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Proton-proton scattering with parity and time-reversal noninvariance
Proton-Proton Scattering with Parity Time-Reversal Noninvariance” A. Department
of Physics
E.
and
WOODRUFF
a.4 Astronom,y, Rochester, New
University Pork
of Rochester,
This paper develops a formalism to treat some of the effects of a hypothetical breakdown of parity and time-reversal invariance in the proton-proton interaction. The first two sections show how the usual parametrization of the S-matrix (in terms of phase shifts and mixing parameters) must be extended to include a description of such violations. The third se&ion discusses some of the experiments which could detect these violations, if they existed, and the final part extends the general form of the p-p pot,ential. An appendix apphes the formalism to two experiments testing time-reversal invariance. INTR0DUCTTON
Present experimental results indicate that’ neither parity (P) nor time-reversal (T) invariance is violated to a substantial degree in interactions between nucleans at moderate energies. For parity violation in nucleon-nucleon scattering processes the characteristic upper limit is given by F2 5 lop3 (1)’ (F2 being the square of the ratio of the violating to nonviolating amplitudes), while nuclear processes yield F’ 5 3 X IO-* (2) Ia; for time-reversal noninvariance the corresponding upper limit is given by F” 5 5 X 10e3 (see Section III). The present analysis was undertaken partly in conjunction with one of the latter exprriments. From the t,heoretical point. of view, it is important that, experiment.s searching for a possible breakdown of parity in strong interactions be performed with as great an accuracy as possible and at higher energies than hitherto. It is known that parity breaks down in weak interactions, whereas T is apparent,ly good, so that to some small estent (via virtual processes) all int,eractions must show some parit,y breakdown. Even if the nucleon current interacted directly with itself in weak interactions, this would yield a very small F* - g2 - lo-“. On the other * This work was 1 In connection la Note added in R. Almyvist have Rell., to be published).
supported in part by the U. S. At,omic Energy Commission. with these experiments see the discllssion in Heer et al. (1). proof: D. A. Bromley, H. E. Gove, J. A. Kuehner, A. E. Litherland obtained an upper limit of I I’* -< 2 X 10-O in nuclear reactions 65
and (Phys.
AA
WOODRUFF
TAULI!: TRANSFORMATION
PROPERTIES
P
-P
ii P
1;
I OF
\‘E(
1’0~s
1s
;I[
-P’ -P
-P -P’ -K
K
-P -n
-P
n dl
n dl
d?
dz
-P n
-dl
dz
-d2
6
hand, Soloviev and Gupta (3) have pointed out t,hat one could simply assume universal invariance under the product CP (C is charge conjugation), or, using the TCP theorem, T, and obtain parity-conserving electromagnetic and nucleonpion elementary interactions as a consequence of other principles (gauge invariance and charge independence). According to this hypothesis, parity violations would appear in strong interactions involving the strange. particles, and, via virtual processes, in nucleon-nucleon interactions. If the virtual strange particles appear mainly in the core region of the nucleons, the breakdown of parity in nucleon-nucleon interactions would be small at low energies and become more pronounced at higher energies. In the following sectlions we show how additional parameters may be introduced to allow for the possibility of a breakdown of parity and/or time-reversal invariance in the proton-proton interaction, and how experimental quantities may be related to these additional parameters. I. GENERAL
FORM
OF
THE
TRANSITION
MATRIX*
In this section we give the general form of the proton-proton transition matrix in the msl , ms, representation; that is, in terms of the Pauli spin matrices for the two protons. This form is a useful starting point for obtaining the expressions which correspond to various experiments, example of which will be given in Section III. Let p, p’ be the incoming and outgoing relative momenta of the proton-proton system in the center-of-mass frame and define the unit vectors K, P, n, wit)h directions : K = p’ - p
P=p’+p
(1)
n=PxK.
K, P, and n are mutually orthogonal. Under space reflection (P), time reversal (T), and exchange of the two particles (X), these vect’ors and the Pauli spin mat,rices have the transformation properties shown in Table I. 2 The derivation of the parity and 6ime-reversal by Wolfenstein and Ashkin (4) nnd Dalitz (4).
invariant,
transition
matrix
was given
SCATTERING
WITH
P
T
AND
67
NONINVARIANCE
The general expression for the transition matrix, not invariant under P or T, but invariant under X, may be constructed readily by reference to this table, forming all possible scalars and pseudoscalars which are invariant under X. It may be written in the form M = B(B,E)s + C(e,E)(dl
+ ds).n
+>;G(B,E)(dl.Kdz.K
+ &.Pdz.P)J
+>$3(O,E)(dl.Kdz.K
- dl.Pdz.P)5
+N(B,E)dl.ndz.n5
(2)
+P(B,E)(dl
- &)*K + Q(B,E)dl
X dz-P
+R(B,E’)(d,
-
X dz*K
&).P
+ S(@)&
+T(B,E)(dl.Kde.P
+ dl.Pdz.K)Z,
where B( e,E) through T(tV,E) are ten complex functions of scattering angle e = 6 .$ and energy E = p’,f’ilf = pf2/M, and where S and ci are the singlet and triplet spin projection operators, respectively: S
= x(1
-
dl.dz)
3 =
!:;(3
+
dl.dz).
,4n equivalent form, more useful for calculating t,races when the total spins of the initial and final states are not specified, is obtained by inserting the above expressions for s and 3 in (2) and simplifying: dl = +;(I? + G + N) + +),i(
-B
+
G
-
-
d2)
N)(d,.Kdz.K
+$$H(dl.Kdz.K
+34(--B
+
C(dl
+
+P(&
-
62) .K + &dl
X 6z.P
+R(dl
-
dz).P
X
+T(dl.Kdz.P
dl.Pd2.P)
dl.Pd2.P)
G + 3N)dl.ndz.n
+
.n
8dl
(3)
d2.K
+ dl.Pd2.K).
The terms containing B, C, G, H, and N are invariant both under P and under T; R and S are invariant under T, but’ not under P; T is invariant under P, but not T; and P and Q are invariant under neither P nor T. Since violations of p and T by the above terms are always complete (involving n change in sign), the behavior of the terms under charge conjugation G follows at once from the TGP theorem. The six more possible independent matrices in the ms,msz representation are forbidden by invariance under X (identity of particles).
68
WOODRUFF
I
I I I
I I I FIG.
II.
PARAMETRIZATION
OF
1 THE
SCATTERING
MATRIX
This section contains the equations relating the functions B(B,E) through T( B,E) of the previous section to the usual partial-wave phase shifts and mixing parameters, and to the new parameters which must be introduced to describe the proton-proton scattering processeswhen P and/or T invariance is violated. First we introduce these new parameters [for time-reversal noninvariance this has been done by Phillips (5)]; next, we express the transition matrix in the s, m, representation (where s is the total spin) in terms of the phase shifts and other parameters in a partial wave expansion; finally, the functions B through T are related to the components of the transition matrix in the s, m, representation. The proton-proton system can exist in the states shown in Fig. 1. If P and T are conserved, transitions can occur between the same states and between triplet states with the same total angular momentum (heavy arrows). The mat,rix cx= X-XCOUl. in the 1s j representation is diagonal in I, S, and j except for these coupled triplet stat,es, in which case it is a 2 X 2 matrix in 1. Using the BlattBiedenharn conventions (6)) 01is writt,en as: singlet states: al=9
2iSl -e 2+2,
uncoupled triplet states: cilj = e2i62--e %l
(I = .i).
coupled triplet states:
&
=
$5 si*
2Ej(e2isi-l.i
_
e2’6’-L
)
(4)
SC-4TTERING
WITH
P
T
AND
69
NONINVARIANCE
The 61 and 61,j are the Blatt-Biedenharn phase shifts, Biedenharn mixing parameters. cpzis defined by
and the ej the Blatt-
2
pr = C tan-’ r=l
n 0x ’
where n q : e2/hv, v being the speed of the incoming proton in the laboratory. The matrix (Y is symmetrical due to the invariance under T. Phillips (5) has shown how to parametrize the LYmatrix if T is not a good operator but P is. The unitarity of X restricts cx to the same form as above except that S and CYare no longer symmetrical:
a+‘=ae
j
2iXj
,
CL’ =ae
j -2iXi
.
