Electromagnetic Pion Form Factor in a deformed background...

Electromagnetic Pion Form Factor in a deformedbackground

Based on: ArXiV 2104.04640

Miguel Angel Martın Contreraswith E. F. Capossoli, D. Li, A. Vega and H. Boschi-Filho

Instituto de Fısica y Astronomıa, Universidad de Valparaıso, Chile

19th International Conference of Hadron Spectroscopy and Structure,July 29th

M. A. Martın Contreras (IFA–UV) Electromagnetic Pion FF HADRON-2021 1 / 16

https://inspirehep.net/literature/1857795

Outline

1 Motivation

2 Deformed Background

3 Holographic πFF

4 Numerical Results I: ∆ = 3 issue

5 Numerical Results II: ∆ = 3 issue solution

6 Conclusions


Motivation

From holographic Grounds

When we consider ∆ as the scalingdimension of the operators creatinghadrons, can we calculated aconsistent pion form factor?

Figure: Taken from Efremov 2009

From phenomenological grounds

The behavior of the πFF at moderate q2 (5-20 GeV2) is expected to exhibit aSudakov suppression up to the asymptotic region where counting rules apply.See Efremov 2009.

Lattice-QCD analysis for the transitions from soft to hard πFF shows asuppressed behavior at intermediate q2 regions. See Choi 2006 and Gao etal. 2021.


https://doi.org/10.1142/S0217732309001029

https://doi.org/10.1142/S0217732309001029

https://doi.org/10.1103/PhysRevD.74.093010



Pion form factor

CERN

JLab 2006

JLab 2007

CEA 1978

Cornell A

Cornell B

Bebek 1978

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q2(GeV2)

q2FExp

π(q2)

Pion Form Factor with Δ=3

It is one of the most valuable QCD quantities related to the transition fromthe non-perturbative to the perturbative regime.

It is defined by the matrix element

〈π±(p2) |Jµπ (0)|π±(p1)〉 = gπ± (p1 + p2)µ Fπ(q2).


Deformed background AdS/QCD model

In a general bottom-up model

Confinement is realized via the Schrodinger-like holographic potential

V (z) =1

4B ′(z)2 − 1

2B ′′(z) + M2

5 R2 e2 A(z) for integer spin hadrons

where B(z) = Φ(z) + β A(z).

Φ(z) is the KKSS dilaton (static or dynamic generated).

h(z) is a metric deformation.

Both proposals induce confinement. However, the latter works better withfermions than the former. See Eduardo’s talk about proton DIS.

General Idea:

Induce confinement geometrically with a quadratic deformation and Φ(z) = 0(See Capossolli et al. 2020):

dS2 =R2

z2e

kH z2

2

[dz2 + ηµ ν dx

µ dxν] where kH sets confinement for a

given hadron.



https://indico.nucleares.unam.mx/event/1541/session/14/contribution/12


Pion in the context of Deformed Background Model

Pion dual to a bulk massive scalar field living on its background defined by kπ:

Sπ =

∫d5x√−gπ [gmn

π ∂mX∂nX + M2

5X2]

M25 R2 = (∆− S)(∆ + S − 4)

∆ = 3→ M25 R2 = −3

β = −3 and kπ = −0.04252

GeV2.Vπ(z) =

15

4 z2− 3

z2e−

kπ z2

2 +k2π z

2

4+kπ

1S

2S

3S

0 20 40 60 80 100

-0.1

0.0

0.1

0.2

z

ψn(z)

Holographic Pion Eigenfunctions

Vπ (z)

0 20 40 60 80 100

0.02

0.04

0.06

0.08

0.10

0.12

z

V(z)

Holographic Potential for Pions

Our pion will be defined by the ground state of Vπ(z), i.e., ψ0(z).


Virtual photon in the background model

Since the photon γ is virtual, the associated deformation slope kγ does not setconfinement. This kγ is associated with the kinematics of the elastic process.

