ON THE SCATTERING OF SCALAR MESONS

BY HARISH-CHANDRA(J. H. Bhabha Student, Cosmic Ray Research Unit, Indian Institute of Science, Bangalore)

Received January 15, 1945(Communicated by Prof. H. J. Bhabha, F.R.s.)

IN a recent paper (Harish-Chandra, 1944, referred to as A in this paper) theequations of motion of a point particle interacting with a scalar mesonfield have been derived. The object of this note is to use these equationsto calculate the scattering of scalar mesons by neutrons (or protons) on theclassical theory, taking into account the radiation damping. This calcula-tion is entirely similar to the corresponding one, done by Bhabha (1939,1941) for the case of the vector-mesons. On account of the neglect of thequantum effects and the charge of the meson these calculations are subjectto the same limitations as those of Bhabha. Since it is as yet not at allcertain whether the actual meson has a spin of 1 or 0 unit, the scatteringformula obtained here are to be looked upon as possible alternatives to thosegiven by Bhabha.

§1. We shall keep to the notation of the previous papers (Bhabha andHarish-Chandra, 1944, Harish-Chandra, 1944). T is the proper time at anypoint on the world line of the neutron measured from some fixed point onit. zµ (T) are the co-ordinates of this point. xµ denote the co-ordinatesof any field-point. The fundamental metric tensor is taken to be

gµ.y = 0 ,. 4- V. g00 = — g11 = — 922 = -- g33 1A dot denotes differentiation with respect to T • vN = k. (T) and u. = x,4 —z,4 (T).

We shall assume that the neutron has a 'charge' and a 'dipole'.Following Bhabha we consider the scattering due to each of these separately.First we calculate the scattering due to the charge alone.

In the notation of A the equation of motion of the neutron is

m71µ — g1 a (U'mean v,4) = — g1 U.'mean (1)[LT l!

where m is the mass and g1 the 'charge' of the neutron. U'mean andU'µmean are the modified mean-potential and field respectively. It has beenshown in A [Eqs. (3.17) and (3.18)] that

U'mean = Uin + U,

U A lmean = Uµin _. g1 (I v,,, + 3 vi µ (v) 2 + I .Y 2 vµ) + UK (2a)Al 135

136 Harish-Chandra

where

U = — Sl x f Ji (u u) dT', U# = g, X 2 f uN, J2(2u) d,-' (2b)

Uin is the ingoing potential and U Ain _ , U'n is the ingoing field. Weassume the ingoing field to be a plane wave travelling in the x,-direction.Assuming that the amplitude of oscillation of the neutron is small comparedto the incident wave length we can write

Ulan = y sin w0t, Uin = 0, U 3in = 0 (3)where t = zo (T). We assume that the velocity of the neutron is smallso that

_ dzkVk dtfor k = 1, 2, 3. In conformity with (3) we put (cf. Bhabha, 1939)

zl = P sin (wo( + 8), z2 = 0, z3 = 0 10

vl = ,B cos (coot + S), il l =— #coo sin (coot + 5) } (4)vo .^ 1, v2 = v3 = 0 1

We assume the amplitudes B and y to be small so that quantities quadraticin them may be neglected. Putting x, = z. (T) we get for (2)

U1 = z, (T) -- z, (T') _ [sin (wo t + 5) — sin (wot' + 6)],0

u9 = u3 = 0 (5a)

U -, t' = uo (5b)From (2b), (4) and (6) we get

U = _ g^ x f Ji (Y u) dT'u-00

00

_ — gix f Jl(u du =—g,x (6)U

a (X u) ,Up =ix f U0 2 dTu

00

= g^ x 8 ` JZ (X u) du =' .^3 2 (7)./ U

On the Scattering of Scalar Mesons 137

T

Uy = gl X 2 f ul J2 (x Cdr'-CO

U2

CO

gi x 2 r J2 (u u) u, du0J CO

= S wo 3 J2 ( s) [sin a — sin (a — vs)] ds (8)0

where s = Xu, v = X and a = wot + 8. Using the well-known result thatDO

f ds J2(s)e -ji[(1- v2)312+ivs-I iv]0<v<1

o `r2 [— 1(v.— 1)312+ ZV 3 — g iv] v>1 (9)

we get

U1= - /3 g: w (P cos a + Q sin a) (JOa)where

I —JV 2 0<v<1P= (v2_1)32 1

l- v3 ++2 V> 1

(1 — V 2)S j 2 1---- 3v3 0< v< 1

Q = 1 (lob)^ v3 v>1

Substituting the value of the various quantities in (1) and putting m+ glX= Mwe get up to terms of the first order in g and y