(5)
One new parameter, Xj , a function of energy but not of scattering angle, just as the phase shifts and mixing parameters are, must be introduced for each set of coupled triplet st.ates [just as one new function, T( B,E), must be introduced in the transition matrix]. If parity is not conserved, the transitions indicated by the dashed arrows in Fig. 1 can occur. This introduces two new mixing parameters, lj and qi , for each coupled set of two triplet and one singlet states; if T is not good for t,hese transitions, two more parameters, Xlj and Xsj , must be included. Since each pair of coupled triplet states is now coupled to a singlet state, a 3 X 3 matrix is required for each set. The S mat.ris for such a set may be expressed in terms of a Hermitian matrix Q by3 8”
=
e2iQJ
(6)
since Ls1.lis unitary. The phase shifts of t,he three coupled states are defined to be the eigenvalues of Q”, SO that &” = UJ-‘A,U.,
,
(7)
where
1,
A, = 3 This J-value;
argument see Ref.
follows 6.
the usual
method
for treating
the coupled
(8) triplet
states
of a single
70
WOODRUFF
and the form of the unitary
mat’rix U., is chosen to be:
U., = B,C.,D, sin 17eix, cos 71
cos q -sin 9 e8’”
B =
1 C= 1) ’
(
cos e --iA -sin E e
(
cos [ -sin remix’
D =
sin E eZXz , cos E .) sin teix’ cos {
(9)
11 . That this form for UJ is sufficiently general follows from the facts that Q” contains 9 = 3’ real parameters, and that its components are linearly independent of one another, and so Q” defined by (7)-(g) is an arbitrary Hermitian matrix. Using the notation (
which corresponds we obtain
to what we had earlier, with the addition
of the CY*’ and a*“,
aj = [cos’ p cos” q + cos’ e sin’ { sin’ 9 +[cos’
$5 cos E sin 2{ sin 27 cos(Xr - X3)]e2’6j
[ sin’ 7j + co? e sin2 { cos2 v + 95 cos e sin 2{ sin 271cos(X1 - X3)]e2”6j-‘~j
+ sin2 E sin2 { e2i6i+Lj _ e2iPj, Cyj-1,j
= [sin’ [
v +
C0S2
COS’
c CO2
{ sin2 f
+ 55 cos E sin 2{ sin 27 cos(X1 - X3)]e2”*’ +[sin’
1 sin’ q + cos” e cos2 [ cos2 f
- $5 cos E sin 25- sin 2~ cos(X1 - X3)]e2’*j-‘~j + sin” e cosz { e2i6i+Li _ e2icPi-1, “j+1,j cY+j
= sin2 e sin2 q e2iai + sin2 e cos2 q =
e2igi-l,j
+
cosz
E e2iG+l.i
_
e2iri+l,
[$$j sin E sin { sin 27j e--i(x1--x3) + lh sin 2~ cos c sin” g]e”2e2i6j + [ - 35 sin e sin { sin 27 e-io--x3) + 35 sin 2~ cos l cos2 q]eix2e2i6j-1*i -
lh
sin
ze
~0s
{ eixzeziaj+Lj,
SCATTERISG
WITH
cYdJ = same as c,+j with a+
fi
P
Ah3
T
71
NONINVARIAKCE
signs of X’s reversed,
(11)
= [>h sin 2{ cos’ 7 - $6 cos 6 sin’ { sin 2~ ei(hl-Xp) + $4 cos c cos’ j- sin 27 e-i(x1--x3) -
$1 ~0s’ c sin 2{ sin’ ~]czh1cz2*J
+ [f,$ sin 2[ sin’ 7 + ,1$ co.3 E sin’ j- sin 29 ei’x’-x3) - f,i ~0s 6 cos2 j- sin 211p-i(h1-A3) - ,$s ~0s’ c sin t4 ~0s’ q]eiX1e2i6’-1.~ &‘i
$5 sill2 E sin &- eiXle2iai+Lj,
= same as CY+‘~with
signs of X’s reversed,
a+ 113== [pi sin E cos j- sin 217eTAa- >i sill 2~ sill { sin” q e”x’]eix2e2’6j + [ -44
sin E cos j+sin 2~ eix3 - $5 sin 26 sin j- ~0s’ 11eixl]eixz$iai-l~i
+ 44 sill zE sill { ~iAleax~e2i6f+~~i, a!- ‘I3 == same as a+”
with
signs of X’S reversed.
where the subscript j has been dropped from the parameters E, r, 7, x1 , x2 , and Xa . If the amount of parity violation is small, { and r] are small. Keeping terms linear in j- and 7, the above relations become approximately crj = $iaj _ e2iqj, CYj+l,J
=
COS”
E e2i6i+laj +
sin2
cY+J = $5 sin zE e+x2(e2i6j-1,i
a+‘li
tj
=
25- +
J&sin
=
cofj
_
E sin
>h[cos
e sin
217e--i(A1-x3)
_
$,$ sin”
E sin
21
t sill
2,,
_ e2iPi-l,
e2iSi+l.i),
+
~0s’
(12) &-]eix~eziSi-l,i
e%+h)e2iJj
$$[sin E sin 27 eih3 + sin ze
+
~5 sin
sin
~ sin
eiAlezi6i+l,i,
-
ze
e2i(oj+l,
2.q e-i(A1-A3)]e’A1e2iai
-
$4 sill
_
+ COSee e*%-l,l
ocj-1.j = sin* E ezi6j+l,i
a+
E e2i6j-l,j
sin
* e’Al]eiA2e2i6j-l,~
1 ei(A1+We2i6i+l,ja
Thus, to terms linear in { and q, the parity-conserving terms are the same as before C(4) and (5), with 12 = 2X]. From the experiments on time-reversal invariance, we expect X2 also to be small, but nothing can be said about X1 and x3 , which represent the relative amount of T-violation in the parity-violating transitions.
72
WOODRUFF
The problem of expressing the transition matrix M in the s, m, representation in terms of the parameters (Y is equivalent to finding the relation between the s, m, representation and sums over the 1, s, j, rnf representation, properly weighted by means of Clebsch-Gordan coefficients. An outline of the method of derivat,ion is given in Ref. 7. If T is not good, Table III of Stapp et al. (7) is still correct, if the changes (Y1+1 + ff+ ‘+’ and 01’~’+ a- ‘-’ are made. The complete set of relntions, including the parity-violating transitions, follows:
- ; d(Z + l)(Z + 2) O+l+l - f 2/(Z - 1)Z -‘-‘1)
M1-l(e,p)
= 2(ik)-‘P~
+ii
az,z-1
-
.gl pl”(e>
41 2/u
-
+ l)(l 1
21 + 1 ’ 4(Z + 1) az91+1 4Z(Z + 1) az*z i + 2) a+ z+* _ 41/ma1 1
Z-l
1’
SCATTERISG
nf*ll(e,$o) = - i(ik)-’
WITH
P
AND
c~lPLm42z+
I’
73
NOiYIiYVARIISCE
1)(1+
1) Q+NL + d(2Z
M,,(B,p)
= i(ik)-l
.g,
Pl(c9>{~(2
+ 1)(21 + 3) cL’L+l + 1/1(2Z -
-
M-i-j(l9,'p)
=
( -)i-jilI,j(f3j
flI+j(O,p>
=
(-
flf-is(e,p>
= (- )iMi,(e,
+ 1)Z Ly+‘ll,
nJ
1) OIP--I},
(2Z - 1) 21
a-l'z-l
‘,
7
-Cp),
)jAlsj(e,
-cp),
-P>.
