Looking at hadrons in the backward direction with hard ...Looking at hadrons in the backward...

Looking at hadrons in the backward direction with hard photons:the Transition Distribution Amplitudes

& DVCS on virtual pions

J.P. LansbergCPHT – Ecole polytechnique

Seminaire de physique des particules

LPT – Paris-Sud, Orsay

March 2, 2010

Collaborative work with S.J. Brodsky, M.Diehl, B. Pire and L. Szymanowski

J.P. Lansberg (CPHT – Ecole polytechnique) Transition Distribution Amplitudes March 2, 2010 1 / 31

Outline

Part 1: Reminders

1 Reminder on DIS and DVCS

Part 2: Backward regime

2 Definition of the Transition Distribution Amplitudes

3 Backward electroproduction of a pion

Part 3: Extensions

4 TDA studies at GSI/FAIR

5 TDA and Intrinsic Charm

Part 3: DVCS on virtual pions

6 A few words on DVCS on virtual-pion target

Part 4: Outlooks

Part I

Reminders

Reminder on DIS and DVCS

Looking into the proton . . .

ß Study of the proton content via (deeply) inelastic scattering (DIS):

γ⋆, µ

γ⋆, νq2

proton proton

x = x′

usual parton distributions

Wµν = (−gµν +qµqνq2

)F1(x , q2)

+PµPνP.q

F2(x , q2)

P = Pµ − P.q

q2 qµ

ß Factorisation in the Bjorken limit: Q2 →∞, x fixed

ß Probability Distribution, since being an amplitude squared

Sum over spect.

ß Probability to find a parton with a momentum fraction x : q(x)F2(x , q2) = x

e2q q(x , q2)

γ⋆, µ

γ⋆, νq2

proton proton

x = x′

Factorisation

)F1(x , q2)

+PµPνP.q

F2(x , q2)

P = Pµ − P.q

q2 qµ

Sum over spect.

e2q q(x , q2)

γ⋆, µ

γ⋆, νq2

proton proton

x = x′

Factorisation

)F1(x , q2)

+PµPνP.q

F2(x , q2)

P = Pµ − P.q

q2 qµ

Sum over spect.

e2q q(x , q2)

γ⋆, µ

γ⋆, νq2

proton proton

x = x′

Factorisation

)F1(x , q2)

+PµPνP.q

F2(x , q2)

P = Pµ − P.q

q2 qµ

Sum over spect.

e2q q(x , q2)

Interferences in the proton. . .

ß Study of interferences in the protonvia Deeply Virtual Compton Scattering (DVCS):

x x′

hadron hadron

Non-pert. object

x 6= x′

GPDGPD

For Q2 t, described in termsof 4 generalised parton distri-bution: GPDs

idem for meson electroproduction

ß Factorisation in the generalised Bjorken limit: Q2 →∞, t, x fixedß The GPDs are not probability distributions

x x′

x 6= x′

p p′

p′p×

but are universal !ß Interpretration only at the amplitude level

Amplitude of probabilityfor a proton to emit a quark with x & to absorb another with x ′

ß Study of interferences in the protonvia Deeply Virtual Compton Scattering (DVCS):

x x′

proton proton

Non-pert. object

x 6= x′

GPDGPD

W 2Factorisation

idem for meson electroproductionß Factorisation in the generalised Bjorken limit: Q2 →∞, t, x fixedß The GPDs are not probability distributions

x x′

x 6= x′

p p′

p′p×

ß Study of interferences in the proton

via Deeply Virtual Compton Scattering (DVCS):

x x′

proton protont

Non-pert. object

x 6= x′

ρ, π, . . .

W 2factorisation

x x′

x 6= x′

p p′

p′p×

x x′

proton protont

Non-pert. object

x 6= x′

ρ, π, . . .

W 2factorisation

ß Factorisation in the generalised Bjorken limit: Q2 →∞, t, x fixed

ß The GPDs are not probability distributions

x x′

x 6= x′

p p′

p′p×

x x′

proton protont

Non-pert. object

x 6= x′

ρ, π, . . .

W 2factorisation

x x′

x 6= x′

p p′

p′p×

but are universal !

ß Interpretration only at the amplitude levelAmplitude of probability

for a proton to emit a quark with x & to absorb another with x ′

x x′

proton protont

Non-pert. object

x 6= x′

ρ, π, . . .