- P wa M sin a = - gi v cos (a - S)

- I- fl gi wo [P cos a + Q sin a]

- gi $coocosa +I X 2 gi COsa (ll)Equating coefficients of sin a and cos a on both sides of (11) we get

- —.- --- ---M 2t (12a)gl wo {P,2 Wo 1 [Q cuo g_2 1

JJI

cos S (12b)CPf2wo+(Qw0+g^)J

where the positive value of the square root is to be taken. P' = P - } + v$Ala

138 Harish-Chandra

so that0 v <l

(12c)p^ (V2 — 1)3(2

v$ v]1

It is clear from (12) that the quantity M + gi Qw o behaves as the effective

mass. For very slow oscillations wo < X (y < 1) Q ,,: — Zy

M+giQwo =M_Jfix= ^̂t+ SixTherefore in this case the field contributes a positive mass 4.g. On theother hand ip the vector meson case the field adds a negative inasr — 3gz xfor slow oscillations (ej: Bhahha, 1939).

To calculate the scattering we have to calculate the retarded field at avery distant point x (X, Y, Z, T) lying on the future light cone from thepoint -r on the world line.

U^xµ' _,^ (T — x g1 JI (u u) dT

T1 Jo (x u) f 1 dr'

— gi a (K>

1 K' )

= 91 Jo (x u) l—2 — 2̂}

dT,

(13)

where K -- u'2 (T') zvµ, (T') and K' = UIL (T') •µ (T'). For a very distant point

we can neglect the first term 12 and write

T 00

Ure'— gi J Jo (Xu) K2 di-' = — gl j Jo (Xu) K3 dtu (14)

—00 o

Writing R = VX 2 ± YL +Z we get up to terms of the first order in 8

K' = — Xv^ = ,Pwo X sin (wot' + a)

U2 = (T — t') 2 — R 2 + 2XP sin (wot' +6) (15)wo

K =(T—t')—Xv i =(T— t') —fiX cos (w„t+S) Jso that in the same approximation

K' —_ Swo X sin (wot' + S)K $ (U2 + R 2)32

_ fl o0 X sin (w„T + S — wo V/u- + R 2) (u2( + R2)312

#wo X sin (a' — wo -\/u + R2)

— (u2 + R2)312 (16)

On the Scattering of Scalar Mesons 139

if we write n' for w0T + . Thus from (14) and (16)

Uret = — gi N wo X f J0 ( u) sm (a — to o '/t + t<-) u du(u2 + R2)3i2

00

gt 9wo X r Jo (r) sin (a' — v s` + r- s ds (17) J (S2 + r2)312 ( )

0

where s = xu, r = RX and v =X On evaluating (17) and retaining only

the terms of the lowest order in 1R we find

f X 1X2 — w2 — R 1/x9 —ma.

gl 9 R Ru,e e ° sin (woT + 8), coo < XUri _ _ (18)

gi X 0X2 cos (w oT + 8 — R Yo )> uoo > X

For the case c0 0 <x the potential falls off exponentially with distance andobviously there is no scattering. We therefore consider the case w> X.In this case the incident wave is consistently with (3)

Uzn = y'w2 y X2 sin (wo t — ti/u,o — x x i). (19)0

The flow of energy per unit area per unit time in the x 1 direction as calculatedfrom the energy-momentum tensor

4Tr U U i, — 4 g, (U0 UP — X 2 U 2) (20)is

U0 Ul — V 2 `t'—o (21)

— 8^ two — xzwhere the bar denotes the average over time. Similarly the now of energydue to (18) in the direction of the radius R across an element of surfacesubtending a solid angle dQ is

gr r 2 Cos 2 B `OJ°__

x21312 dQ(22)877

( UO 1

where 0 is the angle between the incident and the scattered wave i.e.,X/R = cos A. The differential-scattering cross-section in the direction0 is therefore

gi __2 COS 2 0 (W ° 2x2) dQY Wo

cos 9(23)