The indices i and j refer t’o the final and initial t,riplet states, and s refers to the singlet state. along the dir&ion of t’he incoming momentum, of spherical coordinates with angles (O,(p). .f?( 0) tude,
.fde) = --n k( 1 -
cos
--in e)
values, respectively, of m, in t,he The direction of qunntization is which is t’he polar axis of the set is the Coulomb scatt,ering umpli-
In [$(l-
co9 e)]
'
The relations between the functions B - T in t,hc m,, , ‘~1,~representation of Ill and the functions N;j in the s, m, representations are readily found by using the idemities
Here Gi is the i component
#i*jt
= $iS”Sij - Sj*S; ,
#&it
= sA;*,
of the triplet
(14)
spin function,
and fiS the singlet spin
74
WOODRUFF
function; S = $s(di + I$), A = fs(& operator. We obtain
- &), and S is the singlet spin projection
(15) R = ‘4 (AI,, + M,,) cos; - &
T = : (ill,,, - AI,, + M-,)
i
(Ml, + Msl) sin f ,
sin e + &
(MUI + MO,) cos e
It is to be noted that there is a correspondence between the number of functions in the 1cI matrix and the number of independent parameters at a given allowed j-value. III.
PARITY
AND
TIRIE-REVERSAI,
VIOLATIO-lj
IXPERIhlENTS
If there were a breakdown of parity and/or time-reversal invariance in strong interactions, certain symmetries would remain in the proton-proton experimental functions, due to the identity of the protons and to invariance under rotations. These include, of course, that the cross section should be an even function of angle about 90” c.m., and also that the polarization (and t’he asymmetry produced by polarized protons) should be odd about 90” c.m. However, there are many experiment’s, someof which will be discussedin t,his section, which can give clear evidence for an existing parity or time-reversal violation. The parametrization of the transition matrix developed in the previous sections can be used to determine the angular dependence associated, in any given experiment, with the existence of a violating parameter, and to determine the relative sensitivity of various experiments in the search for such a violation. If me assumethat such violations are small, as is evidenced by various experiments (1, 8) ranging up to around 300 Mev, we may express violation-seeking experi-
SCATTERING
WITH
P
AK‘D
I’
75
NONINVARIANCE
ments by means of terms linear in {, q, and X2 , using the formalism of Sections I and II. The coefficients of these terms will contain the ordinary parameters, i.e., the phase shifts 6 and mixing parameters c In this approximation one may use for the ordinary parameters the values which have been obtained by a direct phaseshift analysis of the ordinary (single, double, etc.) scattering data, or from a potential chosen to fit the data, such as those of Signell and Marshak (9), or Gammel and Thaler (IO), even though these analyses were made and potentials constructed assuming that there was no violation of parity or T-invariance. Examples of the type of result one may expect are given in the Appendix, containing calculations developed for two experiments probing T-invariance which have been completed or are in progress at Rochester.4 [See Abashian and Hafner (S).l A.
TIME-REVERSAL
KONINVARIAPXE
The experimental evidence for conservation of parity in nucleon-nucleon interactions is rather good at relatively low energies (see B, below). Until the recent experiments performed at Uppsala and Rochester (8), however, the possibility remained that T might not be a good operator in strong interactions. According to Henley and Jacobsohn (II),“’ ‘a the previous experimental upper limit on Tviolating interactions was lo-20% of the strength of the ordinary interactions. If T-invariance is good, certain relations hold among the ordinary experimental functions (taken at a given angle and energy). Up to and including experiments involving at most two polarizations, they are
P = a,
(1616
where P’ is the polarization normal to the scattering plane produced by an unpolarized beam on an unpolarized target, and @ is the right-left asymmetry produced when a beam polarized normal to t’he scattering plane bombards an unpolarized target;
(A + R’) cosz - (A’ - R) sin: Y where A, A’, R, R’ are the triple-scattering
= 0,
functions
cm with
polarizations
(17) in the
4 The four triple-scattering experiments involving polarizations in the scattering plane are presently being carried out at Rochester by A. tingland, W. Gibson, E. Heer and J. Tinlot (J. Tinlot, private communication). 5 These authors did not consider the results of Osley et al. (1). 6a Note added in proof: Further discussion of possible experiments to test for time-reversal violation in nuclear reactions has been presented by E. M. Henley and B. A. Jacobsohn, Phys. Rev. 113, 225, 234 (1959). 6 This relation is discussed in the papers listed in Ref. 4.
76
WOODRUFF TABLE
II
TRIPLE SCATTERING FUNCTIONS Initial
polarization
A A’ R R’
direction
Final polarization
P P ir i
direction
K P K P
onFIG.
2
scattering plane, as defined by Table II and Fig. 2, and7 apb + cta = CPP + CKK, 0
0
(18)
e
e
a@ cos - + c&p sin - = Cpp cos + CKp sin -, 2 2 2 2 @+,a sin’ i - an;, cos’ 2 = Cgp sin O - CKK
cos
8,
(19) (20)
where the C;? are the correlation functions, i and j referring to the directions of the final particle polarizations, and the a;? are asymmetry functions with a proton polarized in the i direction incident on a t,arget proton polarized in the j direction. These functions are defined in terms of traces by IOp = x Tr [Mt&.nM], I& = J$ Tr [MtMd,.n],
(21)
7 Similar
relations
have
been
derived
by Stapp
(la).
SCATTERING
WITH
P
AND
T
7’7
KONINVARIANCE
The relations (16) through (20) break down if time-reversal invariance is not good. For simplicity we shall write down the modified relations in the linear approximation in the violating amplitudes. The parity-violating amplitudes will then not enter at all, and Eq. (18) remains valid. The extension to the general case is st,raightforward. We obtain Io(P
- a) = -4
Im (H*T),
=I,
[
(.4 + Ii’)
cos i -
= -Re
I,
[
(atih -
(A’ - R) sin i
1
[(B - G + N)*Tl
Cpp) cos 4 + (@+*a - CK~) sin f
1
x (Lso -
(22)
sin i-
== i-8Im(C*T)cosi-4Re(N*T)
M,, + ilI1-1) sin 19+ ti
1
(MN + Mar ) cos 0) ,
1
IO apb sin’ i - @s ~0s’ i - CKP sin 0 + CgK cos e = - -1Re(N*T) = k Re [( fi
sin0+8Im
(C*T)
cosfl
(n/r,, - M1,,) * COYe - (Moo + Mn - M&*
x ((JIoo - Ml1 + Jfl-1)
sin e + 1/2 (M,,
sin 0)
+ MO,) cos e)] .
These equations mere obtained by performing the traces, using expression (3) for M, and proceeding to the s, m, representation by using the relations (15). By cutting off the expansions (13) at the value for I appropriate to the energy of interest, and substituting for the LY’Sthe expressions (12) with j- = 0 = 7 and (A,)~ = 2Xj , one obtains expressions linear in the Xj for the above combinations of experimental functions. In the Appendix the results of such an analysis for
78
WOODRUFF
the first two relations, using the Gammel-Thaler potential (10) phase shifts at 220 Mev, are presented. The experiments performed to date on the polarizationasymmetry differences yield x+2 = -2.0”
f 2.0”.
