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PION POLARIZABILITY AND BREMSSTRAHLUNG

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PION POLARIZABILITY AND

BREMSSTRAHLUNG

Göran Fäldt

Department of physics and astronomy

Uppsala university, Uppsala

G. Fäldt and U. Tengblad, Phys. Rev. C76, 064607 (2007); C78, 014606 (2008);

C79, 014607 (2009).


PION COMPTON SCATTERING

Pion-nucleus bremsstrahlung is at small momentum transfers to the nucleus driven by

the pion Compton amplitude,

M(γ(q 1 )+π − (p 1 ) → γ(q 2 )+π − (p 2 )) = M μν ɛ μ 1 (q 1)ɛ ν 2(q 2 ) .

Gauge invariance requires that, for real as well as virtual photons,

M μν q μ 1 = M μνq ν 2 =0.

The Compton tensor M μν is conveniently decomposed as

M μν = ie 2 [A(s, t)A μν + B(s, t)B μν ] ,

Mandelstam kinematic variables

s =(p 2 + q 2 ) 2 , t =(p 1 − p 2 ) 2 , u =(p 1 − q 2 ) 2 .


The gauge-invariant tensors A μν and B μν are defined as

A μν

= 2g μν − (2p 2 + q 2 ) ν (2p 1 + q 1 ) μ

s − m 2 π

− (2p 1 − q 2 ) ν (2p 2 − q 1 ) μ

u − m 2 π

,

B μν = q 1 · q 2 g μν − q 2μ q 1ν .


BORN AMPLITUDES

For pions there are three Born amplitudes

Õ½

Õ¾

Õ½

Õ¾

Õ½

Õ¾

Ô½

Ô¾

Ô½

Ô¾

Ô½

Ô¾

They describe Compton scattering by point-like pions. The corresponding invariant

functions are, for the sum of amplitudes,

A(s, t) = 1 ,

B(s, t) = 0 .


POLARIZABILITIES

In chiral Lagrangian theory structure dependent contributions are represented by the

diagrams like

The corresponding structure functions, expressed in terms of the electric and magnetic

polarizabilities, α π and β π , become

A(s, t) = 1+ α π + β π

4m π α (s − m2 π)(u − m 2 π) ,

B(s, t) = 2m πβ π

.

α

In the one-loop approximation α π =2.7 · 10 −4 fm 3 and α π + β π =0.


In the lab system (pion at rest)

M = i8πm π

[


α

m π

ɛ 1 · ɛ 2

+α π ω 1 ω 2 ɛ 1 · ɛ 2 + β π (q 1 × ɛ 1 ) · (q 2 × ɛ 2 ) ]

Structure dependent contributions are called, Rayleigh terms, and structure-independent

ones Thompson terms.


MESON-EXCHANGE MODEL

A model valid also in the GeV region is obtained by invoking, in addition to the Born

terms, the contributions from σ(0 + ), ρ(1 − )anda 1 (1 + ) exchanges. We parametrize


· ½

· ½

the structure dependent part of the Compton amplitude by dimensionless functions

λ 1 (s, t) andλ 2 (s, t). In terms of these generalized polarizabilities

A(s, t) = 1+ (s − m2 π )(u − m2 π ) λ

4m 4 1 (s, t) ,

π

B(s, t) = 2 mπλ 2 2 (s, t) .

At the pion-Compton threshold s = u = m 2 π and t =0and

λ 1 (s, t) =0.0006 , λ 2 (s, t) =−0.0131 .


PION-NUCLEUS BREMSSTRAHLUNG

The Born approximation to the coherent nuclear reaction

π − + A → π − + γ + A

is simply the one-photon exchange graph. The small blob in the graph represents the

full pion-Compton amplitude; the large blob the nucleus electromagnetic vertex. With


Ô ½

Ô ¾


Õ ½

Õ ¾

­


Ô

Ô ¼


q 1 the virtual photon four-momentum, the matrix element becomes

M B = −i

q1

2 M μν (p 2 ,q 2 ; p 1 ,q 1 )(−iZe)(p + p ′ ) μ ɛ ν 2 .


Require transverse momenta be much smaller than their longitudinal counterparts;

|p 2⊥ |≪p 2 , |q 2⊥ |≪q 2 .

In addition, sum of the transverse momenta must remain

q 1⊥ = p 2⊥ + q 2⊥

in the Coulomb region, even though the transverse momenta themselves need not.

Introduce x as the fraction of the incident energy that goes into the final-statebremsstrahlung

photon,

x = q 2z

= ω 2

.

p 1 E 1

Then, the cross-section distribution can be written as


d 2 q 1⊥ d 2 q 2⊥ dx = 1

32(2π) 5 E 2 ω 2 MA

2

|M C + M N | 2 ,

with M C the Coulomb and M N the nuclear amplitude. M B is the Born approximation

to M C .


In coherent nuclear reactions q 10 = 0, and the cross-section mainly determined by the

one-photon-exchange factor

q 2 1⊥

(q 2 1⊥ + q2 1‖ )2

which vanishes when the transverse-momentum transfer q 1⊥ to the nucleus vanishes

and has a maximun at q 1⊥ = q 1‖ ,thePrimakoff peak.