W 2factorisation

x x′

x 6= x′

p p′

p′p×

Success of the factorised framework . . .

Angular dependence of measured asymmetries comingfrom Bethe-Heitler/ DVCS interferences

JLab data at Q2 = 2.3 GeV2, t = −0.28 and = −0.23 GeV2

Success of the factorised framework . . .

Angular dependence of measured asymmetries comingfrom Bethe-Heitler/ DVCS interferences

JLab data at Q2 = 2.3 GeV2, t = −0.28 and = −0.23 GeV2

Part II

Looking in the backward region

Definition of the Transition Distribution Amplitudes

Hard limit for backward exclusive processes

ß Let us analyse Hard Electroproduction of a meson

but backward !l

meson nearly atrest in thetarget rest frameproton

x x′

proton

GPDproton

t → u

x2 x3x1

proton

ß The kinematics imposes the exchange of 3 quarks in the u channel

ß Factorisation in the generalised Bjorken limit: Q2 →∞, u, x fixedB. Pire, L. Szymanowski, PLB 622:83,2005.

ß The object factorised from the hard part is a Transition DistributionAmplitude (TDA)

=p p′

ß Interpretation at the amplitude level in the ERBL region (for xi > 0)

Amplitude of probability to find a meson within the proton !

but backward !l

x x′

proton

GPDproton

t → u

x2 x3x1

proton

=p p′

but backward !l

x x′

proton

GPDproton

t → u

x2 x3x1

proton

=p p′

but backward !l

x x′

proton

GPDproton

t → u

x2 x3x1

proton

=p p′

but backward !l

x x′

proton

GPDproton

t → u

x2 x3x1

proton

=p p′

Amplitude of probability to find a meson within the proton !J.P. Lansberg (CPHT – Ecole polytechnique) Transition Distribution Amplitudes March 2, 2010 8 / 31

p → π: parametrisation

ë p → π (at Leading twist)ß ∆T = 0: 3 TDAs (3× p(↑)→ uud(↑↑↓) + π)

TDA DA (Chernyak-Zhitnitsky)

4〈π0| εijkuiα(z1n)uj

β(z2n)dkγ(z3n) |p, sp〉 ∝h

1 (xi , ξ,∆2)(p/ C)αβ(N+

sp )γ+

1 (xi , ξ,∆2)(p/ γ5C)αβ(γ5N+

sp )γ+

1 (xi , ξ,∆2)(σρpC)αβ(γρN+

sp )γi

4〈0|εijkuiα(z1n)uj

β(z2n)dkγ(z3n)|p〉 ∝h

V (xi )(p/C)αβ(γ5N+sp )γ+

A(xi )(p/γ5C)αβ(N+sp )γ+

T (xi )(iσρp C)αβ(γργ5N+sp )γ

B. Pasquini et al., PRD 80:014017,2009.V π0

1 → D↑↑↓,↑ + D↑↓↑,↑Aπ

1 → D↑↑↓,↑ − D↑↓↑,↑T π0

1 → D↑↑↑,↓

ë p → π (at Leading twist)ß ∆T = 0: 3 TDAs (3× p(↑)→ uud(↑↑↓) + π)

TDA DA (Chernyak-Zhitnitsky)

4〈π0| εijkuiα(z1n)uj

β(z2n)dkγ(z3n) |p, sp〉 ∝h

1 (xi , ξ,∆2)(p/ C)αβ(N+

sp )γ+

1 (xi , ξ,∆2)(p/ γ5C)αβ(γ5N+

sp )γ+

1 (xi , ξ,∆2)(σρpC)αβ(γρN+

sp )γi

4〈0|εijkuiα(z1n)uj

β(z2n)dkγ(z3n)|p〉 ∝h

V (xi )(p/C)αβ(γ5N+sp )γ+

A(xi )(p/γ5C)αβ(N+sp )γ+

T (xi )(iσρp C)αβ(γργ5N+sp )γ

iB. Pasquini et al., PRD 80:014017,2009.V π0

1 → D↑↑↓,↑ + D↑↓↑,↑Aπ

1 → D↑↑↓,↑ − D↑↓↑,↑T π0

1 → D↑↑↑,↓

ß ∆T 6= 0: 8 TDAs ( 12 × 2× (2× 2× 2)× 1)

