Fortschritte der Physik / Progress of Physics: Band 27, Heft 10 1979 [Reprint 2021 ed.] 9783112522783, 9783112522776

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHI.UHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 10 • 1979 • B A N D 27

A K A D E M I E - V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der D D R an das Zeitungsvertriebsamt, an eine Buchhandlung oder an den A K A D E M I E - V E R L A G , D D R - 108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der B R D und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle K U N S T U N D W I S S E N , Erich Bieber, 7 Stuttgart 1, Willielmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle K U N S T U N D W I S S E N , Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, Postfach 160, oder an den A K A D E M I E - V E R L A G , D D R - 108 Berlin, Leipziger Straße 3 - 4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. D r . F r a n k Kaschluhn, Prof. D r . A r t u r L6sche, Prof. D r . Rudolf Ritsehl, Prof. Dr. R o b e r t Rornpe, im Auftrag d e r P h y s i k a l i s c h e n Gesellschaft der D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k . V e r l a g : A k a d e m i e - V e r l a g , D D R - 108 Berlin, Leipziger S t r n ß e 3 - 4 ; F e m r u f : 22 3 6 2 2 1 u n d 22 3 6 2 2 9 ; T e l e x - N r . 114420; B a n k : S t a a t s b a n k der D D R , Berlin, K o n t o - N r . 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift, d e r R e d a k t i o n : S e k t i o n P h y s i k d e r H u m b o l d t - U n i v e r s i t ä t zu Berlin, D D R - 104 Berlin, Hessische S t r a ß e 2. V e r ö f f e n t l i c h t u n t e r der L i z e n z n u u i m e r 1324 des P r e s s e a m t e s b e i m V o r s i t z e n d e n des ¡Vliuisterrates der D e u t s c h e n D e m o k r a t i s c h e n Republik. G e s a m t h e r s t e l i u n g : V E B D r u c k h a u s „ M a x i m G o r k i " , D D R - 74 A l t e n b u r g , C a r l - v o n - O s s i e t z k y - S t r a ß e 30/31. E r s c h e i n u n g s w e i s e : Die Z e i t s c h r i f t „ F o r t s c h r i t t e d e r P h y s i k " e r s c h e i n t m o n a t l i c h . Die 12 H e f t e eines J a h r e s b i l d e n einen B a n d . B e z u g s p r e i s j e B a n d 180,— M zuzüglich V e r s a n d s p e s e n ( P r e i s f ü r die D D R : 120,— M). P r e i s j e H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M) B e s t e l l n u m m e r dieses H e f t e s : 1027/27/10. © 1979 b y A k a d e m i e -Verlag B e r l i n . P r i n t e d in t h e G e r m a n D e m o o r a t i c R e p u b l i e . A N ( E D V ) 57 618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 27, 4 6 3 - 4 8 7 (1979)

Polarization Phenomena in Elastic Electron-Proton Scattering MASAAKI

National

Laboratory

KOBAYASHI

for High Energy Physics, Oho-machi, Ibaraki, 300-32, Japan

Tsukuba-gun,

Abstract A unified treatment is given by using the scattering matrix in the Pauli spin spaces and polarization density matrices. Various quantities such as polarization P, asymmetry A, spin correlation parameters double polarization asymmetry Ax/5, polarization transfer parameters Dt, At, Rt and depolarization and spin rotation parameters D, A, R are expressed in terms of six coefficients which appear in the scattering matrix. Some explicit expressions as well as numerical results for these quantities are also given in the one photon exchange approximation.

Contents 1. Introduction

463

2. Formulation

464

3. Observables

467

A. Differential Cross Section B . Polarization and Asymmetry C. Spin Correlation Parameters D. Double Polarization Asymmetry E . Polarization Transfer Parameters F. Depolarization and Spin Rotation Parameters G. Higher Rank Spin Tensors Appendix 1. High Energy Approximation

