Sil, N C

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37CLASSICAL THEORY OF CHARGED MESONB y N. C. S ILDlil’ ARTMIiNT OK TH E(^KKTI^t PHYSICS,I ndian Association kok tiik CuMHvation ok SciivNCi;, J adavkuk, CalcuBCa.'(Received for publication, February 15, jqs3)ABSTRACT. The equations of motion of the nucleon in interaction willi the charged scalar meson field have bei n solved olat-sically and the value.s of total scattering cro.ss sections of charged mesons by nucleons h ive been calculated. The inllnence of radiation damping has also been investigated.t . It is wellknown that the scattering cross sections of various processes become increasingly large with increasing energies if no account is taken of radiation reaction in the quantum mechanical calculations. However, it is not always possible to find exact solutions tor scattering processes when the influence of radiation reaction is included in the mathe­matical formalism of quantum mecluinics ; it is often very difficult to estimate the degree of inaccuracy involved in the approximate solutions. Hence arises the necessity of approaching the problem from the classical theory in which the treatment of radiation reaction is much simpler. 'J'his scheme was originally introduced by Dirac {1038) in his ‘classical theory of radiating electrons’ . Bhabha (1939, 1941) mid Harish Chandra (1946) extended the same method to the case of neutral meson field. It was not possible in Bhabha’s theory to consider the charged mesons which are experimentally observed. The charge formalism was introduced by Fierz (1941) for the case of vector mesons, but he did not consider the dipole coupling. The complete treatment of electrically charged vector meson field was given by I^e Couteur (1949). Recent experimental findings indicate* however, that the mesons involved in the nuclear interaction should be described by the pseudoscalar field. Now in a classical theory there should be no distinction between the scalar and pseudosealar interactions. Hence it is worth while to investigate the case of electrically charged scalar meson field.2. In the charge formalism each field quantity is considered as a vector in a three dimensional space which we may call -‘charge space’ , denoted by three unit vectors y The electrical charge density -  an observable quantity “ is represented as the y component of a vector (I*e Cduleur, 1949).We shall mostly keep to the notation of L»e Gouteur’s paper. We take3 1 2 N . C. Silthe fundamental tnetric tensor .^Vv (the suffixes /u, v, p, o- as usual run from o to 3) defined bywith all the other components \aiiishing. Then vvc may describe the charged scalar meson field in interaction with the nucleons by a potential U and field strengths G,x which statisfy tlic following equations :G% —c^U ~ 4 ;rS;,==G;*—47rS^, ... (i)+ x“U — 47rS, ... (2)where x=n!ci1t,  in being the meson mass c and ft having their usual meanings._  band 8a /*The source densities S and are associated with the mesonic charge and dipole moment of the nucleons. Tiie source densities arising from a single point particle may be assumed to be given byS =  / s{t)f>ix~zU/)dl,— QC(3)or3, .-  I s ^ i lW x -  z(n)dt, (4)where are the co-ordinates of the particle when its proper time is t(having the dimension of length) and S({) and s^ /^) are continuous and differentiable funetions of /.From (ij and (2) we have0 8^ ,. + J U -  ^  7T (S + . ... (5)Pieviously, in  Fhabha’s theory (Bhahha, 19 .4 0 )  it was not possible to consider a charged meson field classically because of the commutability of all quantities in the classical Ihcoiy. This difficulty is removed by defining the charge current vector 7-y^  as the 3rd. component of a vector product denoted by the symbol A  -of tv\o vectors in the charge space. Thuswith the help of (i) and (2) we getA  U - s ;  A  G,.}.The energy momentum tensor is given by4^rr,„=G^G„ -  -x = U “}.Hence we get after use of (i) and (a)9 'T^.