P H Y S I C A L R E V I E W - V O L U M E 1 1 1 , N U M B E R 4 A U G U S T 1 5 , 1 9 5 8

Mass Effect in P-State n-p Scattering* HONG-YEE CHIU

Laboratory of Nuclear Studies, Cornell University, Ithaca, New York (Received February 27, 1958; revised manuscript received May 26, 1958)

The mass difference between charged and neutral mesons has been correlated to a change of the pseudovector coupling constant, giving rise to an additional term in the interaction Hamiltonian. The mass effect in P-state iT-p scattering is then calculated, using Chew's fixed-source theory. A correction to the usual scattering formula for the ir~-p system is obtained. This correction fails to explain the quantitative feature of the observed difference (5i3—631).

I. INTRODUCTION

IN view of several accurate pion-nucleon scattering experiments,1,2 it is interesting to compare in a more

detailed manner the predictions made in Chew's theory and results of experiments. Such comparison is best done by analyzing experimental data into phase shifts and then comparing with those predicted by theory. Several investigations3-4 on this line have been made before the above-mentioned accurate experiments, but, because of poor experimental ac­curacy, no information about small P-wave phase shifts could be drawn. Recently several attempts5 ,6 have been made to obtain some information about the small P-wave phase shifts.

One of the investigations6 leads to a result which contradicts one of the main predictions made in Chew's theory, namely, the equality 5i3=53i. Experimentally, 53i is small and negative, and has a smooth behavior (its magnitude at 150 Mev is about —5°), but 8u is positive and rises rapidly from 5° at 150 Mev to attain a value of about 15° above the resonance. This deviation is important, because the positive value of 8n may fit in with the suggestion that the T = J , / = § state shows a resonance at higher energies.7 A resonance around 750 Mev has been observed,8 but its nature is still not clear.

Possible sources of deviation from <5i3=63i may be classified according to whether (1) they do or (2) do not conserve isotopic spin. The former include meson exchange terms, nucleon recoil,9 possible virtual processes involving strange particles, and isobaric states of the nucleon (if any exist). The second type

* Supported in part by the joint program of the Office of the Naval Research and the U. S. Atomic Energy Commission.

1 Ashkin, Blaser, Feiner, and Stern, Phys. Rev. 101,1149 (1956). 2 Ashkin, Blaser, Feiner, and Stern, Phys. Rev. 105, 724 (1957), 3 Anderson, Fermi, Martin, and Nagle, Phys. Rev. 91, 155

(1953). 4 De Hoffmann, Metropolis, Alei, and Bethe, Phys. Rev. 95,

1586 (1954). 5 Gabrielli, Iernetti, Clementel, and Villi, Suppl. Nuovo

cimento 3, 496 (1956). 6 H. Y. Chiu and E. L. Lomon (to be published). 7 R. R. Wilson, Phys. Rev. 110, 1212 (1958). 8 Cool, Piccioni, and Clark, Phys. Rev. 103, 1082 (1956). 9 S. F. Edwards and P. T. Matthews (private communication).

Their value for 5i3—5Si due to nucleon recoil is about 1° at 150 Mev.

includes the electromagnetic interaction (the direct Coulomb scattering of charged pions by the proton), the radiative capture of TT~ by a proton to form a neutron and a 7 ray (Panofsky effect), the charged- and neutral-pion mass difference that appears both in the kinetic-energy term of the total Hamiltonian and in the interaction Hamiltonian as a modification to the pseudovector coupling constant (this will be explained in Sec. I I ) . Sources of deviation in category (1) presum­ably have a greater effect on ir-p scattering, but they are beyond the scope of Chew's theory and at present we do not have reliable methods to treat them. Among the sources of deviation in category (2), the direct Coulomb scattering has been subtracted out of the experimental results to give the "nuclear" scattering,1,2

and can be disregarded. The radiative capture of a pion by a nucleon and the 7^—T° mass difference that appears in the kinetic-energy term of the total Hamil­tonian make a large contribution to the low-energy scattering. Their effects on 5-state scattering have been discussed in detail by Noyes.10 In the intermediate energy range, i.e., pions of 150-Mev kinetic-energy, they are not expected to be important. For example, the mass difference term can be written as

