PHYSICAL REVIEW D 1 APRIL 1998VOLUME 57, NUMBER 7 Background thermal depolarization of electrons in storage rings A. C. C. Guimarães,* G. E. A. Matsas,† and D. A. T. Vanzella‡ Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, R. Pamplona 145, 01405-900–São Paulo, São Paulo, Brazil ~Received 3 March 1997; published 17 February 1998! We discuss the influence of the background thermal bath on the depolarization of electrons in high-energy storage rings, and on the photon emission associated with the spin flip. We focus, in particular, on electrons at CERN LEP. We show that in a certain interval of solid angles the photon emission is enhanced several orders of magnitude because of the presence of the thermal bath. Notwithstanding, the overall depolarization induced by the background thermal bath at LEP conditions is much smaller than the one induced by plain acceleration at zero temperature and can be neglected in practical situations. Eventually we discuss under what conditions the background thermal bath can enhance the overall depolarization by several orders of magnitude. @S0556-2821~98!03505-X# PACS number~s!: 41.60.2m, 12.20.Ds, 12.20.Fv a irs ob ar ica rs ec n ic th u n ia u n ha n by th in f t m th fted - cts, true ate and a ed ude we to wer he ath We o- EP lain re- o- is or- y s: p as ing di- on I. INTRODUCTION Evidence of polarization in a single circulating beam w detected unambiguously in the early 1970s at Novosib and Orsay@1#. Later, a polarization ofP'76% was ob- served in the storage ring SPEAR at Stanford@2# and more recently a polarization ofP'90% @3#. The first observation of transverse beam polarization at the CERNe1e2 collider LEP was in 1990@4#, reaching further,P'57% @5#. Trans- verse and longitudinal polarization signals have been served since then~see, e.g.,@6# and references therein!, and their utilization to test possible extensions to the stand model constitutes a source of excitement~see, e.g.,@7#!. In spite of the peculiarities of the different machines, theoret calculations indicate that the maximum natural transve polarization possible to be reached by ultrarelativistic el trons moving circularly in storage ringsat zero temperature is P'92% @8–10#. The main reason why the polarizatio obtained is not complete is the high acceleration under wh these electrons are subjected. However, there are o sources of depolarization which should be taken into acco ~see, e.g.,@11#!. Here we discuss the contribution of thebackground ther- mal bath on the depolarization of high-energy electro beams at storage rings and on the photon emission assoc with the spin flip. We focus on electrons at the LEP, but o conclusions will remain basically the same in most situatio of interest. Theoretical results call attention to the fact t depending on the electron’s velocity, the backgrou thermal-bath contribution can be enhanced~or damped! by several orders of magnitude@12#. This result was obtained in a simplified context by modeling the electron’s spin flip the transition of a two-level scalar system@13# coupled to the background thermal bath. The influence of the velocity in thermal depolarization rate can be understood by notic that because of the Doppler effect the energy spectrum o background photons is shifted in the electron’s proper fra Thus, depending on the electron’s velocity, photons of *Email address: acandido@ift.unesp.br †Email address: matsas@axp.ift.unesp.br ‡Email address: vanzella@axp.ift.unesp.br 570556-2821/98/57~7!/4461~6!/$15.00 s k - d l e - h er nt ted r s t d e g he e. e background thermal bath can have their frequency shi into or off the absorbable band, implying thus anenhance- mentor dampingof the excitation rate. Although the two level model is a satisfactory approximation in many respe this is incomplete in some other ones@14#. Here we aim to analyze the influence of the background thermal bath on fast-moving spin-1/2 fermions. The paper is organized as follows: In Sec. II, we calcul the angular distribution of emitted and absorbed photons, radiated power induced by the spin flip. We show that in certain interval of solid angles the photon emission induc by the spin flip is enhanced