pion form factor from the covariant spectator theory · central ideas: •work in physical...

Pion Form Factor from the Covariant Spectator Theory

Franz Gross, Jan.16, 2014

Franz Gross - JLab/W&M

What is the CST?

How is confinement described inthe CST?

Dynamical chiral symmetrybreaking in the CST [through theNambu-Jona-Lasinio (NJL)mechanism]

Results from a simple model:• Quark mass function• Pion form factor

Overview

Discussion

Based on two recent papers publishedin PRD89 (2014):

“Confinement, quark mass functions,and spontaneous chiral symmetrybreaking in Minkowski space,” 016005

“Pion electromagnetic form factor inthe CST,” 016006

in collaboration with

Elmar Biernat

Teresa Pena

Alfred Stadler

Central ideas:• work in physical space-time (Minkowski- real time, not Euclidean- imaginary time)• replace the Bethe-Salpeter (BS) equations (integrates over fourth-component of

momentum)

What is the covariant spectator theory (CST)?

BS

d 4k(2π )4∫

p2 = p02-p2 p2 = -p0

2-p2


momentum)

• by the CST equations (integration replaced by a sum over particle poles)


BS

d 4k(2π )4∫

=

12

d 3k(2π )3∫

mEk

CST-BS

p2 = p02-p2 p2 = -p0

2-p2


momentum)

• by the CST equations (integration replaced by a sum over particle poles)

Advantages:• Efficient reorganization of the series for the irreducible kernel;

• Smooth non-relativistic limit when particle masses approach infinity• Four-channel equation preserves both charge conjugation and particle exchange

symmetry


BS

d 4k(2π )4∫

=

12

d 3k(2π )3∫

mEk

CST-BS

cancellations between crossed graphs and off-shell terms (cancellation theorem) m1

limm1 →∞

p2 = p02-p2 p2 = -p0

2-p2

Applications of the CST and selected references:

NN and 3N systems• FG, PR 186, 1448 (1969); D10, 223 (1974); C26, 2203

(1982); C26, 2226 (1982)• FG, J.D. Van Orden, K. Holinde, PRC45, 2094 (1992)• M.T. Pena, FG, Y. Surya, PRC54, 2235 (1996)• A. Stadler & FG, PRL78, 26 (1997)• J. Adam, FG, C. Savkli, J.W. Van Orden, PRC56, 641

(1997)• A. Stadler, FG, and M. Frank, PRC56, 2396 (1997)• FG, A. Stadler, M.T. Pena, PRC69, 034007 (2004)• FG & A. Stadler, PLB 657,176 (2007); PRC78, 014005

(2008); PRC82, 034004 (2010)

NN and 3N currents and form factors• R.G. Arnold, C.E. Carlson, FG, PRC21, 1426 (1980)• FG & D.O. Riska, PRC36, 1928 (1987)• J.W. Van Orden, N. Devine, FG, PRL75,4369 (1995)• J. Adam, J.W. Van Orden, FG, NPA640, 391 (1998)• S.A. Pinto, A. Stadler, FG, PRC79,054006 (2009)

!N scattering and N→N* form factors• FG & Y. Surya, PRC47, 703 (1993)• Y. Surya & FG, PRC53, 2422 (1996)

NA scattering• FG & K.M. Maung, PRC42, 1681 (1990)• FG, K.M. Maung, J.A. Tjon, L.W. Townsend, J.A.

Wallace, PRC43, 1378 (1991)

eA• FG & S. Liuti, PRC45, 1374 (1992)• S. Liuti & FG, PLB356, 157 (1995)

Quark models of N and N*• FG & P. Agbakpe, PRC73, 015203 (2006)• FG, G. Ramalho, M.T. Pena, PRC77, 015202

(2008)• G. Ramalho, M.T. Pena, FG, EPJA36, 329 (2008);

PRD78, 114017 (2008); PLB687, 355 (2009)• FG, G. Ramalho, K. Tsushima, PLB690, 183 (2010)• FG, G. Ramalho, M.T. Pena, PRD85, 093005

(2012); 093006 (2012)

mesons as qqbar bound states• FG & J. Milana, PRD43, 2401 (1991); 45, 969

(1992); 50, 3332 (1994)• M. Uzzo & FG, PRC59, 1009 (1999)• C. Savkli & FG, PRC63, 035208 (2001)• E. Biernat, FG, M.T. Pena, A. Stadler, FBS, DOI

10.1007 (2012); PRD (2014)

How is confinement described in the CST?