9 (w0 — x2) + (u)o —° x 2) 2 [a + X 3cu2^

140 Harish-Chandra

from (12a) if we put gl = a. Integrating over all directions 0 we find

that the total cross-section is12ir — (0, 1 -- dQ (24)

(u,2 — x 2) I^OJ0 ° X2)2 [3a + 3x -- z lof

For the spin 1 case the scattering of the transverse mesons is given by (Bhabha,1939)

,Y216^(1 t CO —3x —1X43a+1X z 2,o

C 4w0 4 c ) k2 2 ^For high energy this is half of (24) independently of the value of g,.For x = 0 (24) becomes

12ir9a2 + wo

which is of the same form as the corresponding formula of Dirac (1938),247T Z for the scattering of light by an electron. As found inviz., 9a 2 + 4w,

other cases the cross-section (24) decreases with increasing wo as for0

high frequencies.§2. We shall now treat the scattering by the neutron due to the dipole-

moment alone. We denote the dipole-vector by ,; 2S". As already pointedout in A we have to assume

Sa Va = 0 (25)

The equation of rotational motion is [Eqn. (5.36b) of A]I E,,.- VP Sa = g2 Si, Uv — S, Uµ.1 (26)

where U = Uy'mean — v, (vµ U,,'mean). is the tensor which is anti-symmetric in each pair of indices and E0123 = — 1. Following Bhabhawe put m equal to infinity to simplify the problem. In this case we find fromthe translational equation that v _ _ .... = 0, so that we can considerthe dipole in the rest system. In the usual three-dimensional vector notation(26) can be written as

I S = g2 IS . U ' mean j (27)

where the components of S and U lmea ' are Sk and U 'kmean (k = 1, 2, 3)respectively. The bracket denotes the vector product.

Using (25) we find from (3.19) of A that2 p

U'µmean = U^i + g2 F 3 S,, +2 S'^) + U,. (28)

On the Scattering of Scalar Mesons 141

wheret

Uu =g2 X2 J s^ J2() di' (29)u

Here we have used the fact that T = t and u, = u2 = u3 = 0 so that

Sa ua = 0. Also

u=U0 =t — t'

so that

U = S2 X2 J S (t - u) J2 u2 U). (30)

0

It is to be noted that the right side of (30) is the same as - G of Bhabha[1941, Eqn. (68)] if we replace M by S there. (27) can now be written as

I S = g2 [S - U'°] +g [S.S] + i S'2 X 2 [S SJ +S2 [S . U] (31)

This equation is the same as Eqn. (66) of Bhabha (1941) if we replaceH by 2 U1°., M by S and I by 2I in the latter and put K = 0. Thus corres-ponding to equations (69) and (70) of Bhabha we put

U'° =I Ho cos wo t = I H (32)S (t) = So + Sl sin (.V + S2 sin (wot + 6) (33)

where So is the initial direction of the dipole and So. Sl and S2 are mutuallyperpendicular and such that S 2 is along [S° • S, ]. We assume S0 2 = 1.Substituting (32), (33) in (30) and (31) we get corresponding to Eqn. (74) ofBhabha

wo Sl [a cos wo t + i { sin (w° t }- 8) — cos (wot + 6)}li J

+ wo S2 [a cos (wot + 8) — 1 Sl ^ { 6 sin wot — cos wo t} 12 J

3 [So . H0] cos w ot (34)

3Iwhere a = 31 and°^ X3 (X 2 - w2)3/2_ wo + w w°<X

_ X 3° (35)

w w0> Xl 0

0 w"<X

_ tm — Xz)3/2w° > X (36)

(00

142 Harish-Chandra

Here the value of a is twice that used by Bhabha [Eqn. (73) l.c.], while E and Care the same as in his case. Since (34) is identical with Eqn. (74) of Bhabha,all the further results derivable from it in terms of a, , C remain the same.Thus we get

tan 8 = — (37a)

I S^ I = — a--- (37b)S21 /&+IS^ I= 3aIHO I sin0

2S2 W0 [(a2 — t2 —

S

Y2)2 + 4a 2]} (37C)

A being the angle between S o and H0 .. The work done by the external forceon the dipole is on the average

3 H2 sine 9 (a 2 } e 2 } C 2) )— 382 g2 (H' S) = 8 (a 2 — 2 — ^ 2) 2 14a^ .2 (

To obtain the scattering we have to calculate as before the retardedpotential U'et at a very distant point X, Y, Z, T lying on the future lightcone from the point z, (T) _ (0, 0, 0, t).