(23)
A nice feature of the above-mentioned experiment is that there can be no (P - a) effect contributed by the carbon target (or by any spin 0 target) to complicate the interpretation of the results, if parity is conserved. The transition matrix for a nucleon on a spin 0 target has Dheform MO = Aa
+ Bo(B,E)d-n + Co(B,E)d.P + D,(O,E)d.K,
(24)
where n, P, and K are defined similarly to before. COand DOare parity-violating terms, and Do violates T-invariance as well. There is no term violating T but no P. Therefore, P - @ on carbon must give a null result, except for a term quadratic in the parity-violating amplitudes. B. PARITY
VIOLATION
Perhaps the simplest direct test to detect the violation of parity is the experiment which was performed by yarious investigators (1) as a check on their experimental alignment, that is, to measure the asymmetry between protons scattered parallel and antiparallel to the direction of the polarization of the incident beam, which is a pseudoscalar quantity. All of the parity-violating amplitudes enter into this experimental quantity, the time-reversal violating ones as well as those which do not violate time-reversal invariance. The experiment measures (we use the linear approximation throughout this section) :
= Re ((B + Im (C-B
+iV)*(Pcosi+Rsinl))
-I- N)* (8 sin i + S cos i))
+ 2 Im C* P sin i - Rcos%))+2Re(C*(Qcosi-Ssini)) ( ( - Re(H*(Pcosi-
Rsin~))+Im(H*(&sin~-Scosl)).
SCATTERING
This expression lo@; = -- -&
+&
WITH
P
AND
in the s, m, representation Re (M,,*M,J
- &
T
79
NONINVARIANCE
is
Re ((MOO + MI, - MI-I)
*MI,)
Re((MUI- MoI)*Mo~)
+ pi Re ([(MOO -
MU + MM)
* cos 13-
1/2
(Mel + MK,) * sin 0)
(26)
x (MO, sin 8 + 43 ill,, cos 01) Oxley et #nZ.obtained an asymmetry of lo-’ at 230 Mev and a scattering angle of 54” c.m.; for the square of the ratio of the parity-violating to nonviolating amplitudes, F, this sets the upper limit F” 5 lo@ [see the remarks by Heer et al. (1 ).] Chamberlain and co-workers obtained data yielding a higher upper limit. Taken together with the many experiments involving complex nuclei (Z), it seems clear that P is a quite good operator at relatively low energies. On the other hand, from the theoretical point of view, the contemplation of a possible violation of parity in strong interactions is perhaps more pleasing than a possible time-reversal noninvariance, for reasons discussed in the introduction. Therefore, it may be of value to write down analogous expressions to (25) and (26) for the three other parity-violating experiments involving only one polarization. For example, if parity is conserved, one expects no effect on the forwardbackward asymmetry of the scattered proton (i.e., on the ordinary angular distribution ) due to a longitudinally polarized incident beam. If parity breaks down, the following expression must be added to or subtracted from the ordinary cross section, depending on the senseof the initial polarization: Ioag = 34 Tr [MtMd,.fi]
= [+2 Im (C*P)
+ Im ((-B
+ N + H)*Q)
+ Re ( (B + N + H)*R)
- 2 Re (C*X)l cos f
+ [Im ((B - N + H)*X)
+ 2 Im (C*R)
+ Re ((-B
- N + H)*P)
(27)
- 2 Re (C*Q)] sin f ,
which yields I&,
= $4 Re (Mss*Ms~) + X Re( (Moo + MU - Ml-,) *MO,)
+ $$iRe ((Mlo - Mol)*Ml,) + ?d Re ([(Moo - Ml1 + ML-~)* cos /3 - & X ( Mos cos 0 - 4
Mh sin 0) )
(28) (MOM+ n/r,,)* sin 01
80
WOODRUFF
Parity violation would also give rise to polarizations in the scattering plane from initially unpolarized beams and targets. One might have a longitudinal polarization (in the laboratory system) : IOpp = ~~ Tr [Mtdl.PM] = -2
Im (C*P)
+ Re ((B
+ Im ((B
- N - H)*&)
+ N + H)*R)
(29)
- 2 Re (C*S).
giving
+ x Re ((X00*( -
1 + cos 0) + (11111- MM)*(
42
For the transverse I$,
(Mm + Ml,)*
sin 0)
(
M,o cos X -
polarization
in the scattering
= Re ( (B + N - H)*P)
+ 2 Re (C*4?) + 2 Im (C*R)
(30)
1 - cos 0) 4
MS1 sin 5
>>
plane, the expression
+ Im ((B
- N + H)*X)
is (31)
so that IoPK = - k Re
(
2cfs9* Mos sin i + d2 (
- &
Re
- 3; Re ((M,*(
1 -
+ q5
(
(14410- M,,)*
cos
0) + (Mr,
(Mm + Ml,)*
Mls cos i
M,o cos i - l/s
(
- MM)*(
sin 0)
>>
(
1+
cos
M,o sin i + 4
Ml, sin 5
>>
(32)
0) M,r cos i
>>
These expressions may be developed in terms of the phase-shift parameters in the same way as in Section A. Here, however, if T is a good operator, one has two sets of mixing parameters, {i and qj, rather than simply one set (Xi). If both parity and time-reversal invariance break down, then in the linear approximation the four sets of parameters {j, qi , (XX)~ , and (X3) j would appear. We note that modified “P - a” experiments could be performed with the polarization vector in the scattering plane, to test for parity violations. We have
SCATTERING
WITH
P
T
AND
TABLE TRANSFORMATION
III
PROPERTIES
OF VECTORS
P
1:
81
NONINVARIANCE
IN V
T
X
r
-r -P
-r
-P -L -61
L fdl +dz
-P L dz
-dz
dl
(in the linear approximation) yi Tr [lMt[&*K,M]]
= 4 { + Im (C*R)
+ $5 Im ( (B - N + H)*S)
94 Tr [J!J’[dl.P,M]]
= 4 { -1m
+ $5 Im ((B - N - H)*Q)}.
(C*P)
], (33)
Thus the former experiment measures the P-violating T-nonviolating amplitudes and the ‘latter the P- and T-violating amplitudes. However, similar violation effects would take place on the carbon target in these cases, making interpretation of positive results difficult. In fact, with the carbon target, 42’ Tr [U,i[d.K,M,]]
= +4 Im (BO*Co),
$6 Tr [n/r,t[d*P,M,]]
= -4
IV.
GENERAL
FORM
OF
THE
(34)
Im (Bo*Do). POTENTIAL
By simply extending the argument of Okubo and Marshak (14), the general form of the proton-proton potential when P and T are not good operators may be written down. The potential V is a function of d1 , dz , r, and p. Making a table similar to that in Section I, we have Table III. The expressions invariant under rotations and exchanges of the particles are: Independent of d: 1, r”, p”, L2, rep + per (the last expression violates T invnriance).8 Linear in d: (&+ 62) .L, (dl - 132).r, (& - 62) .p (the second expression violates both P and T invariance, the third P). Biline:ar in d: dl.de, dl.rd2.r, &.p&.p, c&.Ldz.L + &.Ld,eL, dl.rdn.p + dz-rdl-p + Hermitian conjugate (h.c.), dl.rd2.L - &.r&.L + h.c., dl.pd2.L - dn.pd1.L + h.c. (the last three terms violate, respectively, T, P and T, and P invariance). The terms dl.rdz.L - &.r&.L + h.c. and dl.pdz.L - d2.pdl.L + h.c. are equivalent to terms in dl X &.r and d, X d2.p, and r.p + p.r. Thus the general form 8 See, however, 677 (1954).
G. Mopurgo,
L. A. Radicati,
and B. F. Jouschek,
Nuovo
cimento
Ser. 9,12,
82
WOODRUFF
of the potential V(h
$2
,r,p)
between =
VO +
[(h
two protons +
dd
is
-LVI + h.c.1 + dl.dfV2
+ [dl.rdz.rV3 + h.c.] + [dlapd2.pV4 + h.c.] + [+i(&.Ldp.L
+ 132.Ldl.L)Vg + h.c.]
(35)
+ [(4 - dd .rV6 + h.c.1 + [(dl - d2) .pV? + h.c.1
+ [(dl.rd2.p
+ d2.rdl.p + h.c.)Vs
+ [dl X &.rVg
+ h.c.]