= G^(S + 8'-S J  =  G^S -  d'-G^S p + 9  "(G^S ,).(6)(7 )(8 ) Cq)Classical Theory of Charged Meson 313It is seen from (7) and (9) that in a source free region9 ^/y'‘ =  o and 0^Tpv =  oThe angular momentum of the meson field is given by»A “  /t f*) A A1- f /lA ■ ... (10)Ilencc8 -^M,„..a =  A  A — v,,'- ' ' 7 ', ,a . . . .  ( I  I )With the heli» of (0) we getSG„ -  .A „C 8"G „ -  S,.G„-  a : s G ^  +  A . S  a9 ' G ^  +  S  , G+ 9 {^.v^G „Sa---iCvG^Sa}. ... (12)I^et us now determine the electric charge current density and the average value of the energy momentum tensor for the special case of the plane wave as we shall require these afterwards for the evaluation of the scattering cross section. Following lyC Couleur (1949) we have=  sin (w .^v'' +  <r;),  ^=  /3 or y. ... (13)Then from equation (6)/y« =  /la/lfl sin (Ta-crp), • (14)and with the help of equation fS) the average value of the energy momentum tensor is given byT,lv—  ^Htt ••• (15)... {14a)Por a stream of positive or negative mesons we assume Aa — yis, Ay — O and —and in the case of neutral mesons3. The equations for the rates of change of the electric charge Q, the energy momentum Af^  and the angular momentum carried by the derived from the method of inflow calculation given by llarish  Chandra (1946). In these calculations we have to use modified mean fields defined byy  / mriui =  U J U fajjjj G , / = G ' »  + iG '™ ''. ... (16)Hereafter we shall write U for U ane G^ for G% Then from (7), (9)and (12).40 = ^ . { - s  A  U + fi^  A  G/*}d i  n c(1 7 )314 N . C. SildA^I t  '=  - s G „  + 6p 0 pG^. (i8)% :  = -  z b^G „ + . + s„G,a t+  z „  sO ,. -  z ~ B  xS^ G ^ -  8 v G ^ .The spin angular momentum B^ » is given bydT ~  di ''(iq )where“  (® /<G 8 ,,G ju)" dt ■(20)Wc may split any vector into two parts (i) one orthogonal to velocity Vf, denoted by the sign ~  overhead, and (ii) the other parallel to V/i. Thus the dipole source function s,, is split as followss,* =  s  ^4-where B,.at/* =  o andIt has been shown by llarish  Chandra (1946) that with respect to the field producing properties the contribution of the part S* can be included within that of the charge function s. Hence without loss of generalitywe can lake 8,,=8,,.Therefore6 v^>‘ =  o ... (21)We now express the source functions 8 and 8  ^ in terms of t  the classical analogue of the quantum mechanical isotopic spin vector. We assume8 =  ... (22)... (23)where g  and / are coupling constants.g has the dinienson of electric charge and f has the diinenson of electric dipole.From (21) we see that SuV>* — o. ... (21)Thus in the rest system of co-ordinates v^—(x, o, o, o), So~o and the spatial parts are different from zero. In analogy with quantum mechanical results we assume that■e* =  I so that =  o, diand S/iS/I Sf^whence Sf, —  = 0 .dt... (24)••• (35)Classical Theory of Charged Meson 315f^iypiT bs 3 tensor which is sntisyniinetric in esch pnir of indices flnd<»0l88“  ■ " I .Then i  pS,r =» i .HenceFurtheratT'’5"’«p,.po-D'‘ =  D .pTl’ =  />"•*" =«•After use of U2) and (23), equation (17) is given by From this we get following I^e Couteurf r i c ^ ^  k U - / 5,G^ + q}.lyCt US write^  p * "  A /I ^  iV/ V ft f(26)(27)2^8)(201(30)(31)where /4 pV'*=o and M* ~  A pv>^ , andwhereG^ =  Gp + G VGpT'‘ “ o andG* =  GpT'*s dUdi (33)Then contracting equation (20) with v '  and using (21), (31) and (32), we get7 Ti »¥T(3O... (34)ai, ai urhelp ol equations (33), (18) and (32), equation (3,}) may bewritten asdAi* dU dQf ,dB p„ dv>‘ 1 dU du" ,d:  d i -  ■ ■ ■ ' « 'An y « ‘iU V '’ — Bm - ,dt dtdM* = v . i ddt dt dtFrom (28) and (20) we see thata tOn account of (27) and (28) we may takeat at. . .  (36)3 1 6 N . C . Silwhere f is a constant and V is an arbitrary vector. From (20) and (36) we have■1+ ~ ~  ^ =  Spi(f‘fQ , +  F — s JfvQf, +  V )^— iv^A y—v^Ap).Multiplying both sides by and using (Harish Chandra 1946) we geti l f  ^dt dt j  \  dt dt !