Model Independence, Dispersion Approach and Nucleon Form-Factors

(1)

Model Independence, Dispersion Approach and Nucleon Form-Factors

Krzysztof M. Graczyk

Instytut Fizyki Teoretycznej Uniwersytet Wrocławski

Polska

Seminar of Neutrino Physics Division

3d November 2014, Wrocław

(2)

References

Model Independent Extraction of the Proton Radius:

Electric: R. J. Hill and G. Paz, PRD82, 113005 (2010)

Magnetic: Z. Epstein, G. Paz, J. Roy, Phys.Rev. D90 (2014) 074027 Axial mass problem:

R J. Hill and G. Paz, PRD84, 073006 (2011)

Model Independence in the Statistical Sense

K. M. Graczyk, C. Juszczak, arXiv:1408.0150

(3)

References

Model Independent Extraction of the Axial Form Factor, Axial Radius?

Recent M

A

measurements ∼ 1.3 GeV old 1.0 GeV

(4)

References

Mep

Born = −ie2 q2u(k¯ 0

)γµu(k)hµ ep

iMνn

Born ≈ i g2 cos θC 8(q2 − M 2

W)u(k0

)γµ(1 − γ5)u(k) hµ cc

hµ

ep/νn = u(p0 )Γµ

p,W(q)u(p) Q2= −q2= −t

Γµ

N = γµFN

1 (Q2)+iσµν qν 2Mp FN

2 (Q2), N = proton, neutron

F1(Q2) = 1

1 + τ GE (Q2)+ τGM (Q2)

F2(Q2) = 1

1 + τ GM (Q2)−GE (Q2)

τ = Q2 /4M 2electric,magnetic

After CVC and PCAC

Γµ W =Γµ

p (q)−Γµ n (q)

| {z }

vector

−γµγ5FA(q)−qµ γ5 2M FP (q)

| {z }

axial

,

FA(q) = 1.267 (1 − q2 /M 2

A)2

FP (q) = 4M 2 FA(q) m2π − q2

Axial mass MA problem: ν-deuteron scattering data (∼ 1 GeV), ν-Carbon ∼ 1.3 GeV ? Need of more sophisticated nuclear models?

(5)

References

Model Independent Extraction of the Proton Radius?

ρ

E,M

(r) = 1 (2π)

³

Z

d

³

qe

^−iq·r

G

E,M

(q)

r

_E,M²

= − 6 dG

E,M

(q) d|q|

²

|q|²=0

G

E,M

(Q

²

) = G

E,M

(0)− 1 6

r

_E,M²

Q

²

+...

0.85 0.90 0.95

r_p (fm) Pohl et al. (µ-atom) CODATA (H,D atom and ep data)

MAMI (ep) Sick (ep, with tpe) Bayesian analysis, with tpe

(6)

References

(7)

References

S-matrix theory → Hadron Physics

Lorentz invariance of the theory and other symmetry principles unitarity of the S-matrix

anlyticity

scattering amplitudes, when expressed as functions of certain kinematic variables, can be analytically continued into the complex domain and resulting analytic functions, at least near the physical regions, have the simplest singularity structure which is consistent with the other general principles of the theory

crossing

(8)

References

Analytic Function

Single-valued function of z is said to be analytic at point z

0

if it has a derivative at z

0

and at all points in some neighbourhood z

0

.

If function is not analytic at point z

0

we say it is singular there.

Property

All derivatives of an analytic function are analytic.

(9)

References

Cauchy’s theorem

If the function f(z) is analytic through the region enclosed by the closed contour C in the complex z-plane then

I

C

f (z)dz = 0

The residue theorem

If f(z) has no singularities other then poles in the interior of the closed contour C,

then I

C

f (z)dz = 2πiR where R is the sum of residues of these poles and the integration is taken in anticlockwise sense.

Cauchy’s integral formula

If f (z) is analytic through the interior of the closed contour C, then at any interior point z of this region,

f (z) = 1 2πi

I

C

f (z

⁰

) z

⁰

− z dz

⁰

The Schwarz reflection principle If f (z) is analytic in a connected region which includes part of the real axis and f (z) is real-valued on this part of the real axis, then

f (z

^∗

) = f

^∗

(z)

(10)

References

Laurent’s theorem

Let f(z) be analytic through the closed annular region between the two circles C

1

and C

2

with common centre z

0

. Then at each point in this annulus

f (z) =

∞

X

n=−∞

A

n

(z − z

0

)

ⁿ

,

with series converging uniformly in any closed region, R, lying wholly within the annulus. Here

A

n

= 1 2πi

I

C

f (z

⁰

) (z

⁰

− z

₀

)

ⁿ⁺¹

dz

⁰

Taylor’s Theorem

If f (z) is analytic at all points interior to a circle C centered about z

0

then in any closed region contained wholly inside C

f (z) =

∞

X

n=0

1 n! f

⁽ⁿ⁾

(z

0

)(z − z)0)

ⁿ

and the series converges uniformly.