4〈π0(pπ)| εijkuiα(z1n)uj

β(z2n)dkγ(z3n) |p(p1, s)〉 =

fπ×h

1 (xi , ξ,∆2)(p/ C)αβ(N+)γ + Vπ0

2 (xi , ξ,∆2)(p/ C)αβ(∆/T N+)γ

1 (xi , ξ,∆2)(p/ γ5C)αβ(γ5N+)γ + Aπ

2 (xi , ξ,∆2)(p/ γ5C)αβ(γ5∆/T N+)γ

1 (xi , ξ,∆2)(σpµC)αβ(γµN+)γ + Tπ0

2 (xi , ξ,∆2)(σp∆T

C)αβ(N+)γ

3 (xi , ξ,∆2)(σpµC)αβ(σµ∆T N+)γ + Tπ0

4 (xi , ξ,∆2)(σp∆T

C)αβ(∆/T N+)γi

B. Pasquini et al., PRD 80:014017,2009.

2 → D↑↓↑,↓ + D↑↑↓,↓ Aπ0

2 → D↑↓↑,↓ − D↑↑↓,↓T π0

2 → D↑↑↑,↑ + D↑↓↓,↑ T π0

3 → D↑↑↑,↑ − D↑↓↓,↑T π0

4 → D↑↓↓,↓

ß ∆T 6= 0: 8 TDAs ( 12 × 2× (2× 2× 2)× 1)

4〈π0(pπ)| εijkuiα(z1n)uj

β(z2n)dkγ(z3n) |p(p1, s)〉 =

fπ×h

1 (xi , ξ,∆2)(p/ C)αβ(N+)γ + Vπ0

2 (xi , ξ,∆2)(p/ C)αβ(∆/T N+)γ

1 (xi , ξ,∆2)(p/ γ5C)αβ(γ5N+)γ + Aπ

2 (xi , ξ,∆2)(p/ γ5C)αβ(γ5∆/T N+)γ

1 (xi , ξ,∆2)(σpµC)αβ(γµN+)γ + Tπ0

2 (xi , ξ,∆2)(σp∆T

C)αβ(N+)γ

3 (xi , ξ,∆2)(σpµC)αβ(σµ∆T N+)γ + Tπ0

4 (xi , ξ,∆2)(σp∆T

C)αβ(∆/T N+)γi

B. Pasquini et al., PRD 80:014017,2009.

2 → D↑↓↑,↓ + D↑↑↓,↓ Aπ0

2 → D↑↓↑,↓ − D↑↑↓,↓T π0

2 → D↑↑↑,↑ + D↑↓↓,↑ T π0

3 → D↑↑↑,↑ − D↑↓↓,↑T π0

4 → D↑↓↓,↓J.P. Lansberg (CPHT – Ecole polytechnique) Transition Distribution Amplitudes March 2, 2010 10 / 31

Backward electroproduction of a pion

More quantitatively: the pionic content of the proton

JPL, B. Pire, L. Szymanowski,PRD 75:074004,2007

ß Let us analyse the soft pion limit

〈πa(k)|O|p(p, s)〉 =− i

fπ〈0|[Qa

5 ,O]|p(p, s)〉

4fπp · kXs′

〈0|O|p(p, s ′)〉u(p, s ′)k/γ5τau(p, s)

ß Direct relation between the TDAs, 〈πa(k)|O|p(p, s)〉, andthe proton wavefunction (DAs), 〈0|O|p(p, s)〉

proton

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξV` x1

2ξ,x2

2ξ,x3

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξA` x1

2ξ,x2

2ξ,x3

´Tπ0

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 3

4ξT` x1

2ξ,x2

2ξ,x3

´ß Similar relations obtained for the proton-pion DAs 〈0|O|π(k)p(p, s)〉

V.M Braun et al. PRD75:014021,2007

ß Note that Eπ = mπ in the proton r.f. ⇔ ξ = M−mπM+mπ

' 0.74 6= 1

〈πa(k)|O|p(p, s)〉 =− i

fπ〈0|[Qa

5 ,O]|p(p, s)〉

4fπp · kXs′

〈0|O|p(p, s ′)〉u(p, s ′)k/γ5τau(p, s)

proton

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξV` x1

2ξ,x2

2ξ,x3

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξA` x1

2ξ,x2

2ξ,x3

´Tπ0

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 3

4ξT` x1

2ξ,x2

2ξ,x3

' 0.74 6= 1

〈πa(k)|O|p(p, s)〉 =− i

fπ〈0|[Qa

5 ,O]|p(p, s)〉

4fπp · kXs′

〈0|O|p(p, s ′)〉u(p, s ′)k/γ5τau(p, s)

proton

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξV` x1

2ξ,x2

2ξ,x3

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 1

4ξA` x1

2ξ,x2

2ξ,x3

´Tπ0

1 (x1, x2, x3, ξ,∆2)