Modeled by a singular, phenomenological, qqbar kernel• screened non-relativistic potential

• generalization (relativistic KERNEL replaces non-relativistic potential)

Features:• non-relativistic limit (m ⇒ ∞) reduces to a pure linear potential• ALL scattering cuts are cancelled; equations have only bound state poles• Propagators can have real poles, but NO asymptotic states• constraint:

σ r = σ limε→∞

d 2

dε 2

e−εr

r⇒ lim

ε→∞

d 2

dε 2

σq2 + ε 2 = σ lim

ε→∞

−2

q2 + ε 2( )2 +8ε 2

q2 + ε 2( )3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

VLφ ( p) = σ d 3k

(2π )3

mEk

φ(k̂) −φ( pR )⎡⎣ ⎤⎦( p − k̂)4∫

k̂ p⇒

X k̂ p̂

1991 2014 k̂ = (Ek ,k); p = ( p0 ,p)

VL ( p) = d 3k

(2π )3

mEk

VL( p, k̂) =∫ VL( p, k̂) =k∫ 0

Four-channel equation (review)

Dynamical chiral symmetry breaking in the CST (1)

=

12

d 3k(2π )3∫

mEk

CST-BS µ≠0


Zero mass limit (for pion)


limµ→0

µ

p − 12 µ

p + 12 µ

⇒

=

12

d 3k(2π )3∫

mEk

CST-BS µ≠0

G( p2 )γ 5

G0γ5

G0γ5


Zero mass limit (for pion)

If the kernel has the form:

then the zero mass pion equation reduces to the constraint


limµ→0

µ

p − 12 µ

p + 12 µ

⇒

=

12

d 3k(2π )3∫

mEk

CST-BS µ≠0

G( p2 )γ 5

G0γ5

G0γ5

Dressed quark propagator iswith


Σ( p ) = A( p2 ) + p B( p2 )

S( p ) = 1

m0 − p + Σ( p )=

Z( p2 ) M ( p2 ) + p⎡⎣ ⎤⎦M 2 ( p2 ) − p2 − iε

M ( p2 ) =

m0 + A1 − B

; Z( p2 ) = 11 − B

A0 = A(m2 )

B0 = B(m2 )

Z0 = Z(m2 )


Self-energy Σ is computed from the kernel:


Σ( p ) = A( p2 ) + p B( p2 )

S( p ) = 1

m0 − p + Σ( p )=

Z( p2 ) M ( p2 ) + p⎡⎣ ⎤⎦M 2 ( p2 ) − p2 − iε

M ( p2 ) =

m0 + A1 − B

; Z( p2 ) = 11 − B

m0 − p + Σ( p ) m0 − p Σ( p )

12

12+ +=

A0 = A(m2 )

B0 = B(m2 )

Z0 = Z(m2 )


Self-energy Σ is computed from the kernel:

The gap equation M(m2) = m gives the constraint obtained from A0

compare with the zero mass pion condition


Σ( p ) = A( p2 ) + p B( p2 )

S( p ) = 1

m0 − p + Σ( p )=

Z( p2 ) M ( p2 ) + p⎡⎣ ⎤⎦M 2 ( p2 ) − p2 − iε

M ( p2 ) =

m0 + A1 − B

; Z( p2 ) = 11 − B

These are thesame if:• m0 = 0•

VS ( p̂, k̂) = 0

k∫

m0 − p + Σ( p ) m0 − p Σ( p )

12

12+ +=

+ +

A0 = A(m2 )

B0 = B(m2 )

Z0 = Z(m2 )

The gap equation and the equation for the existence of a zero mass pion areidentical (dynamical chiral symmetry breaking) if• m0 = 0• the scalar part of the kernel is either

zero (the usual assumption) or

or confining, which automatically satisfies the necessary constraint• we choose VS ⇒ VL

Conclusions:• if quark mass can be dynamically generated (i.e. m ≠ 0 when m0 = 0), then a zero

mass pion state exists• The linear confining interaction does not contribute to the χSB condition, and hence

confinement can have a scalar component• if the linear confinement is of the form

with λ = 2, then it makes no contribution to B either, and hence ΣL = 0 !