, J ( M ^ - A M W I « £ , * ( * ) ! . (1)

L ZEXk)J J

At 140-Mev kinetic energy, i.e., £^=280 Mev=2/x, the deviation of E** from E^ is about 2.5 Mev. This difference would not produce a noticeable effect. The part of the mass difference that causes a modifica­tion in the pseudovector coupling constant has not been investigated before and it is the intention of this paper to discuss its effect in P-state scattering. In out treatment we shall limit ourselves to the scope of Chew's fixed-source theory.

In Sec. I I we shall derive an interaction Hamiltonian and in Sec. I l l we shall perform calculations based on this interaction Hamiltonian and shall discuss our results in Sec. IV.

10 H. P. Noyes, Phys. Rev. 101, 320 (1955).

1170

M A S S E F F E C T I N P - S T A T E w-p S C A T T E R I N G 1171

IL GENERAL CONSIDERATIONS

According to the equivalence theorem, the PV coupling constant / is connected to the PS coupling constant g by the relation

/ = (Mr/2M)g. (2)

If we believe that the mesonic charge g is fundamental, then / must assume different values for processes involving w± and 7r°. The PS interaction is renormaliz-able whereas the PV interaction is not; therefore it seems reasonable to assume that g is fundamental. With Eq. (2) and with Kemmer's form of interaction Hamiltonian, we write the interaction Hamiltonian as

Hi^ fHt$Ti(<r • grad)<l>jfr+aff>Tz(<r • g r a d ) ^ , (3)

where the constant a is (^0—jU7r±)/yu^-±= — 0.03. In

Eq. (3) we have neglected the mass difference between proton and neutron, because it corresponds to a change in a of about 1/30 of 0.03. Though the total isotopic spin is not a good quantum number, its third component MT is still conserved. The additional term in the inter­action Hamiltonian introduces an MT dependence into the phase shifts as well as giving a transition between states having different T values and the same MT. The usual meaning of the six pure (T,J) phase shifts can still be retained, if the mixing between states with different T is small, and if the scattering formula is properly modified. This will be shown in the Appendix.

We shall use perturbation methods in our calculations. This is because of the small size of the mass-dependent term. We use the same wave functions as in Chew's treatment with the mass-dependent term neglected. This has the consequence that the renormalization of the coupling constant will not be altered. This is expected in perturbation theory because the modifica­tion of wave functions can appear only in second- or higher-order approximations.

III. CALCULATIONS

In the following we follow the notation of Wick.11

I t is easily seen that the Low equation is still justified in our case. The "zero energy'' limit of the scattering amplitude is the renormalized Born approximation. Following Wick's treatment, the Born term can be written as

M=M!+M2,

Mx= 27r(//M)2co-Ux(r cr) (p-tr)\ v(p) \ \ (4)

J f 8 = -2Tr(f/»)2co-2Q^(p-<r)(q-<0 \v(q) \\

Q=l-(l+a)t-T+akT3+I,

g = l + ( l - a ) * - r - a / , T 3 + 7 , (5)

[0 0 0 1= 0 0 0

(0 0 2aj

where

The matrix I will disappear in M because of cancellation due to subtraction. We shall be mainly interested in the off-diagonal elements in the TMT representation for hrs, which can be worked out by using the wave functions for the total iso topic spin. Such wave functions have been listed in a number of papers.12 The diagonal elements will give a "fine structure" type of splitting in the phase shifts which cannot be detected by present experimental techniques.

The values f or (f, — § | hn | J, — §) and (f, \ \ hrz \\,\) are — |V2 and |V2, respectively. (Here the notation (T, MTI hr* I T', MT') is used.)