Eπ→mπ→ 3

4ξT` x1

2ξ,x2

2ξ,x3

' 0.74 6= 1

Beyond the soft limit

ß The factorised framework goes beyond the soft limit

ß One needs input from models, such asthe pion cloud model, ... B. Pasquini, et al.PRD 80:014017,2009.

1.21.6

0.81.2

1.21.6

x1 x20

0.81.2

0.40.8

0.81.2

1.21.6

0.81.2

1.21.6

x1 x20

1.21.6

0.40.8

1.21.6

ß The factorised framework goes beyond the soft limitß One needs input from models, such asthe pion cloud model, ... B. Pasquini, et al.PRD 80:014017,2009.

1.21.6

0.81.2

1.21.6

x1 x20

0.81.2

0.40.8

0.81.2

1.21.6

0.81.2

1.21.6

x1 x20

1.21.6

0.40.8

1.21.6

ß The factorised framework goes beyond the soft limitß One needs input from models, such asthe pion cloud model, ... B. Pasquini, et al.PRD 80:014017,2009.

1.21.6

0.81.2

1.21.6

x1 x20

0.81.2

0.40.8

0.81.2

1.21.6

0.81.2

1.21.6

x1 x20

1.21.6

0.40.8

1.21.6

Backward Electroproduction of a pion: II

ß First evaluation: backward electroproduction of a pion for Eπ → mπ

using the soft limit for the TDAs

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

π(pπ)

ß The amplitude at the Leading-twist accuracy:

Mλs1s2

= −i(4παs)2

√4παemf 2

54fπQ4u(p2, s2)ε/(λ)γ5u(p1, s1)

×1+ξ∫−1+ξ

7∑α=1

Tα +14∑α=8

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

π(pπ)

Mλs1s2

= −i(4παs)2

√4παemf 2

54fπQ4u(p2, s2)ε/(λ)γ5u(p1, s1)

×1+ξ∫−1+ξ

7∑α=1

Tα +14∑α=8

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

π(pπ)

Mλs1s2

= −i(4παs)2

√4παemf 2

54fπQ4u(p2, s2)ε/(λ)γ5u(p1, s1)

×1+ξ∫−1+ξ

7∑α=1

Tα +14∑α=8

Hard part: Mh for γ?p → pπ0 at ∆T = 0

JPL, B. Pire, L. Szymanowski,PRD 75:074004,2007.

Backward Electroproduction of a pion: III

At ξ = 0.8 and using CZ Distribution Amplitudes, one gets:

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

π(pπ)

0 2 4 6 8 10

dσ /d

Ω∗ π|

θ∗ π=π

Q2 (GeV2)

NOTE: the result with asymptotic DAs is not zero !

Backward Electroproduction of a pion: III

At ξ = 0.8 and using CZ Distribution Amplitudes, one gets:

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

π(pπ)

0 2 4 6 8 10

dσ /d

Ω∗ π|

θ∗ π=π

Q2 (GeV2)

NOTE: the result with asymptotic DAs is not zero !

Backward Electroproduction of a pion: IV

á Model-independent predictions

ß Scaling law for the amplitude:

M(Q2) ∝ α2s (Q2)

ß Approximate Q2-independence of the ratios

M(γ?p → pπ)

M(γ?p → pγ),M(γ?p → pγ)

M(γ?p → p)and

dσ(pp→`+`−π0)dQ2

dσ(pp→`+`−)dQ2

(see later)

ß Dominance of γ?Tp → pπ, . . .