VL( p, k̂) = 0

k∫

the famous Nambu-Jona-Lasinio (NJL) mechanism

Results from a simple model (proof of principle) --1

Motivation:• near the chiral limit (pion and light quarks with small m0)• neglect the confining interaction• reasonable first test of the ideas

Vector interaction only• constant in coordinate space (regularized by quark from factors h)

⇒

VL( p, k̂) = 0

k∫

VV →VC = 2C(2π )3 Ek

mδ 3 ( p − k)h( p1

2 )h( p22 )h(k1

2 )h(k22 )

Mass function (simple model) --2

Solution for the running mass

Choose h to fit lattice data (in the Euclidean region, p2 ≤ 0)

M ( p2 ) = C(m0 )h2 ( p2 )h2 (m2 ) + m0 mχ h2 ( p2 )⇒m0=0

h( p2 ) =

(Λ2 − mχ2 )2

(Λ2 − p2 )2

fit the parameters to thefirst 50 points (p2=-1.94)with χ2/dof = 0.61:

Λ = 2.042 GeV

mχ = 0.308 GeV

+

12

Σ( p2 ) = 12

Γχ ( p) = G0h( p2 )γ 5 Γχ ( p−

2 ) Γχ ( p+

2 )

−k̂

p− p+

Pion form factor (simple model) --3

Approximate pion vertex function

CST: keep ONLY propagator polesin the triangle diagram• complete calculation requires all 6

poles (2X3) from each diagram• RIA requires only the spectator pole

from each

Need dressed quark current• we used prescription of FG & Riska• Ball-Chiu prescription is different

• RIA reliable at high Q2 for any pion mass• Errors in RIA at small Q2 and µ2

(omitted propagator poles)• µ = 0.42 ---- gives best fit for ALL Q2 µ = 0.28 ---- µ = 0.14 ----

Γχ ( p) = G0h( p2 )γ 5 Γχ ( p−

2 ) Γχ ( p+

2 )

−k̂

p− p+

Pion form factor (simple model) --3

Approximate pion vertex function

CST: keep ONLY propagator polesin the triangle diagram• complete calculation requires all 6

poles (2X3) from each diagram• RIA requires only the spectator pole

from each

Need dressed quark current• we used prescription of FG & Riska• Ball-Chiu prescription is different

Results

Overview

Advantages:• χSB and quark confinement, even though quark propagators can have

real poles• Very simple model can explain the physics near the chiral limit --

and it can be improved• Way is open to the study of the meson spectrum and other

phenomena in the time-like (as well as space-like) regions

Problems and disadvantages• Breaking of the confining flux tube leads to particle production,

which can be handled using coupled channel equations.• Singularities (some unphysical) still exist in the time-like region and

must be treated carefully (distasteful, but not a show-stopper).• Form factors include both physical effects and regularization. A

better regularization scheme should be developed

Discussion

Comparison between CST and DSE

Issue #1: Mass-shell quarks?• CST allows single quarks to be on-shell, with real masses;• DSE does not• Comment: The CST can be generalized to include complex poles

Issue #2: Existence of flux tubes (or confining strings)• All agree that flux tubes break (is the glass half full or half empty?)• CST can handle particle production (string breaking) through coupled channel

equations• How does DSE treat particle production?

Issue #3: Construction of bound state equations• CST starts with bound state equations and constructs, through the NJL mechanism,

a consistent quark mass function• No simple way for the DSE to get the bound state kernel from the mass function.

A new kernel must be introduced.

Issue #4: Study of excited states• The CST bound state equations can be studied for large time-like p2

Evidence from the LQCD mass function

No evidence for an inflection point near p = 0 (there is alwaysone if a function of p2 is plotted vs. p)

1/p4 falloff does not rule out real mass poles (?)

(0.32 − c) Λ2

Λ2 + p2

⎛

⎝⎜⎞

⎠⎟

2

+ ccurve fit =

Screened linear potentials break confinement

Correct treatment of string breaking requires coupled channels

BUT, confinement is broken by regularized “linear” potentials withfinite ε. Example for σ = 0.2 GeV2 ∼ 1 GeV/fm:

• π→3π requires energy of 0.42-0.14 ≈ 0.28 GeV• J/Ψ→DD requires 3.4-3.1 ≈ 0.3 GeV• In both cases, string breaks when it is stretched to about r0 ≈ 0.3 fm.• Choose r0ε ≈ 1, or ε = 3.33 fm-1

σ r

σε

1 − e−εr( ) σ r e−εr

Bound-state equations

In CST-BS equation has the usual structure

The DSE-BS equation requires a new kernel and a new anzatz

=

Known from thegap equation

New kernel

pion form factor from the covariant spectator theory · central ideas: •work in physical...

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