(11)

References

Casuality

In non-relativistic physics the recruitment of causality follows the analyticity of the f (E) scattering amplitude in the upper half-plane of the complex plane (in E).

see (Bjorken & Drell, 1998)

(12)

References

Electromagnetic Vertex

(13)

References

Electromagnetic Vertex

F (q

²

) ∼

Z

d

⁴

k 1

(k

²

− m

²_π

+ i)((k + q)

²

− m

²_π

+ i)((p − k)

²

− M

²

) + i) We easily get,

F (q

²

) ∼

Z

1

0

dx

Z

1−x

0

dy

Z

d

⁴

k

⁰

1 [k

⁰²

+ ∆ + i]

³

where

∆ = q

²

xy−M

²

(x+y−1)

²

−m

²

(x+y)+i → ∆ = (q

²

xy+i

⁰

)−M

²

(x+y−1)

²

−m

²

(x+y) (1) In practise,

F (q

²

+ i

⁰

) ∼

Z

1

0

dx

Z

1−x

0

dy 1 [∆]

It is easy to see that ∆ vanishes along q

²

> 4m

π

! Hence F (z) is analytic in complex

plane but without cut along Rez axis starting from z = 4m

π

(14)

References

q

²

= t, for elastic scattering t < 0!

Large distance from singularities implies the existence of the expansion parameter

Conformal Mapping on unit circle

z(t) =

√ t

cut

− t − √ t

cut

− t

0

√ t

cut

− t + √ t

cut

+ t

0

t

cut

= 4m

²_π

, t

0

= t

cut

(1 − p

1 + Q

²_max

/t

cut

)

(15)

References

Notice that lim

|t|→∞

z(t) = 1

as well as

z(t = −Q

²_max

) = −z(t = 0) = z

max

Line segment (−Q

²_max

, 0) transforms to (−z

max

, z

max

)

Expansion:

G(t) =

∞

X

k=0

a

_k

z

^k

(t)

The main result:

|a

k

| can be bounded by the knowledge of the Im(G) in timelike region.

(16)

References

E-M Form Factors are real on

−q

²

> 0 line lim

|t|→∞

|G(t)| → 0

G

^p_E

(t) = 1 π

Z

^∞

4m²_π

dz ImG

^p_E

(z)

z − t

(17)

References

(18)

References

Points just above the cut project onto the upper half of the circle (with unit radius)!

z(x) = e

^iθ(t)

→

t(θ) = t

0

+ 2(t

cut

− t

0

) 1 − cos θ a

k

must be Real! Now

Re

Z

π

0

dθG(t)e

^ikθ

= πa

k

Hence

a

k

= 1

π

Z

π

0

dθReG(t + i0) cos(kθ)

− 1 π

Z

π

0

dθImG(t + i0) sin(kθ)

(19)

References

G(t) is analytic in almost everywhere (cut). Hence a

−n

= 0, for n integer, hence, if one change k → −k then sin part is related with cos part.

a

0

= G(t

0

), z(t

0

) = 0.

a

k≥1

= − 2 π

Z

π

0

dθImG[t(θ) + i0] sin(kθ)

= 2

π

Z

^∞

t_cut

dt t − t

0

r t

cut

− t

0

t − t

cut

ImG(t) sin(kθ)

(20)

References

Norm

k G k

p

=

∞

X

k=0

|a

_k

|

^p

!

^1/p

In the case of p = 2,

(k G k

2

)

²

=

∞

X

k=0

|a

_k

|

²

=

Z

π

0

dθG(z)G

^∗

(z) =

Z

π

0

dθ|G(z)|

²

=

I dz z |G(z)|

²

= 1

π

Z

∞

tcut

dt t − t

0

r t

cut

− t

₀

t − t

cut

|G(t)

²

|

Z

π

0

dθ (z

^k

a

k

+ z

ⁿ

a

n

)(z

^∗k

a

k

+ z

^∗n

a

n

) =

Z

π

0

dθ (2 cos[(n − k)θ])a

k

a

n

+ π(a

²_k

+ a

²_n

) (2)

(21)

References

Below two nucleon production

(22)

References

Below two nucleon production

(23)

References

(24)

References

Below two nucleon production

(25)

References

(26)

References

(27)

References

(28)

References

(29)

References

(30)

References

(31)

References

(32)

References

K. M. Graczyk, C. Juszczak, arXiv:1408.0150

(33)

References

(34)

References

Bjorken, J., & Drell, S. D. (1998). Relativistic quantum mechanics. McGraw-Hill

Education.