by several orders of magnit because of the presence of the thermal bath. In Sec. III exhibit the frequency distribution. Section IV is devoted calculating the total emission rate and total radiated po induced by the spin flip. In Sec. V we use results of t previous sections to calculate the background thermal-b influence on the depolarization of electrons at the LEP. show that in spite of the results of Sec. II, the overall dep larization because of the background thermal bath at L conditions is much smaller than the one because of p acceleration at zero temperature. Finally we discuss our sults in Sec. VI. Natural units\5c5k51 will be adopted throughout the paper. II. PHOTON ANGLE DISTRIBUTION In order to calculate the angle distribution of emitted ph tons induced by the spin flip of a fast-moving electron, it useful to define from the beginning spherical angular co dinates (u,f) in an inertial frame at rest with the laborator and with its origin instantaneously on the electron as follow u is the angle between the electron’s three-velocityv and the three-momentumk of the emitted photon, whilef is the angle between the projection ofk on the plane orthogonal to v and the electron’s three-accelerationa. To calculateat the tree levelthe angular distribution of emitted and absorbed photons associated with the spin fli well as the corresponding radiated power, rather than us the thermal Green function approach, we will introduce rectly the proper thermal factor~Planck factor! in thevacuum probability distribution previously calculated by Jacks @10#: The photon emission rateper laboratory time dPvac em per 4461 © 1998 The American Physical Society a ch r- th th l, to s b r, ia - e h -flip . In n p - o- i- 4462 57GUIMARÃES, MATSAS, AND VANZELLA solid angledV5sinfdudf, and frequencydv induced by the spin flip of an electron circulating in a storage ring zero temperatureis d2Pvac em~u0! dVdv 5 3A3 40p3 n3~11t2! t0g2v0 H sin2u0K1/3 2 ~h! 1 1 2 ~11cos2u0!~11t2!@K1/3 2 ~h!1K2/3 2 ~h!# 12cosu0A11t2K1/3~h!K2/3~h!J , ~2.1! whereg51/A12v2, t5gusinf, v0 is the electron’s orbital frequency, t05F5A3 8 e2g5 m2r3G21 ~2.2! is the typical time interval for the electron beam to rea polarization equilibriumP0, i.e. P(t)5P0@12exp(2t/t0)#, m is the electron’s mass,r is the bending radius of the sto age ring, andh5n(11t2)3/2/2 with n[ 2v 3g3v0 . ~2.3! The variableu0 is defined as being the angle between initial spin directions and the magnetic fieldB. After any transition the spin state changes fromus& to u2s&. Deexcita- tion processes are characterized by the fact that 00(h@1) ;Ap/2h e2h implies that the integral has its main contr bution for 0,h,10, and in this rangev!1/b. Hence, after some algebra we obtain dPem~u0! dV 5 dPvac em~u0! dV 1 dPther em ~u0! dV , ~2.11! where 57 4463BACKGROUND THERMAL DEPOLARIZATION OF . . . dPther em ~u0! dV U utu,50 5 G2~2/3!G~8/3!z~8/3! 53481/6p3 g27 t0 S b21 v0 D 8/3H G2~1/3!G~10/3!z~10/3! 62/3G2~2/3!G~8/3!z~8/3! sin2u0 g2 S b21 v0 D 2/3 1S 32 3 D 1/3 G~1/3!z~3! G~2/3!G~8/3!z~8/3! cosu0 g S b21 v0 D 1/3 1 ~11cos2u0! 2 F G2~1/3!G~10/3!z~10/3! 62/3G2~2/3!G~8/3!z~8/3! ~11t2! g2 S b21 v0 D 2/3 11G J , ~2.12! dPther em ~u0! dV U utu.93102 5 A3 48p ~11t2!27/2 t0v0bg2 H sin2u01 6 5 ~11cos2u0!~11t2!1 64 5A3p A11t2cosu0J , ~2.13! n he for r- ed - , we and dPvac em~u0! dV 5 16 45p2 g~11t2!25 t0 H sin2u01 9 8 ~11cos2u0! 3~11t2!1 105A3p 256 A11t2cosu0J . ~2.14! Equations~2.12! and ~2.13! are plotted in Fig. 1 over the result obtained through explicit numerical integration, a are in perfect agreement. Figure 2 plotsdPther em(u0)/dV againstdPvac em(u0)/dV, and shows that for ‘‘large’’uusinfu, the spin-flip photon emission is largely dominated by t ob d presence of the thermal bath. In particular at the LEP utu'105 (u5f5p/2), we have @dPther em(u0)/dV#/ @dPvac em(u0)/dV#'108. This shows that the background the mal bath must not be always overlooked here. The angular distribution of the radiated power is obtain by multiplying Eq.~2.4! by v and integrating over frequen cies. By using the same approximations described above obtain dW em~u0! dV 5 dW vac em~u0! dV 1 dW ther em ~u0! dV , ~2.15! where~see Figs. 3 and 4! dW ther em ~u0! dV U ut,50 5 A3 45p3 v0 t0g7 S b21 v0 D 11/3H S 3 4D 2/3 G2~1/3!G~13/3!z~13/3! sin2u0 g2 S b21 v0 D 2/3 1S 9 32D 1/3 ~11cos2u0! 