The longitudinal-momentum transfer to the nucleus fixed by

q 1‖ = q min = m2 π

2E 1

·

x

1 − x .

Numerically, m 2 π /E 1 =0.1 MeV/c at E 1 = 200 GeV/c.


COULOMB AMPLITUDE

In Born approximation pions are point like, A(s, t) =1and B(s, t) =0,andthe

nuclear bremsstrahlung amplitude,

M B =

2Zα

q 2 ⊥ + g 0 · q ⊥ ,

q2 ‖

with q = q 1 and g 0 a vector in the impact parameter plane,

[

]

4xE 2

q 2⊥ · ɛ 2

g 0 =4πiM A e

q 2 2⊥ + ɛ x2 m 2 2⊥ − 2q 2⊥

π

q 2 2⊥ + .

x2 m 2 π

Generalize to include multiple-Coulomb scattering,

M C (q) = −1 ∫

d 3 re −iq·r g

2πi

0 · ∇V C (r) e iχC(b) .

Here, V C (r) is the Coulomb potential and χ C (b) the Coulomb phase function,

χ C (b) = −1

v

∫ ∞

−∞

dzV C (b,z)


The amplitude for point-charge-Coulomb scattering is

M C = g 0 · q ⊥ F C (q ⊥ ,q ‖ )

Factorize into point-Coulomb amplitude and form factor

F C (q) = 2Zα(aq)iη e iσ η

q 2

h C (q)

with a the Coulomb cut-off radius, η = 2Zα/v, andq =

scattering q ‖ vanishes.

Analytic expression for the form factor,


q 2 ⊥ + q2 ‖ .

In elastic

h C (q) =Γ(2− iη 2 )Γ(1 + iη 2 ) F (iη 2 , 1 − iη 2 ;2;

q 2 ⊥

q⊥ 2 + ).

q2 ‖

Remark: Nuclear charge distribution may be considered point-like, sinceCoulomb

production mainly occurs far away from the nucleus.


0

z=1

−0.1

(A, Z) = (208, 82)

Im h C

(z)

−0.2

−0.3

z=0.5

−0.4

z=0

0.6 0.7 0.8 0.9 1

Re h (z) C

Plot of h C (z) for lead with z = q 2 ⊥ /(q2 ⊥ + q2 ‖ ).


NUCLEAR AMPLITUDE

The pion-nuclear radiation occurs either before or after a pion-nucleus scattering


­




­




The radiation vertices are the same as before. Replacing there the Coulomb potential

V C (r) bythesumV C (r)+V N (r) we may identify the nuclear contribution as

M N (q) = g 0 · q F N (q ⊥ ),

F N (q ⊥ ) =

iv



d 2 be −iq ⊥·b e iχ C(b)

[ ]

1 − e iχ N (b)

.

with χ N (b) the eikonal-hadronic-phase function. F N (q ⊥ )istheelastic pion-nucleus

scattering amplitude.


The integration in F N (q ⊥ ) is over the nuclear interior. Thus, we may set q ‖ = q ⊥ =0

in the Primakoff region. Neglecting the Coulomb phase as well and

F N (q ⊥ =0)=iσ πA (1 − iα πA )/4π

Coulomb−nuclear part

|F| 2

10 6 q tr

2

10 5

10 4

10 3

10 2

10 1

(A, Z) = (208, 82)

|F C (q) +F N (q)| 2 , |F C (q)| 2 ,

and |F N (q)| 2 calculated for

q ‖ = 1.0 MeV/c. The first

point on the curves at q⊥ 2 =

0.0002 (GeV/c) 2 , well outside

the Primakoff peak.

10 0

0 0.002 0.004 0.006 0.008 0.01


POLARIZABILITY AMPLITUDE

The polarizability contribution is due to the non-point-like structure of the pion. The

amplitude that emerges is

M P (q) = −1 ∫

d 3 re −iq·r g

2πi

P · ∇V C (r) e i(χ C(b)+χ N (b))

= g P · q ⊥ F P (q ⊥ ,q ‖ )

with g P a vector in the impact parameter plane

g P = 16πieM A xE 2

[ B(x, q

2

2⊥

)

4(1 − x) ɛ 2⊥

+ A(x, q2 2⊥ ) − 1

q 2 2⊥ + x2 m 2 π

{

ɛ 2⊥ − 2q 2⊥

q 2⊥ · ɛ 2

q 2 2⊥ + x2 m 2 π

}]

.

Due to hadronic distortion the form factor F P (q) differs slightly from the pure Coulomb

form factor F C (q).


Remember that A(s, t) andB(s, t) are structure functions of the pion-Compton amplitude.

Values of s and t in the pion-Compton system

[ ]

s − m 2 π = 1

q 2 2⊥

x(1 − x)

+ x2 m 2 π ,

t =

−1 [ ]

q 2 2⊥

1 − x

+ x2 m 2 π

The complete pion-nucleus bremsstrahlung amplitude is

]

M = g 0 · q 1⊥

[F C (q 1 )+ F N (q 1⊥ ) + g P · q 1⊥ F P (q 1 ),

.