In the Born approximation, the diagonal elements including the "fine structure" splitting for 5i3 and 631 are —2— 8a/3 and — 2 + 4 # / 3 , respectively. Since a is negative, the Born approximation we have 8n~5n positive but small.

The transition amplitude from the ap to the f3q state will be written as11

Rp(q)=<qP-\vp\ap) = -h\jv(p)v(q)/(tf<*PW)y*. (6)

To obtain r^ for the transition from J T = § to T—\ in the same MT state, we use the Chew-Low type of integral equation:

<73 r r^x (<aq) = M^ (coQ) -) I

7T J,,

' v2(k)do)k

¥

X (rV)x

\ C0k~C0q~irj COyt + WgJ (7)

also, rMx= (fx/i)*. Defining (f, • 2 / — C ( 2 , 2 y

— J), we have

c=<!,-i|Af|i,-4>+-f v2(k)da>k

¥

ra*(|)C+C*a(i) XI -+crossed terms , (8)

L o)k~coq—ir) J

where a (T) is the transition amplitude for the pure T states. If we neglect the crossed terms and the terms small in comparison with the T = f term, we have

C=<!,-i |Jf |i-4)+-f f /•« tf{k)dm a*(f)C

-• (9) 0)k~0)q~t7J

C gives ( | , —§|r|f, —|) as well as (§, —i |f | | , —J). This can be seen as follows: (f, -§> = Her-mitian conjugate of (f, — §|f|f, —J) with respect to variables other than isotopic spin. Also, C is Hermitian in this sense (see Appendix), therefore (§, —J|f | J, —J)

= < * , - 1 _ i \ 3? 2 / -

Ji G. C. Wick, Revs. Modern Phys. 27, 339 (1955).

12 E.g., H. A. Bethe and F. de Hoffmann, Meson and Fields (Row, Peterson and Company, Evanston, 1955), Vol. II, p. 60.

1172 H O N G - Y E E C H I U

Now we split C into the 7 = f and the / = § states and use subscripts 3 and 1 to denote the / = f and the / = J states, respectively. The resulting integral equation for C3 is

C3— (§> " " J l ^ l i j ~~h)s

+± r 0*(*)dcO* g33*C3

& (10)

cofc — c o 2 — # 7

where g33=sin533exp(i533). A similar integral equation for Ci can also be derived, but Ci is small and we shall ignore it. I t is easily seen that (f, —- i |M| i , —\)z will differ from the Born approximation term for #33 by a real constant multiplier, B. Writing

s(h -h\M\h -1)3= <§, -m\h -*>, (ID

and multiplying both sides of Eq. (10) by B, we have

BCz={h-h\M\^-h)z

+ If •*(k)da„ g3s*(BC3)

¥ (12)

a)k—o)q—iy

The integral equation for g33 is

£33= (§ < i - 4 1 ^ 1 1 , - * > «

+-3 - c > V2(k)d0)k g33*#33

F , (13)

m—Uq—tv

If we regard #33* as a known kernel, then Eq. (12) and Eq. (13) should yield the same solution. Therefore the solution for C3 is just

C s = (1/B)gi9. (14)

The value of B has been calculated as

l / £ = i a v 2 = -0 .008 .

Thus the scattering formulas for irp-^irp and irp ir°n are modified:

(ir p\w p)

= iC0.98fl(t)P|+a(f)5>Pi+2a(i)5,Pi fp i],

(TT-^ITT0^) (15)

= t^ [0 .99a( f )p f +f l ( t )5 ,P i -a ( i )5 i Pt ,p l ] .

In the above formula, the usual definition of S and P phase shifts is still retained. Justification of the use of pure T-state phase shifts will be found in the Appendix.