=  SpU‘sGy+ Vy)ei‘ '”^ P-VpA(37)(38)Omitting the parts proportional to Vp or v ,  the equation (37) may be written as follows :I S p y p ^ V  '• =  4>l / x  G  V +  F  „ I — S v[ /  +  V p ^39)dUContracting equation (38) with ( / t +  F„ — / x v —  jwe obtain using (21), and T;/i—— = 0 ,dt, ds" I f .  „ dU \<11 IAlso with the help of equation (28)d s "  , ,  d s ' ^  a  I •d, ' " • - - j r * ' " ' " ' ' "  “d f dv ‘^‘ ,, dv^  ^ dv''. v s - )  ~  =■Hence from (21) and (30) we getdt dt dt diTherefore equation (35) may be written asf = l i i  { +-  i'iGp ^  + h  %  V'* V M v ' ‘ .d t  d t  d t  d tUsing (40) and (30) we finally get-g^^+f^' ^sp U ^ q +  ^  F^di dt I dt di(4^ 0{41)(42)(43)Now q and may be so chosen that ^ A  q, (s^l '. -SvF^) and ‘J ^ q -  Vat dtare perfect differentials.The simplest solution of course is Q =  o, \\ '= o .  Another simple solution isClassical Theory of Charged Meson 317With the help of (,^ 4) the intregration of (43) leads to(.15)Where ni**^  is the constant of integration.Thus the equation of translational motion is solved completely.4. We now evaluate the modified radiation fields U ' a n d  O ' a t  points z^K (/) on the worldline of the nucleon. Following the calculations given in Harish Chandra's paper (1946) we obtainU rad(3,< (b) • dsdl -2 X  f  ^ dt'J  u-^8.-d^ V'dT=x’ / ^ dSo dv"' dl dt ~ d ^u s„ dl'. (46)a n d G ;,.™ to )-_ 2 s d % _ 2  dS d r ,  _ ^ d -S  , __^/.sr„ dt  ^ dt dl dt-J  uH d s . r  I  „  d * r*> _ U 1.* sJ / ‘“ " dtL'l* -1'’dl- dt^d tdr- + dy_^  ^.^t^r \  _ r , r - +" dl dt 1 Hr*- B . r dv'' _ 2 r ,r -dt dl dt5—1832P—6318 N. c. sn+ 2X- f S„ d / '-2 X ” f d f\J  u J  u ... (47)where =  — being the field point and u =  f ).in the above calculations we have neglected terms of the second degree? y 2in —  , and used the relation sJ t "^ =  c).di d eIn the rest system of co-ordinates 7 v = ( i ,  o, o , o ) ,  =  and So =  os o  that =  0 ,t X... (48)—ar 0H e n c e  in  t h i s  s y s t e m  of c o  o r d i n a t e s ,  e q u a t i o n s  ( 4 6 )  a n d  ^47) c a n  bo simplified as follows :and — whence, dt' — J du.' iiui d  ^ jU(=^(/))= - 2 g x J du,u (49)(i ^  ^• 2 g V^~ jeX‘ ‘''ZV + 2 gX‘ J0f du4- -S J -  V *^^~d?  ^  ^dl + 2X j  S,.JjAx-u) (so)I'urther equation (39) may be written in view of equation (44)as (51)Jwhere M is a three dimensional vectot in ordinary space with components •?i» 52. . 3^ ; M / cM  f c =  I ,  and H is a vector in ordinary space with componentsG j, Go and G3. f A , B] denotes vector product of vectors A  and B in ordinary three dimensional space and (A B) denotes their scalar product. Similarly ecpiation (30) becomesWith the help of (48) and (50) w^ e have" ^ 3  J  j2dM ^  d dt  ^ d i  dtxM du.uClassical Theory of Charged Meson Hence equations (51) and (53) become319dMdt(51)andA £ 1dt‘2 ^  A 2 » , f  ,JJXU)J  U du-  I t  A / - j*.* (52)5. We now try to solve the above etjuations assuming the incident field to be weak and simply periodic with frequency w. Equation (51) agrees with Le Couteur’s equation ^6i) (Ee Couteur, 1940). the solution of which is given by Bhabha in his neutral theory. We shall neglect the periodic oscillation of velocity*To solve equation (52) we assume with Le Couteur thatT =  av + «i« sin + crj + sin Uot-h<r.^ )^  (53)where a ^ ± i  depending upon whether the particle is initially a neutron or a proton, further and '■*2 are quantities of smaller magnitude. To evaluate the radiation damping terms we notice that the terms proportional to p  are the same as in Le Couteur's paper except for a factor i* Hence we need evaluate only the terms proportional tog^. Ws shall carry our calculation for the case w >  X as needed for the evaluation of the scattering cross section. Thus we get the radiation damping termwhereand-~PaY-l-R®i« cos (tof + cTj) ' h R e J }  cos (<*>fP = 2 g * X + § fM V ,i? =  -  (to* -  x") -  x")*"'-.