U(XI,) _ ^a 192 r Jo (Xu) (;) dT } (39)

u 2 = (T — t') 2 — R 2 (40a)K =T — t' = -,/U2 -1- R2 (40b)

Retaining only terms of the lowest order in we get

f00

Uret =g2 as Jo (Xu) (u2 + R2) du

000

_ — g2 div j f u2 J+^R2) {S1 cos (cooT — wo A/u;

+ ^)

+ S2 cos (coo T + S — coo \/u2 -}- R2) } • udu (41)

)

where the components of divergence are bk = GX' SY. ). The term

inside the divergence can be evaluated and up to the terms of lowest order in

1R it is equal to

—R VXa_^, 2

e __ _ _ ___ [Sl sin cwoT +S2 sin (woT +)] for w,o < X

{S1 sin (w0 T — R ^ 2 )

+ S2 sin (coo T +- S — R V/w; — x%)} for wo > X

On the Scattering of Scalar Mesons 143

so that to the same approximation92 ^ X Wo

e R 1^"^ _ w° ((RR i

) sin woT

+ (R S2)R

sin (woT +8)] for coo < Xu'=j (42)

g2 a' 0—X

(R S1)Rcos (woT — R -,/u,o — X 2)R

+ ( R2) cos (WOT+ 8 — R \/wo—x=), for w>

wherewhere R = (X, Y, Z). (42) would agree completely with the expression (84)of Bhabha for Uk t if we replace the scalar products with R by vector products.When wo < X there is no radiation. For wo > X the average rate of radia-tion in the direction R inside a solid angle d 9 is from (20)

9-! (moo X2)312 W0 [flRKRSi j)) 2 ..}. (RR 2) 2

2 2

+ 2 (R S) (R SO cos S^ dQ (43)R 2

corresponding to (84) of Bhabha. The total radiation obtained by integrating(43) over all directions is

* g2 (WO — X 2)3l2 WO (!S 1 1 2 + S2I 2) (44)which is the same as (38) due to (37) and (36). The energy flow due tothe incident wave (32) is

Ho 1 2w0 (45)32ir VW 2 — X 2

The total effective cross-section for the scattering of the scalar meson waveis therefore from (45), (44) and (38) and (36)

12 in2 B (WO X2)2 a2 +

S

t2 + Y2 (46

)n s u,2 (a2 — ^ 2 S2)2 -}S 4a2 ^ 2

0

In terms of a, , (46) is the same as Eqn. (82) of Bhabha except for a factor 2.Substituting the value of and 2 the scattering cross-section becomes

12 7 Sln 2 (W0 — X 2) 2 ra2 W2 + ^W2 ± X2 3 '+ X2112± 4a 2 u) 2 X6

(47)

Except for a factor 2, (45) agrees entirely with Eqn. (88) of Bhabha in whichK has been put equal to zero. For small g2 and not very high frequencieswe can expand (47) as a series in ascending powers of g 2. The first term is

sin2 6 I2 (w — X2)2 (48)

144 Harish-Chandra

which is exactly half of the corresponding expression (86) of Bhabha. Thusfor large coo the scattering is double (independently of the value g2) and forsmall w° half that of transverse vector-mesons. The formuhe (47) and (48)have already been discussed by Bhabha in detail.

§3. The above theory will now be compared with the quantum theoryof neutral mesons. For simplicity we put c = 1, * = 1. The totalLagrangian for the neutron and the meson fields together is taken to be

L= 2 (O+ Nyµ bµO — bMO '- Pyµ O) — t'Y'+ N0+g1Y'+ NY' U`F'g20+P Y"O aµ U

i 3 i NYµ Yv Yv Eµyvp ap U + yµ yv ya yp b Eµvop U

- (aµ U.aw U— x 2 U 2) (49)Here yw are the 4-rowed square matrices satisfying

yµ yy + yv = 2gµ'and $ = y°. & refers to the neutron field and 0+ is its hermitian conjugate.y's are related to the usual Dirac-matrices a and $ by

a = (a', a 2, a3) = $ (yl y 2, y$)