3FG2~1/3!G~13/3!z~13/3! 22/3 ~11t2! g2 S b21 v0 D 2/3 1 G2~2/3!G~11/3!z~11/3! 322/3 G 1 36pz~4! A3 cosu0 g S b21 v0 D 1/3J , ~2.16! dW ther em ~u0! dV U utu.93102 5 16 45p2 g~11t2!25 t0b H sin2u01 9 8 ~11cos2u0!~11t2!1 105A3p 256 A11t2cosu0J , ~2.17! the and dW vac em~u0! dV 5 77A3 256p g4v0 ~11t2!13/2t0 H sin2u01 12 11 ~11cos2u0! ~11t2! 1 8192A3 2079p A11t2cosu0J . ~2.18! III. FREQUENCY DISTRIBUTION The frequency distribution of emitted photons can be tained by integrating Eq.~2.4! in the solid angle. By using - the approximation@10# dV'(2p/g)dt which is good for small u, we obtain dPem~u0! dv 5 3 10p n2 g3v0t0 Fsin2u0E n ` K1/3~s!ds 1~11cos2u0!K2/3~n!12cosu0K1/3~n!G 3@11n~v!#. ~3.1! The small-angle approximation above is corroborated by f i- th n a l s t , the ec- , a- h o a nce 4464 57GUIMARÃES, MATSAS, AND VANZELLA last section results~see Figs. 1 and 2!. The unit in the square brackets is related to the vacuum~see Ref.@10#! and ac- counts for spontaneousemission, while then(v) term is related to the background thermal bath and accounts stimulatedemission. The frequency distribution of the radiated power is triv ally obtained from this result by simply multiplying Eq.~3.1! by v, and is introduced for sake of completeness: dW em~u0! dv 5 3 10p n2v g3v0t0 Fsin2u0E n ` K1/3~s!ds 1~11cos2u0!K2/3~n!12cosu0K1/3~n!G 3@11n~v!#. ~3.2! These results will be used in the next section to calculate total photon emission and power radiated. IV. TOTAL EMISSION RATE AND RADIATED POWER In order to calculate the total photon emission rate a radiated power, we integrate Eqs.~3.1! and~3.2! in frequen- cies. The vacuum term is trivially integrated. For LEP p rameters andg.33103, in order to integrate the therma term, we use the approximation Ka.0(n!1) 'G(a)2a21/na, sincen(vb@1);e2vb implies that the in- tegral has its main contribution for 0,vb,10 and in this interval n!1. Now, if 10&g&102, in order to integrate the thermal term we use the approximationn(v!1/b) '1/(vb) sinceKa>0(n@1);Ap/2n e2n implies that the integral has its main contribution for 0,n,10, and in this interval v!1/b. In doing these approximations, one mu keep in mind that Eq.~2.1! and the assumption in our las section, dV'(2p/g)dt, are only valid in relativistic re- gimes. In summary, we obtain, for the total emission rate Pem~u0!5Pvac em~u0!1Pther em ~u0!, ~4.1! FIG. 1. Thermal contribution to the angular distribution of r diation induced by the deexcitation of electrons at the LEP. T dashed line was obtained through numerical integration and is t compared with the solid line obtained through analytic approxim tion. The analogous figure for excitation is very similar. or e d - t where Pther em ~u0!5 8 A3t0~3v0b!8/3g8S 2 5 z~7/3!~3v0b!1/3g 1z~8/3!cosu0D ~4.2! for g.33103, Pther em ~u0!5 1 5t0v0bg3S 2 A3 1cosu0D ~4.3! for 10&g&102, and Pvac em~u0!5 1 2t0 S 11 8 5A3 cosu0D ~4.4! for any g, where we assumeu050 for deexcitation andu0 5p for excitation because hereafter we will suppose polarization to be measured along the magnetic field dir tion. Analogously, we obtain, for the total radiated power W em~u0!5W vac em~u0!1W ther em ~u0!, ~4.5! where W ther em ~u0!5 16v0 405t0~v0b!11/3g8 @7A3z~10/3!~v0b!1/3g 120z~11/3!cosu0# ~4.6! for g.33103, W ther em ~u0!5 b21 2t0 S 11 8 5A3 cosu0D ~4.7! for 10,g,102, and e be - FIG. 2. The dashed line representsdPther em(u0)/dV while the solid line representsdPvac em(u0)/dV. For sufficiently ‘‘large’’ uusinfu, the spin-flip photon emission is dominated by the prese of the thermal bath. ck i la xt - n lid b n - ton ile s- ion, ro the ion her- rgy and ncy in- by . In ath er- - he E d yt la the 57 4465BACKGROUND THERMAL DEPOLARIZATION OF . . . W vac em~u0!5 16v0g3 5A3t0 S 11 35A3 64 cosu0D ~4.8! for any g. In particular, for the LEP we have Pther em ~u0!5~431021616310218cosu0!/2t0 , W ther em ~u0!5~4310221531024cosu0!/2t0 , ~4.9! which are much smaller than Pvac em~u0!5~11931021cosu0!/2t0 , W vac em~u0!5~4310201431020cosu0!