467 468 468 471 473 477 482 482

Appendix 2. Relativistic Transformations of Spins

483

References

487

1. Introduction P o l a r i z a t i o n of recoil p r o t o n s in elastic e l e c t r o n - p r o t o n s c a t t e r i n g has b e e n studied b y m a n y a u t h o r s in order t o see t w o p h o t o n e x c h a n g e c o n t r i b u t i o n s (ARAFUNE (1970)). More r e c e n t l y , high e n e r g y polarized electron b e a m s h a v e b e e n c o n s t r u c t e d a t S L A C a n d a m e a s u r e m e n t of t h e a s y m m e t r y in elastic s c a t t e r i n g of longitudinally polarized elect r o n s on longitudinally polarized p r o t o n s has shown t h e sign of GE/GM t o be positive (ALTTGUARD (1977)). Progresses in r e c e n t y e a r s in o b t a i n i n g polarized electrons a n d pro37

Zeitschrit't „Fortschritte der Physik", Heft 10

464

MASAAKI KOBAYASHI

tons1), and also in increasing accelerator energies show that various type of polarization experiments in wide energy ranges may perhaps be feasible in the not too distant future. The experiments will be done not always with the initially polarized particles, but also by measuring the final polarizations. Kinematical conditions will be also various: fast electrons injected on stationary targets, or high energy protons on low energy electrons, or colliding beam experiments, etc. Besides fundamental researches, practical purposes also require formulations easy to use of elastic lepton-nucleon scattering for example in order to discuss a possibility of producing polarized beams or to estimate depolarization of stored particles. It is well known that the cross section of elastic scattering of unpolarized electrons and protons is described by the Rosenbluth formula in the one photon exchange approximation. S C O F I E L D ( 1 9 5 9 , 1 9 6 6 ) treated a more general case when both initial electrons and protons are arbitrarily polarized. He gave the differential cross section by following the usual trace methods for Dirac spinors and "/-matrices. D O M B E Y ( 1 9 6 9 ) gave a unified treatment for scattering of polarized electrons from nucleons in terms of the polarization density matrix of a virtual photon exchanged. He calculated as an example the scattering cross section of longitudinally polarized electrons. This paper presents a unified treatment of various spin effects in elastic electron-proton scattering, based on the scattering matrix in Pauli spin spaces and polarization density matrices. Various quantities such as polarization P, asymmetry A, spin correlation parameters Cdouble polarization asymmetry Aa/3, polarization transfer parameters Dlt Af, Rt, and depolarization and spin rotation parameters D, A, R, have simple expressions in terms of six coefficients which appear in the scattering matrix. These quantities are also calculated in the one photon exchange approximation. In Appendix II, a formulation is presented for relativistic transformations of spins which may be useful to obtain various spin parameters in an arbitrary reference frame. The formulations presented in this paper are directly applicable for scattering between any two fermions, one having only the Dirac magnetic moment, the other having an additional anomalous moment. 2. Formulation

The differential cross section in the centre of mass system (CMS) of electron (the suffix 1) and proton (the suffix 2) can be written as %

= KxiV

\M\ xa*)\2,

(1)

where Xi a n ( i 7a {i = 1» 2) are two-column spin functions normalized to unity of the initial and final state fermions respectively in their rest systems. If one assumes the parity conservation and time reversal invariance for the electromagnetic interactions, M has a general form: M = a + ib[((ov,(1>)j

:

(26 a)

IoK$ = - Tr ( M a ^ M + o ^ ) . On the contrary, when unpolarized electrons scatter from polarized protons, the polarization vector of scattered electrons is given by pm (0W) =

+

e

1 +

(26b)

I 0 K% = 1 Tr ( M a ^ M ^ W ) . In the one photon exchange approximation where P parameters vanish, we have (oi») = Z K % { a M ) i t f , (f=l,2:

(27)

f ± j )

i.e., K ^ gives the /?-component of polarization transfered to the initially unpolarized j-th particle from the target particle which is initially polarized in the direction ot. Straightforward calculation by substituting (2) m (26) gives I0Kna

= 2[—[6|2 + |c|2 + |gf - \h\* + Re {am*)},

I0Kpp = 2 Re [{a + to) g* + (a - to) h*}, IqKkk = 2 Re [(a + to) g* — (a —

to)

fc*],

(28)

h K f P = IaK% = 4 Re (bh* + eg*), I0K™ =

= 4 Re (-bh*

+ eg*).