+ h.c.1 + [& X d2-pV10 + h.c.]
where Vi = Vi(r2,p2,L2,r.p
+ p-r).
(36)
ACENOWLEDGMENTS This work was performed wishes to thank Dr. Susumu
under Okubo
the guidance for several
of Professor R. E. Marshak. helpful discussions.
The
author
APPENDIX
Using the phase shifts and ordinary mixing parameters (up to L = 5) given by the Gammel-Thaler potential No. 2 at 220 Mev (10)) and neglecting Coulomb effects, we find in the linear approximation (with Xj as in (5)) lo(P - ~3) = ((6.71 + 22.2 co2 0 - 23.2 cos419)sin 2X2 + ( - 15.5 - 12.0 cos219+ 161cos40 - 115 cos60) sin 2 X4) sin 2 0. 10m2* cm.2
(37)
If X2>> X4, lo(P - c%)is a maximum near 0 = 45” c.m. However, it is not much smaller at 30” c.m., which is the angle actually used: 10(P - a) = (1.2 sin 2x2 + 0.45 sin 2X4) mb/sterad at 45” c.m. = (0.89 sin 2x2 + 1.5 sin 2X4) mb/sterad at 30” c.m.
(38)
The Oxley group (8) obtained P - B = 0.01 f 0.06 at around 230 Mev and a c.m. angle of 54’. If the results of Hillman et al. (8), which were obtained at 180 Mev and 31” cm., are combined with those of Abashian and Hafner (8) (210 Mev and 30” c.m.), assuming that the energy dependence of P - a is not rapid, we obtain (see the paper of Abashian and Hafner) P - a = -0.014 f
0.014.
(39)
Substitution into (38), assuming X4 = 0, yields X2 = -2.0” f 2.0”. Using sin 2X2as an indication of the relative size F of the violating amplitude, we obtain F2 5 5 X 10m3.
SCATTERING
WITH
P AND
T
83
NONINVARIANCE
The four triple-scattering experiments, A, A’, R, and R’ are being performed at Rochester.4 Proceeding as above, we find (A + R’) cos;
-
1
8 (A’ - R) sin 2
= {[O.OlO - 0.228 cos 19+ 0.09 cos’ 13+ 0.28 cos3 0 - 0.22 cos4 e
c0s5e] sin 2X2 + [0.143 + 0.491 cos e - 0.79 ~02 e - 1.19 ~0s~e + 1.13 ~0s~e + 0.28 ~0s~e - 0.14 COS” e
-
0.41
(40)
- 0.39 cos’ e] sin 2X4] sin 28 mb/sterad. The X2 term is largest in magnitude expression is (-0.13
sin 2x2 - 0.05 sin 2X4) mb/sterad
at 45’ c.m.,
(-0.21
sin 2hz - 0.17 sin 2X4) mb/sterad
at 34” c.m.,
(-0.24
sin 2Xz - 0.22 sin 2X4) mb/sterad
at 25’ c.m.
January
RECEIVED:
at about 25” c.m. Numerically,
the above
(41)
15, 1959 REFERENCES
1. C. I,. (~XLEY, W. F. CARTWRIGHT, J. ROUVINA, E. BASKIR, D. W. SKILLMAX, Phys. Rev. 91, 419 (1953); 0. CHAMBERLAIN, E. WIEGANI), AND T. J. YPSILANTIS, Phys. Rev. 93,143O (1951);E. AND J. TINLOT, Phys. Rev. 111, 645 (1958). 2. N. TANNER, Phys. Rev. 107, 1203 (1957); D. H. WILKINSON, Phys.
Rev. 109, 1603 (1958) succeeding articles. V. G. SOLOVIEV, J. Exptl. Theoret. Phys. (C!.S.S.R.) 33, 537 (1957); Suelear Physics 6, 618 (1958); S. N. GUPTA Canadian J. Phys. 36, 1309 (1957). L. WOLFENSTEIN AND J. ASHKIN, Phys. fZev. 86, 947 (1952); R. H. DALITZ, Proc. Phys. Sot. A65, 175 (1952). R. J. X. PHILLIPS, Nuovo ciwrato Ser. 10, 8, 265 (1958). J. M. BLATT AND I,. C. BIEDESHARX, Phys. Rev. 86, 399 (1952); Revs. Modern Phys. 24, 258 ((1952). H.P. STSI'P, T.J. YPSILANTIS, AND X.RIETROPOLIS, Phys.Rev. 105,302 (1957);see also S. C. WRIGHT, Phys. Rev. 99, 996 (1955). P.HILLMAS, A. JOHANNSON,.~XI G. TIBELL, Phys. IZev. 110,1218 (1958);A. ABMHIAN ANL) E. 31. HAFSER, Phys. Rev. Letters 1, 255 (1958); see also OXLEY et al., Ref. 1. This experiment was suggested by J. S. BELL AND F. MANDL, Proc. Phys. Sot. (London) ‘71, 272 (1958). P. SIGNELL AND R. E. MARSHAK, Ph,/s. Reu. 106, 832 (1957); Phys. Rev. 109, 1229 (1958). J. L. GAMMEL ANU R. M. THALER, Phys. Rev. 107,291 (1957). E. M. HENLEY AND B. A. JACOBSOHN, Phys. Rev. 108, 502 (1957). L. WOLFEXSTEIN, Phys. Rev. 96, 1654 (1954); Snn. Rev. Nwd. Sci. 6, -13 (1956). H. P. STAPP, University of California thesis UCRL 3098 (1955). S.OKUBO BND R.E. ?rl~~s~~q~lnnal.s ofPhysics 4,166 (1958). and
3.
4. 5. 6. 7. 8.
9. 10. 11.
12. 13.
14.
KLEIN, J. RING AND SEGR&., R. TRIPP, C. HEER, A. ROBERTS,
the
of Physics
E.
and
WOODRUFF
a.4 Astronom,y, Rochester, New
University Pork
of Rochester,
This paper develops a formalism to treat some of the effects of a hypothetical breakdown of parity and time-reversal invariance in the proton-proton interaction. The first two sections show how the usual parametrization of the S-matrix (in terms of phase shifts and mixing parameters) must be extended to include a description of such violations. The third se&ion discusses some of the experiments which could detect these violations, if they existed, and the final part extends the general form of the p-p pot,ential. An appendix apphes the formalism to two experiments testing time-reversal invariance. INTR0DUCTTON
Present experimental results indicate that’ neither parity (P) nor time-reversal (T) invariance is violated to a substantial degree in interactions between nucleans at moderate energies. For parity violation in nucleon-nucleon scattering processes the characteristic upper limit is given by F2 5 lop3 (1)’ (F2 being the square of the ratio of the violating to nonviolating amplitudes), while nuclear processes yield F’ 5 3 X IO-* (2) Ia; for time-reversal noninvariance the corresponding upper limit is given by F” 5 5 X 10e3 (see Section III). The present analysis was undertaken partly in conjunction with one of the latter exprriments. From the t,heoretical point. of view, it is important that, experiment.s searching for a possible breakdown of parity in strong interactions be performed with as great an accuracy as possible and at higher energies than hitherto. It is known that parity breaks down in weak interactions, whereas T is apparent,ly good, so that to some small estent (via virtual processes) all int,eractions must show some parit,y breakdown. Even if the nucleon current interacted directly with itself in weak interactions, this would yield a very small F* - g2 - lo-“. On the other * This work was 1 In connection la Note added in R. Almyvist have Rell., to be published).
supported in part by the U. S. At,omic Energy Commission. with these experiments see the discllssion in Heer et al. (1). proof: D. A. Bromley, H. E. Gove, J. A. Kuehner, A. E. Litherland obtained an upper limit of I I’* -< 2 X 10-O in nuclear reactions 65
and (Phys.
