) Quarks and mesons in CST May 22, 2014 21 / 23

The pion form factor is calculated with different pion massesBest fit with μ=0.42 GeV (somewhat large, but allows us to use RIA at small Q2!)

Fπ shows correct monopole fall-off with this is independent of h(p2)Remarkable scaling relations hold at large Q2

(used in the figure for μ=0.14 and 0.28 GeV)

F⇡(Q2,�µ)

Q2�µ2

' �2F⇡(Q2, µ)

F⇡(Q2)

Q2�µ2

⇠ 1

Q2 + ⌫2 ⌫ ' 0.63 GeV Q: Why does the model work well without a ρ-pole (VMD)?Should calculate the dressed quark current dynamically (to be done soon!)

Pion form factor EB, Gross, Pena, Stadler PRD 89 2014

µ = 0.42 GeV, µ = 0.28 GeV, µ = 0.14 GeV

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Q 2 ! GeV2 "

Λi2F Π!Q2,Μ

i"

0 0.25 0.5

1

1.5

2

Q 2 ! GeV2 "

Λi2F Π!Q2,Μ

i"

ρ pole

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

Q 2 ! GeV2 "

Q2F Π!Q2"

Fπ independent of h form factors !

scaling relation

Fπ(Q2,λµ)

Q2≫µ2

≃ λ2Fπ(Q2, µ)

correct monopole behaviour

Fπ(Q2)

Q2≫µ2

∼ 1Q2+ν2

RIA fails for small pion masses µdata: Amendolia et al 1986; Brown et al 1973; Bebek et al 1974; 1976; 1978; Huber et al 2008

scalingElmar Biernat (CFTP/IST) Quarks and mesons in CST May 22, 2014 21 / 23

simple ρ-pole

our model, μ=0.42 GeV


SummaryFormalism

Model calculationsA very simple model can yield good agreement of the quark mass function with lattice calculations, and of the pion form factor with experimental data

Work in progress or planned for the near future

Developed the CST formalism for a dynamical model for all mesonsCST equations are solved in Minkowski spaceConsistency between bound-state and mass gap equationsDescribes confinement and spontaneous chiral symmetry breakingCan be applied in the time-like region

qq

Solution of the full CST equation and fit to the meson spectrumStudy interaction kernels with different Lorentz structure; add gluon exchangeCalculate quark-photon vertex dynamicallyStudy ππ scattering (constraints from axial-vector WTI)

qq


But what does the pion look like?

Most likely…

the pion form factor in the covariant spectator theoryalfred stadler, university of Évora the pion...

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