ß Spinorial structures at ∆T 6= 0:u(p2, s2)ε/(λ)γ5u(p1, s1) and εµ∆T ,ν u(p2, s2)(σµν + gµν)γ5u(p1, s1)

At ∆T 6= 0, σTT 6= 0 and cos 2ϕ dependence

á Model-independent predictionsß Scaling law for the amplitude:

M(Q2) ∝ α2s (Q2)

M(γ?p → pπ)

M(γ?p → p)and

dσ(pp→`+`−)dQ2

(see later)

M(Q2) ∝ α2s (Q2)

M(γ?p → pπ)

M(γ?p → p)and

dσ(pp→`+`−)dQ2

(see later)

M(Q2) ∝ α2s (Q2)

M(γ?p → pπ)

M(γ?p → p)and

dσ(pp→`+`−)dQ2

(see later)

M(Q2) ∝ α2s (Q2)

M(γ?p → pπ)

M(γ?p → p)and

dσ(pp→`+`−)dQ2

(see later)

Backward Electroproduction of a meson: data

ß Data from JLab exist for the π Analysis approved, out soon (K. Park)

ß “Visible signal in the yield of ω at 180” (G. Huber, Sept. 09)

ß Data for the electroduction of η (V. Kubarovsky, P. Stoler)

To be modelled

ß We are working on the theory(∆T 6= 0, DGLAP-region contribution, other models, ...)

Lattice calculations

Gavela, King, Sachrajda, Martinelli,...a lattice computation of proton decay amplitudes

Nucl.Phys.B312:269,1989

á Calculation of the matrix elements for the GUT decays

p → π0e+ p → π+ν p → K 0 + lepton

á Evaluation of the two moments matrix elements

εijk〈π0|(uiCd j)ukγ |P〉 = A1Nγ εijk〈π0|(uiCγ5d

j)(γ5uk)γ |P〉 = A2Nγ

á Update of this study would be very usefulß would fix the normalisation of the TDAs via Sum Rulesß would give information on their t-dependence

Part III

Extensions

TDA studies at GSI/FAIR

TDAs in exclusive processes at GSI/FAIR

JPL, B. Pire, L. Szymanowski PRD76 :111502(R),2007

ß pp → γ?π0 can be studied by PANDAß Involves the same TDAs as for backward electroproduction

In the GPD case, after crossing, we have to deal with GDAs

p(pp) π(pπ)

ℓ1DA

γ⋆(q) ℓ−

ß GSI-FAIR: Ep ≤ 15 GeV ⇒W 2 ≤ 30 GeV2

6 12 18 24 30

dσ /d

t| ∆T=

W2 (GeV2) (a)

|pzπ|=0

|pzπ|=155 MeV

5 10 15 20

dσ /d

2 | ∆T=

Q2 (GeV2) (b)

W2=5 GeV2

W2=10 GeV2

W2=20 GeV2

ß σ`+`−π0

(7 < Q2 < 8GeV2,W 2 = 10GeV2,∆T < 0.5GeV) ∼ 100fb.ß Expected

∫dtL of about 2 fb−1 for a 100-day experiment

ß Other channels are also of much interest, such aspp → `+`−η or pp → `+`−ρ0

6 12 18 24 30

dσ /d

t| ∆T=

W2 (GeV2) (a)

|pzπ|=0

|pzπ|=155 MeV

5 10 15 20

dσ /d

2 | ∆T=

Q2 (GeV2) (b)

W2=5 GeV2

W2=10 GeV2

W2=20 GeV2

ß σ`+`−π0

6 12 18 24 30

dσ /d

t| ∆T=

W2 (GeV2) (a)

|pzπ|=0

|pzπ|=155 MeV

5 10 15 20

dσ /d

2 | ∆T=

Q2 (GeV2) (b)

W2=5 GeV2

W2=10 GeV2

W2=20 GeV2

ß σ`+`−π0

Future application to charmonium production

ë J/ψ decay in proton antiprotonwell accounted by the perturbative mechanism

ë pp → J/ψπ0 at small tcan be described likewise

ß this process is used to search for new charmonium states (hc ,. . . )ß will be extensively studied at GSIß For now, comparisons are possible with previous calculations

Soft pion limit M.K. Gaillard, et al., PLB 110:489,1982.

T.Barnes, X.Li, PRD 75:054018,2007

TDA and Intrinsic Charm

Heavy quarks in the proton . . .

ß Protons can contain nonperturbative fluctuations of heavy quarks QQfor instance the so called Intrinsic Charm

S.J. Brodsky et al. PLB93:451-455,1980

proton

ß Key point: large-x heavy-quark content at low Q2

does not come from gluon splitting from DGLAP evolution

ß Recent global PDF analyis: one can accomodate more IC than expected〈xcc〉 ' 3% is possible

µ = 1.3→ 100 GeV J.Pumplin et al. PRD75:054029,2007.