AA
WOODRUFF
TAULI!: TRANSFORMATION
PROPERTIES
P
-P
ii P
1;
I OF
\‘E(
1’0~s
1s
;I[
-P’ -P
-P -P’ -K
K
-P -n
-P
n dl
n dl
d?
dz
-P n
-dl
dz
-d2
6
hand, Soloviev and Gupta (3) have pointed out t,hat one could simply assume universal invariance under the product CP (C is charge conjugation), or, using the TCP theorem, T, and obtain parity-conserving electromagnetic and nucleonpion elementary interactions as a consequence of other principles (gauge invariance and charge independence). According to this hypothesis, parity violations would appear in strong interactions involving the strange. particles, and, via virtual processes, in nucleon-nucleon interactions. If the virtual strange particles appear mainly in the core region of the nucleons, the breakdown of parity in nucleon-nucleon interactions would be small at low energies and become more pronounced at higher energies. In the following sectlions we show how additional parameters may be introduced to allow for the possibility of a breakdown of parity and/or time-reversal invariance in the proton-proton interaction, and how experimental quantities may be related to these additional parameters. I. GENERAL
FORM
OF
THE
TRANSITION
MATRIX*
In this section we give the general form of the proton-proton transition matrix in the msl , ms, representation; that is, in terms of the Pauli spin matrices for the two protons. This form is a useful starting point for obtaining the expressions which correspond to various experiments, example of which will be given in Section III. Let p, p’ be the incoming and outgoing relative momenta of the proton-proton system in the center-of-mass frame and define the unit vectors K, P, n, wit)h directions : K = p’ - p
P=p’+p
(1)
n=PxK.
K, P, and n are mutually orthogonal. Under space reflection (P), time reversal (T), and exchange of the two particles (X), these vect’ors and the Pauli spin mat,rices have the transformation properties shown in Table I. 2 The derivation of the parity and 6ime-reversal by Wolfenstein and Ashkin (4) nnd Dalitz (4).
invariant,
transition
matrix
was given
SCATTERING
WITH
P
T
AND
67
NONINVARIANCE
The general expression for the transition matrix, not invariant under P or T, but invariant under X, may be constructed readily by reference to this table, forming all possible scalars and pseudoscalars which are invariant under X. It may be written in the form M = B(B,E)s + C(e,E)(dl
+ ds).n
+>;G(B,E)(dl.Kdz.K
+ &.Pdz.P)J
+>$3(O,E)(dl.Kdz.K
- dl.Pdz.P)5
+N(B,E)dl.ndz.n5
(2)
+P(B,E)(dl
- &)*K + Q(B,E)dl
X dz-P
+R(B,E’)(d,
-
X dz*K
&).P
+ S(@)&
+T(B,E)(dl.Kde.P
+ dl.Pdz.K)Z,
where B( e,E) through T(tV,E) are ten complex functions of scattering angle e = 6 .$ and energy E = p’,f’ilf = pf2/M, and where S and ci are the singlet and triplet spin projection operators, respectively: S
= x(1
-
dl.dz)
3 =
!:;(3
+
dl.dz).
,4n equivalent form, more useful for calculating t,races when the total spins of the initial and final states are not specified, is obtained by inserting the above expressions for s and 3 in (2) and simplifying: dl = +;(I? + G + N) + +),i(
-B
+
G
-
-
d2)
N)(d,.Kdz.K
+$$H(dl.Kdz.K
+34(--B
+
C(dl
+
+P(&
-
62) .K + &dl
X 6z.P
+R(dl
-
dz).P
X
+T(dl.Kdz.P
dl.Pd2.P)
dl.Pd2.P)
G + 3N)dl.ndz.n
+
.n
8dl
(3)
d2.K
+ dl.Pd2.K).
The terms containing B, C, G, H, and N are invariant both under P and under T; R and S are invariant under T, but’ not under P; T is invariant under P, but not T; and P and Q are invariant under neither P nor T. Since violations of p and T by the above terms are always complete (involving n change in sign), the behavior of the terms under charge conjugation G follows at once from the TGP theorem. The six more possible independent matrices in the ms,msz representation are forbidden by invariance under X (identity of particles).
68
WOODRUFF
I
I I I
I I I FIG.
II.
PARAMETRIZATION
OF
1 THE
SCATTERING
MATRIX
This section contains the equations relating the functions B(B,E) through T( B,E) of the previous section to the usual partial-wave phase shifts and mixing parameters, and to the new parameters which must be introduced to describe the proton-proton scattering processeswhen P and/or T invariance is violated. First we introduce these new parameters [for time-reversal noninvariance this has been done by Phillips (5)]; next, we express the transition matrix in the s, m, representation (where s is the total spin) in terms of the phase shifts and other parameters in a partial wave expansion; finally, the functions B through T are related to the components of the transition matrix in the s, m, representation. The proton-proton system can exist in the states shown in Fig. 1. If P and T are conserved, transitions can occur between the same states and between triplet states with the same total angular momentum (heavy arrows). The mat,rix cx= X-XCOUl. in the 1s j representation is diagonal in I, S, and j except for these coupled triplet stat,es, in which case it is a 2 X 2 matrix in 1. Using the BlattBiedenharn conventions (6)) 01is writt,en as: singlet states: al=9
2iSl -e 2+2,
uncoupled triplet states: cilj = e2i62--e %l
(I = .i).
coupled triplet states:
&
=
$5 si*
2Ej(e2isi-l.i
_
e2’6’-L
)
(4)
SC-4TTERING
WITH
P
T
AND
69
NONINVARIANCE
The 61 and 61,j are the Blatt-Biedenharn phase shifts, Biedenharn mixing parameters. cpzis defined by
and the ej the Blatt-
2
pr = C tan-’ r=l
n 0x ’
where n q : e2/hv, v being the speed of the incoming proton in the laboratory. The matrix (Y is symmetrical due to the invariance under T. Phillips (5) has shown how to parametrize the LYmatrix if T is not a good operator but P is. The unitarity of X restricts cx to the same form as above except that S and CYare no longer symmetrical:
a+‘=ae
j
2iXj
,
CL’ =ae
j -2iXi
.
(5)
One new parameter, Xj , a function of energy but not of scattering angle, just as the phase shifts and mixing parameters are, must be introduced for each set of coupled triplet st.ates [just as one new function, T( B,E), must be introduced in the transition matrix]. If parity is not conserved, the transitions indicated by the dashed arrows in Fig. 1 can occur. This introduces two new mixing parameters, lj and qi , for each coupled set of two triplet and one singlet states; if T is not good for t,hese transitions, two more parameters, Xlj and Xsj , must be included. Since each pair of coupled triplet states is now coupled to a singlet state, a 3 X 3 matrix is required for each set. The S mat.ris for such a set may be expressed in terms of a Hermitian matrix Q by3 8”
=
e2iQJ
(6)
since Ls1.lis unitary. The phase shifts of t,he three coupled states are defined to be the eigenvalues of Q”, SO that &” = UJ-‘A,U.,
,
(7)
where
1,
A, = 3 This J-value;
argument see Ref.
follows 6.