The structure-dependent contribution represented by g P is small. It is measured

through its interference with the other terms.


0.1

R=Pol amp/Coulnucl amp

0

(A, Z) = (208, 82)

z=0

−0.1

Im R

−0.2

−0.3

z=10

−0.4

z=5

−0.5

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Re R

The ratio

F P (q

R(q 1 )=

1 )

F C (q 1 )+F N (q 1 )

plotted in the interval 0 ≤ q 2 1⊥ ≤ 0.002 (GeV/c)2 .


LOW-MASS COMPTON REGION

For Compton scattering with s ≪ m 2 ρ the threshold approximation applies, i.e. λ 1 ≈ 0

and λ 2 ≈−0.013 are constants.

Since polarizabilities are visible only in the Coulomb region, we simplify and neglect

all hadronic contributions, i.e. put F P (q 1 )=F C (q 1 )andF N (q 1 )=0.

As a consequence, the bremsstrahlung amplitude factorizes into one factor depending

on q 1⊥ , and another one depening on x and q 2⊥ .

Integrating the cross-section distribution over q 1⊥ gives a numerical factor P Z . Then,

integrating over the transverse momentum component 0 ≤ q 2 2⊥ ≤ q2 2⊥max and


dx = 4Z2 α 3

m 2 π

· 1 − x

x

P Z F (z),

with z = q 2 2⊥max /x2 m 2 π, and the distribution function F (z).


The distribution function F (z)

F (z) =u(z)+

[− x2 1 ]

1 − x 2 v(z)λ 1 + w(z)λ 2 .

The functions u(z), v(z), and w(z), with z = q 2 2⊥max /x2 m 2 π, are all elementary.

Small z-values: u(z) =v(z) =w(z) =z .

Large z-values: u(z) =2/3, v(z) =z, w(z) =1.

Conclusion; λ 2 can be observed only if x ≈ 1,

λ 1 only if z is large and x ≈ 1.

Since the Compton mass √ s and the transverse momentum component q 2⊥ are related,

avalueq 2⊥max corresponds to a value s max , and hence a maximal value of x,

x max =1− m 2 π /s max .


Specialize to √ s max =3.75m π , half way to the rho mass. Maximal x-value is then

x max =0.93. The graph traces the value of the ratio R of the cross-section distribution

to its Born approximation value.

1

b)

0.95

R

0.9

0.85

0.8

0.2 0.4 0.6 0.8 1

x


RHO-MASS COMPTON REGION

At large Compton masses, the pion structure functions are complex functions of s and

x. Exploiting the factorization property of the cross-section distribution we get after

suitable integrations


dsdx = 4Z2 α 3

(s − m 2 π) 2P ZG(x, s),

and

G(x, s) =|1 − YA| 2 + |YA+ B| 2 .

The maximal value of x for a given value of s is x =1− m 2 π /s, andY is

with 0 ≤ Y ≤ 1.

Y (x, s) =

x

1 − x ·

m 2 π

,

s − m 2 π

In Born approximation A =1andB = 0, and hence 1 2 ≤ G B(x, s) ≤ 1. The maximal

value G B (x, s) = 1, obtains at the kinematic boundary x =1− m 2 π /s.


The expressions for the structure functions read

A(x, s) = 1− 1 − x ( ) s − m

2 2

π

λ

4 m 2 1 (s, t),

π

B(x, s) = x ( ) s − m

2

π

λ

2 m 2 2 (s, t).

π

Remember that t = −x(s − m 2 π ) and expressible in s and x.

Clearly, the λ 1 contribution is maximal when x ≈ 0, andtheλ 2 when x ≈ 1.

x

1

0.8

0.6

0.4

0.2

p(Y)

0

0 10 20 30 40 50

s

0.8

0.6

0.8 Contour plot of the Born-

0.6

approximation distribution

G B (s, x). Squared

Compton mass s in units

of m 2 π, and x = ω 2 /E 1 .

The dashed line is the

kinematic boundary x =

1 − m 2 π /s.


1

G/G B

1 0.9

0.8

0.8

0.95

0.95

0.6

x

0.4

0.9

0.9

0.2

0.8 0.8

0

0 10 20 30 40 50

s

The cross-section-distribution ratio G/G B (s, x) as function of the Compton mass s, measuredin

units of m 2 π, and the ratio x = ω 2 /E 1 .


Basic references:

A. S. Gal’perin et al., Sov. J. Nucl. Phys. 32, 545 (1980).

Yu. M. Antipov et al., Phys. Lett. B 121, 445 (1983);

Yu. M. Antipov et al., Z. Phys. C 24, 39 (1984);

Yu. M. Antipov et al., Z. Phys. C 26, 495 (1985).

G. Fäldt et al., Nucl.Phys. B41 (1972) 125;

G. Fäldt, Nucl.Phys. B43 (1972) 591.

Thanks to Jan Frierich for valuable help!

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