ß bb et tt also possible but suppressed as M−2Q

ß Could be uncovered in diffractive Higgs productione.g. S.J Brodsky et al. , PRD73:113005,2006

proton

ß The real impact of IC is controversial: difficult to find an undisputable probe

ß Dedicated test: γ?p → p J/Ψ in the TDA regionS.J. Brodsky, JPL, work in progress

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

〈J/ψ|ǫabcua(x1)ub(x2)d

c(x3)|Proton〉

ß one needs sufficient W 2 to be away from the thresholdÞ at threshold (M + mψ), the proton and the J/ψ have no relative momentum

ß For W M + mψ,Þ t → 0: usual diffractive production: target stays nearly at rest, J/ψ fastÞ u → 0: “backward” region: J/ψ nearly at rest, proton fast

ß JLab 12(11) GeV →Wmax =√

(0.942 + 2 · 11 · 0.94) ' 4.64 GeV:not quite enough

ß Only COMPASS could do itÞ 1 < Q2 < 7 GeV2

Þ 0.03 < xB < 0.2: for large Q2, W ∈ [5 : 15] GeV

ß The real impact of IC is controversial: difficult to find an undisputable probeß Dedicated test: γ?p → p J/Ψ in the TDA region

S.J. Brodsky, JPL, work in progress

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

ß For W M + mψ,Þ t → 0: usual diffractive production: target stays nearly at rest, J/ψ fast

Þ u → 0: “backward” region: J/ψ nearly at rest, proton fastß JLab 12(11) GeV →Wmax =

√(0.942 + 2 · 11 · 0.94) ' 4.64 GeV:

not quite enoughß Only COMPASS could do it

Þ 1 < Q2 < 7 GeV2

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

DAℓ1

P (p1)

P ′(p2)γ⋆(q)

J/ψ(pψ)

Proton J/ψ

c(x3)|Proton〉

Heavy quarks in the proton . . .S.J. Brodsky, JPL, work in progress

ß Modelling the proton to charmonium TDA (pseudoscalar case)

Þ “SU(4)” spin-flavour symmetry:

V p→Q = T p→Q Ap→Q = 0

Þ Non-relativistic approx for QQ: Charmonium DA ∝ fQδ(xc − xc)(xc + xc = xQ)

Þ Light cone inspired form for the 5 particle IC Fock State

ψ(x1, x2, x3, xc ,xc ,Q2) = δ(1−

(m2p − m2

c( 1xc

)− m2q( 1

(Effective masses (from the kT integration) m2c ' 1.8 GeV, m2

q ' 0.45 GeV)

Þ Only in ERBL region (xi > 0)

Þ Stay tuned !

ψ(x1, x2, x3, xc ,xc ,Q2) = δ(1−

(m2p − m2

c( 1xc

)− m2q( 1

q ' 0.45 GeV)

Þ Stay tuned !

ψ(x1, x2, x3, xc ,xc ,Q2) = δ(1−

(m2p − m2

c( 1xc

)− m2q( 1

q ' 0.45 GeV)

Þ Stay tuned !

ψ(x1, x2, x3, xc ,xc ,Q2) = δ(1−

(m2p − m2

c( 1xc

)− m2q( 1

q ' 0.45 GeV)

Þ Stay tuned !

ψ(x1, x2, x3, xc ,xc ,Q2) = δ(1−

(m2p − m2

c( 1xc

)− m2q( 1

q ' 0.45 GeV)

Þ Stay tuned !J.P. Lansberg (CPHT – Ecole polytechnique) Transition Distribution Amplitudes March 2, 2010 25 / 31

Part IV

DVCS on virtual pions

A few words on DVCS on virtual-pion target

neutron

e e′

proton

TDA + GPD

Þ e(`) + p(p)→ e(`′) + γ(q′) + π+(p′π) + n(p′) q = `− `′

pπ = p − p′ xB = Q2

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

p·`(xπ =fraction of energy of the virtual pion from the proton in the ep c.m)

Þ For the πγ subprocess,we further define:

tπ = (pπ − p′π)2 sπ = (pπ + q)2 xπB = Q2

2pπ·q

Þ GPD regime: |tπ| Q2

Þ One pion-exchange approximation: t small.