(54)(55)Substituting (54) in (52) we obtain two e<iuations by equating the co- efiBcients of o and 5 on both sides. We write{  MH"*)}a =  8, sin * » t  + 82 cos t » t ,•|gU '“— tf =  8i sin^u)t± + 83Cos^a>t± - ^ (56)3 2 0 N. C. SilCoiiipartiig with (14 a) we associate the ± signs with the positive or the negative meson, h'urther we take(T.j, =  (r^±7rl2 and <^, =  e2 =  fc’,where (r^  and e are determined l)y the following relationssin ______f)fecc o s  fT 1 ohc± + p )  (  ± + i^) -  K Randc* =  4 (8,^  + V ) { (  ~  + L . r i p )  + ]?*|(57)... (58)^  4 V  +  V  . P \ \ ^ -1(5Q;dM6. The contribution of ---  to the scatteiing cross section is of secondaryimportance and we need consider the contribution of — only. The variationsdtof the charge function and the dipole function are respectively given by and The retraded potentials induced are respectivelyf  [  « sin-^ in i ~  V  X “. r “f j- 4- s in ^  <ot — V' <«>“ —  X * . r  + (60)and+ 3 co s| ;o f- v/ w®-X*. r + o - j j j .  ... (61)The averafje of radial current density over all directions of orientation of M according to (14) and (14a) is given byi - A'^  + fM M .*-X ^ > co s*0  y.* ... (6a)where is the angle between r and M and the bar denotes an average over all possible directions of r with respect to MClassical Theory of Charged Meson I,et the incideut luesou wave he uiveii by321U =  / J « sin -ItoA '' — V — X®. v’ j-  ^ [5 sin|uia " -- <.i'' -  x ' . \ ' ± -the corresponding current density being given by± I —^  1. \/ '"“ --x” .\ 4n-«r / ... {64)Comparing equation (6.3) with (,s6) we get the values of and and substitute these in (59) to calculate Tims from (62) and (64) the total cross section is given byi C m  {  +  c o s= '< ^ }| g - + / - A f " f . . r - x " )  c u s  ‘ 0  |(<15)where 0  is the angle between M and H'".In the (luantum theory of nucleon spin i  we havecos“0 “  cos‘*^= I and - 3. (ho)7. To coniiiare with experimental results w e  i)Ut l. =  o as has been done by Le Couteur (1949). As the values of the coupling constants in the pseudoscalar field are arbitraly to the extent of the cut-off distance, we assume, for our purpose here, their values to be the same as determined for the Mbller-Rosenfeld mixed field from the deuteron binding energies ;j%c= .02A and /*x^/fer =  .096. The theoretical results arc shown in curves. I and II  in figure i .  curve T ( ± a =  — i) gives the scattering of (negative/positive) mesons by (neutrons/protons) and curve II  ( ± a = i )  gives that of (negative/ positive) mesons by Iprotous/neutrons). curve I shows that the maximum value of the scattering cross section is reached when the incident meson energy is 210 Mev ; in case of curve II  the corresponding meson energy is 270 Mev. The ex[ierimental results, though not very reliable, indicate that the m axi­mum value of the cross section of the scattering of positive meson by proton is reached for 180 Mev energy of the incident meson. It may be mentioned here that the maximum values as obtained by he  Couteur (1949) for the above cross sections in the vector field with the same coupling constants occur round about 140 Mev meson en ergy. So it appears that in the322 N. C. SilMeson energy in Mevpseudoscalar field the niaxiniuni value of the cross section is reached at meson energies higher than that in the vector field.A C K N O W L E D G M E N TThe author wishes to thank Dr. D. Basu  for suggesting the problem ami for his kind help and guidance during the course of investigation.R E F E R E N C E SBhabha, IQ39» Eroc. Koy.lSoc A., in ,  384.1940, Proc. Ind. Acad. A.■. tt. 347->> 1941, Proc. Roy. Soc A.,, 178, 314.Dirac, 1938, Proc. Roy. Soc. A.,, 187, 148.F'ierz, 1941, Hclv. Phys. Acta., 11, 257.Ilarish Chandra, 1946, Proc. Roy. Soc. /1 , 188. 269Le Cotttenr, 1949, Proc Cambridge. Phil. Soc., 16,429,

Classical Theory of Charged Meson

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Classical Theory of Charged Meson

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