/2t0 , ~4.10! respectively. This result shows that eventually the ba ground thermal-bath contribution to the total transition rate very small in this case, and can be disregarded for depo ization purposes. This will be explicitly shown in the ne section. Note, however, the strongg dependence on Pther em(u0) andWther em(u0) which makes the thermal contribu tion larger than the vacuum contribution in the 10,g,102 range. As a consequence, the background thermal bath only is important to the photon-emission rate for large so angles at LEP-type accelerators as shown in Sec. II, could be also important for the polarization itself providedg was considerably smaller. V. POLARIZATION Finally, let us calculate the polarization function P5 P↓2P↑ P↓1P↑ ~5.1! for electrons at the LEP taking into account the backgrou thermal bath, where the excitation rate is given by FIG. 3. Thermal contribution to the angular distribution of t radiated power induced by the deexcitation of electrons at the L The dashed line was obtained through numerical integration an to be compared with the solid curves obtained through anal approximations. The analogous figure for excitation is very simi - s r- ot ut d P↑5Pvac em~p!1Pther em ~p!1Pabs~p! ~5.2! and the deexcitation rate is given by P↓5Pvac em~0!1Pther em ~0!1Pabs~0!. ~5.3! Pvac em(p), Pther em(p), andPabs(p) are the excitation rates asso ciated with spontaneous photon emission, stimulated pho emission, and photon absorption, respectively, wh Pvac em(0), Pther em(0), andPabs(0) are the deexcitation rates a sociated analogously with spontaneous photon emiss stimulated photon emission and photon absorption. Now, by substituting Eqs.~5.2! and ~5.3! into Eq. ~5.1!, and using Eq.~2.10!, we obtain P'PvacS 122 Pther em ~0!1Pther em ~p! Pvac em~0!1Pvac em~p! D , ~5.4! wherePvac50.92 is the vacuum polarization obtained at ze temperature. Finally, by using Eqs.~4.9! and ~4.10!, we ob- tain P'Pvac~128310216!, which confirms the statement in the last section that background thermal bath contribution to the depolarizat should be small. VI. DISCUSSION We have discussed the influence of the background t mal bath on the depolarization of electrons in high-ene storage rings, and the corresponding photon emission radiated power. We have calculated the angle and freque distribution of such photons and obtained that in a large terval of solid angles the photon emission is enhanced several orders of magnitude because of the thermal bath addition, we have shown that the background thermal b can be very important to the total photon emission and ov all depolarization in someg interval, although it can be ne P. is ic r. FIG. 4. The dashed line representsdW ther em(u0)/dV, while the solid line representsdW vac em(u0)/dV. For sufficiently ‘‘large’’ uusinfu, the radiated power is dominated by the presence of thermal bath. th fo of t n nd t a ng ts. ien- 4466 57GUIMARÃES, MATSAS, AND VANZELLA glected at the LEP and similar accelerators. In spite of fact that some of these conclusions were anticipated be @12# by modeling the electron’s spin flip by the transition a two-level scalar system, this approximate approach and realistic calculation here developed lead to fairly differe numerical results. This is another indication of the outsta ing role played by the Thomas precession in this contex first called attention to by Bell and Leinaas@13# and further investigated in more detail by Barberet al. @14#. - e R e re he t - s ACKNOWLEDGMENTS We are really grateful to Desmond Barber for readi carefully our manuscript and for his enlightening commen We also thank him for calling our attention to Ref.@15#. A.G. and D.V. acknowledge full support by Fundac¸ão de Amparo à Pesquisa do Estado de Sa˜o Paulo, while G.M. was partially supported by Conselho Nacional de Desenvolvimento C tı́fico e Tecnolo´gico. iz. d, . @1# V. N. Baier, Usp. Fiz. Nauk.105, 441~1971! @Sov. Phys. Usp. 14, 695 ~1972!#; D. Potaux, inProceedings of the 8th Interna tional Conference on High Energy Accelerators,edited by M. H. Blewett ~CERN, Geneve, 1971!. @2# J. G. Learned, L. K. Resvanis, and C. M. Spencer, Phys. R Lett. 35, 1688~1975!. @3# J. R. Johnsonet al., Nucl. Instrum. Methods Phys. 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