This imposes xπ not too large (−t ≥ x2πm2

1−xπ)

Þ On the other hand, xπmin ≈ 1ymax

sπmin+Q2min

s ; s cannot be too small

Þ Neglecting the dependence of the eπ cross section on the pion virtuality t

d4σ(ep → eγπn)

dy dQ2 dtπ dφπ≈Z

dxπ Π(xπ, |t|max)d4σ(eπ → eγπ)

dyπ dQ2 dtπ dφπ

(for |t|max = 0.3 GeV2 (solid) and |t|max = 0.5 GeV2 (dashed))

neutron

e e′

proton

TDA + GPD

Þ e(`) + p(p)→ e(`′) + γ(q′) + π+(p′π) + n(p′) q = `− `′pπ = p − p′ xB = Q2

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

neutron

e e′

proton

TDA + GPD

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

neutron

e e′

proton

TDA + GPD

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

neutron

e e′

proton

TDA + GPD

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

neutron

e e′

proton

TDA + GPD

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

neutron

e e′

proton

TDA + GPD

2p·q y = p·qp·` t = (p − p′)2 xπ = pπ·`

2pπ·q

1−xπ)

sπmin+Q2min

d4σ(ep → eγπn)

dyπ dQ2 dtπ dφπ

D. Amrath, M. Diehl, JPL, EPJC58:179-192,2008.

Þ Finally, we want to avoid the effect of πn resonances: M2nπ (mN + mπ)2

Þ M2nπ ' x−1

π [...] + m2N + m2

π: we prefer low xπ, and tπ away for its min value.

Þ Not so easy

Þ In brief, all those conditions imposes to wait for JLab-12 GeV

Þ M2nπ ' x−1

π [...] + m2N + m2

Þ Not so easy

Þ M2nπ ' x−1

π [...] + m2N + m2

Þ Not so easy

Þ M2nπ ' x−1

π [...] + m2N + m2

Þ Not so easy

A few words on DVCS on virtual-pion targetD. Amrath, M. Diehl, JPL, EPJC58:179-192,2008.

Defining the weighted differences

ScosφπC

Zdφπ cosφπ

»dσ(e` = +1)

dφπ−

dσ(e` = −1)

sinφπL

Zdφπ sinφπ

»dσ(P` = +1)

dφπ−

dσ(P` = −1)

of cross sections for different beam charge or beam polarization.

Q2min sπmin |t|max |tπ|max ymax M2

πn min σBH σVCS σINT ScosφπC

SsinφπL

2 4 0.3 0.7 0.85 — 18.4 0.88 −0.18 0.39 7.57

2 4 0.3 0.7 0.8 — 5.12 0.29 −0.09 0.17 2.17

2 4 0.3 0.7 0.9 — 45.6 1.86 −0.27 0.64 17.9

2 4 0.2 0.7 0.85 — 0.41 0.016 −0.002 0.004 0.16

2 4 0.5 0.7 0.85 — 105 6.52 −2.32 5.00 46.2

2.5 4 0.3 0.7 0.85 — 2.55 0.103 −0.010 0.018 0.96

2 5 0.3 0.7 0.85 — 0.30 0.013 −0.003 0.008 0.12

2 4 0.3 0.5 0.85 — 16.2 0.69 −0.09 0.18 6.30

2 4 0.3 0.7 0.85 1.5 13.4 0.67 −0.19 0.42 5.72

2 4 0.3 0.7 0.85 1.8 5.08 0.31 −0.14 0.30 2.46

(Limiting values of Q2, sπ , t, tπ , and M2πn are given in units of GeV2, and cross sections in units of fb.)

Part V

Outlooks

Perspectives

Perspectives (for the TDAs)

Þ Further quantitative predictions require modelsß Soft pion limit: OKß 4-ple distribution (spectral representation: double distr. for GPD):

to be doneß etc.

Þ Experimental data are necessary to test the pictureand then to extract physics

Þ ...expected from

ß JLab-6 GeV: Backward electroproduction of π, ηß GSI: pp → γ?π0, pp → J/ψπ0, pp → γ?γ, . . .ß JLab-12 GeV: e.g. DVCS on pionß B-factories (γ?γ → MM) possible: TDA γ → Mß COMPASS: γ?p → pJ/ψ ... EIC ?

Perspectives

to be doneß etc.

Þ ...expected from

Perspectives

to be doneß etc.

Þ ...expected from

Looking at hadrons in the backward direction with hard ...Looking at hadrons in the backward...

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