We assume a and R to be Hermitian. u is the mass of the neutron. Theterms containing g, and g2 in (49) represent the usual ` scalar ' interactionsfor the charge and the dipole respectively while those containing g,' and g 2 'represent the corresponding 'pseudo-scalar' interactions. Evidently in theclassical theory there is no distinction between the scalar and pseudo-scalarinteractions since this distinction arises only from the y-matrices. Tosimplify (49) we use the usual representation of a and P through two setsof mutually independent Pauli matrices.

a = Pl (a', a2, a3) , $ = P:i

With this substitution (49) simplifies to

L = ('3 Y^` ^ 0—+ $ yµ O) — + f +- g1 + f '& ' U

+ g2 [(Q "1 + U0J Y' + g2 Y'+ [(a U) + P1 U0J Y' — 91 ' 'V+ P2 Y' U

-i- I (^µ U . bµ U x 2 U 2) (50)where U = ( U, 2 U, a3 U), U° _ i ° U. The Hamiltonian of the systemfor only one neutron present is found as usual and on ignoring the infiniteself-energy of the neutron terms out to be

H= (a p)± $ — ^ E (2k0)- (g, P8 — g;' 2) {ak ez (kx) — ak a 1(k^)

+ 1V E (2k0)4 (g2 p,+ g2') (a k) — Pi k0} {ak ei (kx) + ak e.' (kx)}

+E(N4+1) (51)

On the Scattering of Scalar Mesons 145

where k0 = + '/x2 + 1k I '', Nk is the number of mesons in the momentumstate k and ak and ax are the corresponding absorption and emission opera-tors respectively.

[ak, a]_ = 1As usual V is a large volume in which the field quantities are periodic.

Corresponding to the treatment of §2 we put g, = g 1 ' = 0 and regardthe neutron as an infinitely heavy particle (µ moo). A straightforwardcalculation shows that the scattering due to a pure ` scalar ' dipole-inter-action (g2 ' = 0) is zero in the usual second order (g2) approximation. Thisresult holds even when the calculation is performed taking into account thefinite mass of the neutron. Also the terms containing both g 2 and g2 'vanish for µ -->oo, so that in this limit the contribution to the scatteringcomes purely from the pseudo-scalar interaction. The differential scatteringcross-section is thus found to be for p -->-oo

,tQ= 4 dQ (5;;;)4

E2 sin 2 (52)

where p is the momentum, E the energy and tj the angle of scattering ofthe meson. The total cross-section is therefore

' `P2 (53)33̂ r( 82/ ) Ez

g" is the value of the dipole-moment in the usual units as against its value

g2' in the 'Heaviside-units' which were employed till now. On putting

I = (_ ? since h is put equal to 1) and averaging over all 0, (48) becomes

identical with (53) except for a factor 3. This discrepancy is due to thewell-known (cf. Bhabha, 1941, footnote on page 340; also see Bhabha andMadhava Rao, 1941) difference in the classical and the quantum averageover the direction of the spin of the neutron. In fact (52) also agrees to thesame extent with the corresponding differential cross-section obtained bydividing (43) by (45), expanding in powers of g 2 and retaining only the lowestterm.

I am thankful to Prof. H. J. Bhabha for his criticism and advice.

SUMMARY

The classical formula for the scattering of scalar mesons by a neutronare obtained taking account of the radiation damping. The neutron isassumed to possess a 'charge' and a `dipole moment'. The scatteringdue to each of these is treated separately. It is found that the formulae for

146

Harish-Chandra

the scattering due to the dipole has exactly the same form as the one obtainedby Bhabha for the transverse mesons. Due to numerical factors the scatter-ing for large energies of the incident mesons is double, and for small energieshalf that of transverse vector-mesons.

The scalar and pseudo-scalar charge and dipole interactions are con-sidered in the quantum theory. The scalar dipole interaction does not giverise to any scattering at all, the whole of the scattering being due to thepseudo-scalar interaction. In this case the quantum-theoretical formula'agree with the corresponding classical ones if the effect of radiation reactionis neglected in the latter.

REFERENCES

Bhabha .. Proc. Roy. Soc. A, 1939, 172, 384-409.

----- Ibid., 1941, 178, 314-50.and Harish-Chandra .. Ibid., 1944, in course of publication.and Madhava Rao .. Proc. Ind. Acad. Sci. A, 1941, 13, 9-24.

Dirac .. Proc. Roy. Soc. A, 1938, 167, 148-69.Harish-Chandra .. Ibid., 1944, in course of publication,

referred to as A in this paper.