the usual
method
for treating
the coupled
(8) triplet
states
of a single
70
WOODRUFF
and the form of the unitary
mat’rix U., is chosen to be:
U., = B,C.,D, sin 17eix, cos 71
cos q -sin 9 e8’”
B =
1 C= 1) ’
(
cos e --iA -sin E e
(
cos [ -sin remix’
D =
sin E eZXz , cos E .) sin teix’ cos {
(9)
11 . That this form for UJ is sufficiently general follows from the facts that Q” contains 9 = 3’ real parameters, and that its components are linearly independent of one another, and so Q” defined by (7)-(g) is an arbitrary Hermitian matrix. Using the notation (
which corresponds we obtain
to what we had earlier, with the addition
of the CY*’ and a*“,
aj = [cos’ p cos” q + cos’ e sin’ { sin’ 9 +[cos’
$5 cos E sin 2{ sin 27 cos(Xr - X3)]e2’6j
[ sin’ 7j + co? e sin2 { cos2 v + 95 cos e sin 2{ sin 271cos(X1 - X3)]e2”6j-‘~j
+ sin2 E sin2 { e2i6i+Lj _ e2iPj, Cyj-1,j
= [sin’ [
v +
C0S2
COS’
c CO2
{ sin2 f
+ 55 cos E sin 2{ sin 27 cos(X1 - X3)]e2”*’ +[sin’
1 sin’ q + cos” e cos2 [ cos2 f
- $5 cos E sin 25- sin 2~ cos(X1 - X3)]e2’*j-‘~j + sin” e cosz { e2i6i+Li _ e2icPi-1, “j+1,j cY+j
= sin2 e sin2 q e2iai + sin2 e cos2 q =
e2igi-l,j
+
cosz
E e2iG+l.i
_
e2iri+l,
[$$j sin E sin { sin 27j e--i(x1--x3) + lh sin 2~ cos c sin” g]e”2e2i6j + [ - 35 sin e sin { sin 27 e-io--x3) + 35 sin 2~ cos l cos2 q]eix2e2i6j-1*i -
lh
sin
ze
~0s
{ eixzeziaj+Lj,
SCATTERISG
WITH
cYdJ = same as c,+j with a+
fi
P
Ah3
T
71
NONINVARIAKCE
signs of X’s reversed,
(11)
= [>h sin 2{ cos’ 7 - $6 cos 6 sin’ { sin 2~ ei(hl-Xp) + $4 cos c cos’ j- sin 27 e-i(x1--x3) -
$1 ~0s’ c sin 2{ sin’ ~]czh1cz2*J
+ [f,$ sin 2[ sin’ 7 + ,1$ co.3 E sin’ j- sin 29 ei’x’-x3) - f,i ~0s 6 cos2 j- sin 211p-i(h1-A3) - ,$s ~0s’ c sin t4 ~0s’ q]eiX1e2i6’-1.~ &‘i
$5 sill2 E sin &- eiXle2iai+Lj,
= same as CY+‘~with
signs of X’s reversed,
a+ 113== [pi sin E cos j- sin 217eTAa- >i sill 2~ sill { sin” q e”x’]eix2e2’6j + [ -44
sin E cos j+sin 2~ eix3 - $5 sin 26 sin j- ~0s’ 11eixl]eixz$iai-l~i
+ 44 sill zE sill { ~iAleax~e2i6f+~~i, a!- ‘I3 == same as a+”
with
signs of X’S reversed.
where the subscript j has been dropped from the parameters E, r, 7, x1 , x2 , and Xa . If the amount of parity violation is small, { and r] are small. Keeping terms linear in j- and 7, the above relations become approximately crj = $iaj _ e2iqj, CYj+l,J
=
COS”
E e2i6i+laj +
sin2
cY+J = $5 sin zE e+x2(e2i6j-1,i
a+‘li
tj
=
25- +
J&sin
=
cofj
_
E sin
>h[cos
e sin
217e--i(A1-x3)
_
$,$ sin”
E sin
21
t sill
2,,
_ e2iPi-l,
e2iSi+l.i),
+
~0s’
(12) &-]eix~eziSi-l,i
e%+h)e2iJj
$$[sin E sin 27 eih3 + sin ze
+
~5 sin
sin
~ sin
eiAlezi6i+l,i,
-
ze
e2i(oj+l,
2.q e-i(A1-A3)]e’A1e2iai
-
$4 sill
_
+ COSee e*%-l,l
ocj-1.j = sin* E ezi6j+l,i
a+
E e2i6j-l,j
sin
* e’Al]eiA2e2i6j-l,~
1 ei(A1+We2i6i+l,ja
Thus, to terms linear in { and q, the parity-conserving terms are the same as before C(4) and (5), with 12 = 2X]. From the experiments on time-reversal invariance, we expect X2 also to be small, but nothing can be said about X1 and x3 , which represent the relative amount of T-violation in the parity-violating transitions.
72
WOODRUFF
The problem of expressing the transition matrix M in the s, m, representation in terms of the parameters (Y is equivalent to finding the relation between the s, m, representation and sums over the 1, s, j, rnf representation, properly weighted by means of Clebsch-Gordan coefficients. An outline of the method of derivat,ion is given in Ref. 7. If T is not good, Table III of Stapp et al. (7) is still correct, if the changes (Y1+1 + ff+ ‘+’ and 01’~’+ a- ‘-’ are made. The complete set of relntions, including the parity-violating transitions, follows:
- ; d(Z + l)(Z + 2) O+l+l - f 2/(Z - 1)Z -‘-‘1)
M1-l(e,p)
= 2(ik)-‘P~
+ii
az,z-1
-
.gl pl”(e>
41 2/u
-
+ l)(l 1
21 + 1 ’ 4(Z + 1) az91+1 4Z(Z + 1) az*z i + 2) a+ z+* _ 41/ma1 1
Z-l
1’
SCATTERISG
nf*ll(e,$o) = - i(ik)-’
WITH
P
AND
c~lPLm42z+
I’
73
NOiYIiYVARIISCE
1)(1+
1) Q+NL + d(2Z
M,,(B,p)
= i(ik)-l
.g,
Pl(c9>{~(2
+ 1)(21 + 3) cL’L+l + 1/1(2Z -
-
M-i-j(l9,'p)
=
( -)i-jilI,j(f3j
flI+j(O,p>
=
(-
flf-is(e,p>
= (- )iMi,(e,
+ 1)Z Ly+‘ll,
nJ
1) OIP--I},
(2Z - 1) 21
a-l'z-l
‘,
7
-Cp),
)jAlsj(e,
-cp),
-P>.
The indices i and j refer t’o the final and initial t,riplet states, and s refers to the singlet state. along the dir&ion of t’he incoming momentum, of spherical coordinates with angles (O,(p). .f?( 0) tude,
.fde) = --n k( 1 -
cos
--in e)
values, respectively, of m, in t,he The direction of qunntization is which is t’he polar axis of the set is the Coulomb scatt,ering umpli-
In [$(l-
co9 e)]
'
The relations between the functions B - T in t,hc m,, , ‘~1,~representation of Ill and the functions N;j in the s, m, representations are readily found by using the idemities
Here Gi is the i component
#i*jt
= $iS”Sij - Sj*S; ,
#&it
= sA;*,
of the triplet
(14)
spin function,
and fiS the singlet spin
74
WOODRUFF
function; S = $s(di + I$), A = fs(& operator. We obtain
- &), and S is the singlet spin projection
(15) R = ‘4 (AI,, + M,,) cos; - &
T = : (ill,,, - AI,, + M-,)
i
(Ml, + Msl) sin f ,
sin e + &
(MUI + MO,) cos e
It is to be noted that there is a correspondence between the number of functions in the 1cI matrix and the number of independent parameters at a given allowed j-value. III.
