PION POLARIZABILITY AND BREMSSTRAHLUNG

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
dσ
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
2π
∫
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
dσ
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
dσ
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!