As K'JJ = Kl2J, they are simply written as K aa by dropping the particle index. In the one photon exchange approximation, (28) becomes much simpler: K-nn — CNN, Kpp — Cpp, Kkk — —C, Mi j -fiTjjp = K-fk = C/fP) -S^PK = KRP = —CpK- J

(29)

474

MASAAKI KOBAYASHI

By referring to the laboratory system with the coordinate axes (A 7), (A 15) and (A 21) and by using the relations (A 16) and (A 22) between M+o%9)

Tr

-

{Mo™M+oZ,),

( M ^ M + o f t , ) ,

(31)

j T r

j

Tr

(Ma£M+crZj,

Tr (Ma^M+aZ,),

' W

=

—

)

=

i T

1

Tr

{Mo%M+at]_Pr),

Tr

[Mo™M+o +

(38)

R^KOù

+

D ~ R' are depolarization and spin rotation parameters which can be measured by laboratory experiments as sketched in Fig. 12. They are given by I0D = i / o 4 (D

Tr (McnWM+o%n) r

(Mo R; and Kaß. 01id

Xßi A-1"

R'l"

\A ,,,121 Fig. 12. Depolarization and spin rotation experiments in the laboratory system 38

Zeitschrift „Fortschritte der Physik", Heft 10

480

MASAAKI KOBAYASHI

In matrix representations we have D

=

Dnn

RM A'M

=

M « )

(40)

R'W

T{6,«')

Dependences of depolarization and rotation parameters on the scattering angle, evaluated in the one photon exchange approximation, are shown in Fig. 13 for various energies. The characteristic features at high energies are as follows: (a) Depolarization in the direction perpendicular to the scattering plane increases with the scattering angle from zero at the forward (d — 0) to complete at the backward (6 = n). (b) Helicity of electrons is almost completely conserved. Rotation from the longitudinal to transverse polarization in the scattering plane hardly occurs. Helicity of protons is approximately conserved except at the very forward angles, where the longitudinal polarization almost completely rotates into the transverse one.

Fig. 13 b

Polarization Phenomena in Elastic Electron-Proton Scattering

481

-0,5

-1,0 Fig. 13 c

10

Fig. 13 d

1 -

1 1 R'«>

-

0,5 f> \ 1 \ \

•

•

\

^

\

\

s

Î? j

-

x s-rcef?

Fig. 13. Depolarization and spin rotation parameters referred to the laboratory system and evaluated in the one photon exchange approximation. (a) D, (b) A, (c) R, (d) A' and (e) R' • 6: the CMS scattering angle

38*

482

MASAAKI K O B A Y A S H I

(e) Transverse polarization of electrons is conserved at the forward angles. The depolarization increases with the scattering angle. Transverse polarization of electrons hardly rotates into longitudinal one. Transverse polarization of protons is conserved at small angles but very forward angles. At the very forward angles, transverse polarization rotates almost completely into longitudinal one.

G. Higher Rank Spin-Tensors We can define the third rank spin correlation tensors by IoCafy = T

(41)

(MaVM+a^a/*))

Tr

for the scattering of polarized electrons on unpolarized protons. Similarly the third rank polarization tensors can be defined as W

=

T

(42)

Tr ( M o S W i M + O y W )

for the scattering of polarized electrons on polarized protons. The third rank spin-tensors can be calculated straightforward by using M given by (2). They vanish in the one photon exchange approximation, because all the coefficients a m are real. We do not discuss here the fourth rank spin tensors which correspond to much more complicated experiments. Acknowledgements The author appreciates Professor K. Kondo and J . Arafune for helpful discussions.

Appendix I.

High Energy Approximation

At high energies where k is much larger than the electron mass, the contribution of (m1)n terms are typically by (m 1 /k) n smaller than the main terms. Let us expand a ~ m in power series of mjk and take only up to the linear terms of mx. Writing for example a = a0 + A a where Aa is the correction due to nonzero electron mass, we obtain the following: F,q2

a0

2(V« +

&0 + Op

— C0

]/4fc2 + q2 ' q21s

2riq2~]/s j/,s + m2

+

1/« + W„

TOn

«0 ?n0 =

m2f

F^fs

2{s

+

m2)

g2]/s (s — m 2 2 ) (]/« + m2) _

(A 1)

Polarization Phenomena in Elastic Electron-Proton Scattering =

A (b

+ c)

k(s

—

w22)

—

F

(]/s + m 2 f

2

q

j/47i 2 +