Photon dual to a massless bulk vector field:

Sγ = − 1

c2γ

∫d5x

√−gγ

1

4FmnFmn ,

Photon Bulk to boundary propagator dual to the EM current in the pion vertex atthe boundary:

φµ(z , q) = −ηµeiq·y

2kγ z

2 Γ

[1− q2

2kγ

]U(

1− q2

2kγ; 2; −kγ z

2

2

)≡ −ηµe

iq·y

2B(z , q) ,


Interaction Picture: Pion Form Factor

General Action

S = Sπ + Sγ + SInt

where

SInt = geff

∫d5x√−gπ gmn

π φm[Xp1 ∂m X ∗p2

− X ∗p2∂m Xp1

],

= i (2π)4 δ4(q − (p2 − p1)) ηµνηµ (p1 + p2)ν geff

∫dz e3 Aπ(z) ψ2

0 B,

We define the pion form factor as

Fπ(q2) =

∫dz e3 Aπ(z) v(z)B(z , q2) v(z).

We evaluate πFF in the pion geometry!!


Results πFF try one: Comparison with Exp. Data

kγ=-3.0 GeV2

kγ=-3.8 GeV2

kγ=-4.6 GeV2

CERN

JLab 2006

JLab 2007

CEA 1978

Cornell A

Cornell B

Bebek 1978

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q2(GeV2)

q2FExp

π(q2)

Pion Form Factor with Δ=3

In agreement with low q2 limit.

Not good for intermediate q2

Highly Sudakov suppressed.

Not in agreement withBrodsky-Lepage counting rule.


Results πFF try one: Comparison with Exp. Data

No Brodsky-Lepage counting rule (C.R.)

Let us take the large q2 limit for the πFF:

Fπ(q2)∣∣q2→∞ =

1

8 ∆2 (∆− 1)

(1

q2

)∆−1 [1 + γe (∆− 1) ∆−∆2 (log 4− 3) + . . .

]LF πFF (Brodsky 2008)

Fπ(q2) =4κ2

4κ2 + q2→(

1

q2

)∆−1∣∣∣∣∣q→∞

∆ = 2 → constituent number.

FF independent of hadronic spin.

Other bottom-up models(non-LF)

∆ = 3 → scaling dimension.

Scalar mesons does not have theright C.R.(Ops!)!

Only in the vector meson case,the C.R. is fulfilled.


https://doi.org/10.1103/PhysRevD.77.056007

Our Solution: kγ → kγ(q)

AnsatzSince the geometric slope kγ is related with the kinematic of the scatteringprocess, we propose

kγ(q) = q κγ

with κγ defined as a constant given in energy units.

This choice guarantees that the Brodsky-Lepage C.R. be satisfied.


Results πFF try two: Comparison with Exp. Data

CERN

JLab 2006

JLab 2007

CEA 1978

Cornell A

Cornell B

Bebek 1978

kγ=-2.8 GeV

kγ=-3.3 GeV

kγ=-3.7 GeV

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q2(GeV2)

q2FExp

π(q2)

Pion Form Factor with Δ=3 and kγ→kγ(q)=q kγ


Results πFF try two: Comparison with other models

BSE

Disp. Rel.

Sume Rules

LFQCD 1

LFQCD 2

pQCD

Ours

0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

q2(GeV2)

q2FπTh(q2)

Non-Holographic Theoretical Pion Form Factor

Ours

Extrapolated Sakai-Sugimoto

Sakai-Sugimoto

SWM Δ=2

HWM Δ=2

0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

q2(GeV2)

q2FπTh(q2)

Holographic Theoretical Pion Form Factor


Results πFF try two: low q2, the pion charge radius

Pion charge radius is extracted from the low q2 behavior of the πFF as:

〈r2π〉 = −6

dFπ(q2)

dq2

∣∣∣∣q2=0

.

In our case, with κγ = −3.3 GeV,we obtain

Exp. PDG (fm) Theo. (fm) R. E.(%)

0.659± 0.004 0.671 2.0


Conclusions

We have calculated the πFF in the AdS deformed background model, with∆ = 3.

The calculated pion form factor exhibits a Sudakov-like suppression for theintermediate q2 region.

In order to have the proper C.R. behavior, we impose that kγ = kγ(q), sinceit is not related to confinement.

The pion charge radius in our proposal has 2.0% of R.E in comparison withexperimental data (PDG).

This holographic procedure is an alternative to the L.F. holographycalculation, where ∆ is interpreted in terms of the constituent number.


Thank you!


Electromagnetic Pion Form Factor in a deformed background...

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