IV. DISCUSSION OF RESULTS

Due to the mass effect, near the T=J=% resonance the value of 8U as obtained experimentally through the original scattering formula12 will be changed by about 0.01 (in radians). This is because, e.g., the first of Eqs. (15) can be obtained from the original formula on

replacing ai3 by [ai3— (O.Ol)^] , where OUJ= £exp(2idlj) — 1]. Despite the difference in phases between an and 0:33, the change of $13 will still be of the order of 0.01. The experimental value of 8u will be raised by this amount before resonance, and will be suppressed by this amount after resonance (in the case w~p—>7r~p), because 0:33 changes its phase at resonance. However, this does not explain the rapid rise of 613. The contribu­tion to dn—8n comes mainly from the Born term and amounts to about +0 .7° at 150 Mev. This is certainly too small. I t may be concluded that the mass difference is not the real cause of the discrepancy 5u7*5n. Of course the relation / = (jir/2M)g may not hold. How­ever, if this is the case, some other serious consequence for the foundations of field theory would result. We do not believe so; therefore it is not likely that the 5i3?^3i discrepancy can be explained within the framework of the Chew-Low theory.f

ACKNOWLEDGMENTS

I wish to express my gratitude to Professor H. A. Bethe, Professor J. Hamilton, and Professor E. E. Salpeter for many helpful discussions and suggestions.

APPENDIX

We shall now justify the use of pure T and pure / phase shifts, for a small mixing between different T states, assuming that the scattering formulas are written as in (15). This is valid so long as only one state is large while the other states are small.

The relation C (f,|; — J) = C (J,f; — J) guarantees the time reversibility condition. The most general form of the 2X2 scattering matrix which satisfies both unitarity and time reversibility can be written as13

(cose — sine\ /e2iA* 0 \ / cose s ine\

)( , ) ( I- <16) sine cose/ V 0 e2lAl/ V —sine cose/

where e is the mixing parameter which determines the amount of mixing between the T ,= f and the T=^ states, and A3 and Ai are matrices in / space and are so defined that in the limit e—»0, one has A3—•»5(J'=f) and Ar -»5( r=f ) . Of course, e must also be written as a matrix in / space. If only one pure / state is large we may put e = 0 for all states except the large state.

After carrying out the matrix multiplication in (16), we have

Vsi:

euA3 COs2e+e2iAl sin2e sine cose(e2iA3- e2iAl)

) • sine cose(e2iA3— e2^1) e2iAl cos2€+e2<A» sin2e

(16') t Note added in proof.—Recent results of reference 6 shows that

5i3 has a better behavior at 220 Mev ( ~ + 4 ° ) . Thus the statement made in the last sentence of Sec. 4 is not strictly valid.

13 J. M. Blatt and L. Biedenharn, Phys. Rev. 86, 399 (1952), and 93, 1387 (1954).

M A S S E F F E C T I N P - S T A T E ir-p S C A T T E R I N G 1173

In the limit e—>0, the diagonal elements should tend to the pure-state phase shifts. In fact, if we put sine= e, and neglect terms containing e2, the foregoing form of the S matrix reduces to the form from which Eq. (15) was derived, i.e.,

/ e x p [ > ' 5 ( r = f ) ] { e X p [ 2 i 8 ( r = t ) ] - l } 6 \

\{exp[2 iS(Z , =f) ]~ l}6 exp[2i8(T=i)] /

(17)

I. INTRODUCTION

THE polarization of high-energy protons elastically scattered from nuclei has been calculated by

many authors.1 Most of these calculations are, however, based on phenomenological potentials between incident protons and target nuclei as a whole. Therefore, it is rather difficult to relate their results to individual nucleon-nucleon interactions. Optical-model potentials directly connected to nucleon-nucleon scattering phase shifts have first been studied by Riesenfeld and Watson2

and, more recently, by Bethe3 in estimating the proton polarizations by carbon. Riesenfeld and Watson used the phase shifts derived by Feshbach and Lomon4 and compared the calculated values of the polarization at © = 20°, where © is the scattering angle in the labora­tory system, with experimental data. Bethe evaluated

* This research was supported by the U. S. Atomic Energy Commission and by the Office of Ordinance Research, U. S. Army.

1 See, for example, E. Heiberg, Phys. Rev. 106, 1271 (1957) for detailed references.

2 W. B. Riesenfeld and K. M. Watson, Phys. Rev. 102, 1157 (1956). This paper will henceforth be referred to as RW.

3 H. A. Bethe, Ann. Phys. 3, 190 (1958). 4 H. Feshbach and E. Lomon, Phys. Rev. 102, 891 (1956). This

work will be referred to as FL.

From Eq. (17) and Eq. (14) we also derive e=l/2B = —0.004. The square of e can really be neglected. Eq. (17) also verifies the relation between C3 and g33 which is contained in Eq. (14).

I t may be remarked that the unitarity condition of Eq. (17) is destroyed because of the presence of off-diagonal terms while the diagonal elements are still unimodular. This can be restored by using a constant multiplier in front of Eq. (17); this constant is close to 1.

the scattering cross sections as well as the polarization at £ = 3 1 0 Mev, with E standing for the kinetic energy of incident protons in the laboratory system. He used five sets of phase shifts by Stapp, Ypsilantis, and Metropolis5 for those states with isotopic spin T=l together with phase shifts for those with T= 0 that have been computed by Gammel and Thaler6 from their potential. His conclusion is that it is difficult to dis­criminate between those sets for T=l from his results.

In the present paper, the polarization of high-energy protons with energies E between 90 and 310 Mev scattered from carbon has been calculated as a function of scattering angles ©<20° . Also, the triple-scattering parameter (3 defined by Wolfenstein7 has been estimated at £ ~ 3 0 0 Mev. In evaluating the transition matrix element which has been derived in RW, nucleon-nucleon phase shifts of Signell and Marshak,8 and of Gammel and Thaler6 have been considered. Although far from being the final answer to the problem of nuclear forces,

5 Stapp, Ypsilantis, and Metropolis, Phys. Rev. 105, 302 (1957). 6 J. L. Gammel and R. M. Thaler, Phys. Rev. 107, 291 (1957);

107, 1337 (1957). This work will be referred to as GT. 7 L . Wolfenstein, Phys. Rev. 96, 1654 (1954); 98, 1870 (1955). 8 P. S. Signell and R. E. Marshak, Phys. Rev. 106, 832 (1957);

109, 1229 (1958). This work will be referred to as SM.

P H Y S I C A L R E V I E W V O L U M E 1 1 1 , N U M B E R 4 A U G U S T 1 5 , 1 9 5 8

Nucleon-Nucleon Interactions and Polarization of High-Energy Protons Elastically Scattered from Carbon*

SHOROKU OHNUMA Sloane Physics Laboratory, Yale University, New Haven, Connecticut

(Received April 28, 1958)

The transition matrix element in momentum space derived by Riesenfeld and Watson has been used to calculate the polarization and the triple-scattering parameter /? of high-energy protons elastically scattered at small angles from carbon. As the nucleon-nucleon phase shifts which represent two-body interactions, those by Signell and Marshak, by Gammel and Thaler, and by Feshbach and Lomon have been considered. In evaluating nuclear as well as Coulomb scattering amplitudes, the first Born approximation has been employed with the assumption that, in carbon, the distribution of protons is equal to that of neutrons. Final results are then independent of the assumed distribution of nucleons. It has been found that, while one cannot discriminate between Signell-Marshak and Gammel-Thaler phase shifts, both of them being in semi­quantitative agreement with experimental data, Feshbach-Lomon phase shifts may be ruled out because of the wrong sign of the resulting /?. Since only the first-order transition matrix element in momentum space has been used in the present work, the calculation does not depend on the optical model potential in the usual sense.