PARITY
AND
TIRIE-REVERSAI,
VIOLATIO-lj
IXPERIhlENTS
If there were a breakdown of parity and/or time-reversal invariance in strong interactions, certain symmetries would remain in the proton-proton experimental functions, due to the identity of the protons and to invariance under rotations. These include, of course, that the cross section should be an even function of angle about 90” c.m., and also that the polarization (and t’he asymmetry produced by polarized protons) should be odd about 90” c.m. However, there are many experiment’s, someof which will be discussedin t,his section, which can give clear evidence for an existing parity or time-reversal violation. The parametrization of the transition matrix developed in the previous sections can be used to determine the angular dependence associated, in any given experiment, with the existence of a violating parameter, and to determine the relative sensitivity of various experiments in the search for such a violation. If me assumethat such violations are small, as is evidenced by various experiments (1, 8) ranging up to around 300 Mev, we may express violation-seeking experi-
SCATTERING
WITH
P
AK‘D
I’
75
NONINVARIANCE
ments by means of terms linear in {, q, and X2 , using the formalism of Sections I and II. The coefficients of these terms will contain the ordinary parameters, i.e., the phase shifts 6 and mixing parameters c In this approximation one may use for the ordinary parameters the values which have been obtained by a direct phaseshift analysis of the ordinary (single, double, etc.) scattering data, or from a potential chosen to fit the data, such as those of Signell and Marshak (9), or Gammel and Thaler (IO), even though these analyses were made and potentials constructed assuming that there was no violation of parity or T-invariance. Examples of the type of result one may expect are given in the Appendix, containing calculations developed for two experiments probing T-invariance which have been completed or are in progress at Rochester.4 [See Abashian and Hafner (S).l A.
TIME-REVERSAL
KONINVARIAPXE
The experimental evidence for conservation of parity in nucleon-nucleon interactions is rather good at relatively low energies (see B, below). Until the recent experiments performed at Uppsala and Rochester (8), however, the possibility remained that T might not be a good operator in strong interactions. According to Henley and Jacobsohn (II),“’ ‘a the previous experimental upper limit on Tviolating interactions was lo-20% of the strength of the ordinary interactions. If T-invariance is good, certain relations hold among the ordinary experimental functions (taken at a given angle and energy). Up to and including experiments involving at most two polarizations, they are
P = a,
(1616
where P’ is the polarization normal to the scattering plane produced by an unpolarized beam on an unpolarized target, and @ is the right-left asymmetry produced when a beam polarized normal to t’he scattering plane bombards an unpolarized target;
(A + R’) cosz - (A’ - R) sin: Y where A, A’, R, R’ are the triple-scattering
= 0,
functions
cm with
polarizations
(17) in the
4 The four triple-scattering experiments involving polarizations in the scattering plane are presently being carried out at Rochester by A. tingland, W. Gibson, E. Heer and J. Tinlot (J. Tinlot, private communication). 5 These authors did not consider the results of Osley et al. (1). 6a Note added in proof: Further discussion of possible experiments to test for time-reversal violation in nuclear reactions has been presented by E. M. Henley and B. A. Jacobsohn, Phys. Rev. 113, 225, 234 (1959). 6 This relation is discussed in the papers listed in Ref. 4.
76
WOODRUFF TABLE
II
TRIPLE SCATTERING FUNCTIONS Initial
polarization
A A’ R R’
direction
Final polarization
P P ir i
direction
K P K P
onFIG.
2
scattering plane, as defined by Table II and Fig. 2, and7 apb + cta = CPP + CKK, 0
0
(18)
e
e
a@ cos - + c&p sin - = Cpp cos + CKp sin -, 2 2 2 2 @+,a sin’ i - an;, cos’ 2 = Cgp sin O - CKK
cos
8,
(19) (20)
where the C;? are the correlation functions, i and j referring to the directions of the final particle polarizations, and the a;? are asymmetry functions with a proton polarized in the i direction incident on a t,arget proton polarized in the j direction. These functions are defined in terms of traces by IOp = x Tr [Mt&.nM], I& = J$ Tr [MtMd,.n],
(21)
7 Similar
relations
have
been
derived
by Stapp
(la).
SCATTERING
WITH
P
AND
T
7’7
KONINVARIANCE
The relations (16) through (20) break down if time-reversal invariance is not good. For simplicity we shall write down the modified relations in the linear approximation in the violating amplitudes. The parity-violating amplitudes will then not enter at all, and Eq. (18) remains valid. The extension to the general case is st,raightforward. We obtain Io(P
- a) = -4
Im (H*T),
=I,
[
(.4 + Ii’)
cos i -
= -Re
I,
[
(atih -
(A’ - R) sin i
1
[(B - G + N)*Tl
Cpp) cos 4 + (@+*a - CK~) sin f
1
x (Lso -
(22)
sin i-
== i-8Im(C*T)cosi-4Re(N*T)
M,, + ilI1-1) sin 19+ ti
1
(MN + Mar ) cos 0) ,
1
IO apb sin’ i - @s ~0s’ i - CKP sin 0 + CgK cos e = - -1Re(N*T) = k Re [( fi
sin0+8Im
(C*T)
cosfl
(n/r,, - M1,,) * COYe - (Moo + Mn - M&*
x ((JIoo - Ml1 + Jfl-1)
sin e + 1/2 (M,,
sin 0)
+ MO,) cos e)] .
These equations mere obtained by performing the traces, using expression (3) for M, and proceeding to the s, m, representation by using the relations (15). By cutting off the expansions (13) at the value for I appropriate to the energy of interest, and substituting for the LY’Sthe expressions (12) with j- = 0 = 7 and (A,)~ = 2Xj , one obtains expressions linear in the Xj for the above combinations of experimental functions. In the Appendix the results of such an analysis for
78
WOODRUFF
the first two relations, using the Gammel-Thaler potential (10) phase shifts at 220 Mev, are presented. The experiments performed to date on the polarizationasymmetry differences yield x+2 = -2.0”
f 2.0”.
(23)
A nice feature of the above-mentioned experiment is that there can be no (P - a) effect contributed by the carbon target (or by any spin 0 target) to complicate the interpretation of the results, if parity is conserved. The transition matrix for a nucleon on a spin 0 target has Dheform MO = Aa
+ Bo(B,E)d-n + Co(B,E)d.P + D,(O,E)d.K,
(24)
where n, P, and K are defined similarly to before. COand DOare parity-violating terms, and Do violates T-invariance as well. There is no term violating T but no P. Therefore, P - @ on carbon must give a null result, except for a term quadratic in the parity-violating amplitudes. B. PARITY
VIOLATION
Perhaps the simplest direct test to detect the violation of parity is the experiment which was performed by yarious investigators (1) as a check on their experimental alignment, that is, to measure the asymmetry between protons scattered parallel and antiparallel to the direction of the polarization of the incident beam, which is a pseudoscalar quantity. All of the parity-violating amplitudes enter into this experimental quantity, the time-reversal violating ones as well as those which do not violate time-reversal invariance. The experiment measures (we use the linear approximation throughout this section) :
= Re ((B + Im (C-B
+iV)*(Pcosi+Rsinl))
-I- N)* (8 sin i + S cos i))
+ 2 Im C* P sin i - Rcos%))+2Re(C*(Qcosi-Ssini)) ( ( - Re(H*(Pcosi-
Rsin~))+Im(H*(&sin~-Scosl)).
SCATTERING
This expression lo@; = -- -&
+&
WITH
P
AND
in the s, m, representation Re (M,,*M,J
- &
T
79
NONINVARIANCE
is
Re ((MOO + MI, - MI-I)
*MI,)
Re((MUI- MoI)*Mo~)
+ pi Re ([(MOO -
MU + MM)
* cos 13-
1/2
(Mel + MK,) * sin 0)
(26)
x (MO, sin 8 + 43 ill,, cos 01) Oxley et #nZ.obtained an asymmetry of lo-’ at 230 Mev and a scattering angle of 54” c.m.; for the square of the ratio of the parity-violating to nonviolating amplitudes, F, this sets the upper limit F” 5 lo@ [see the remarks by Heer et al. (1 ).] Chamberlain and co-workers obtained data yielding a higher upper limit. Taken together with the many experiments involving complex nuclei (Z), it seems clear that P is a quite good operator at relatively low energies. On the other hand, from the theoretical point of view, the contemplation of a possible violation of parity in strong interactions is perhaps more pleasing than a possible time-reversal noninvariance, for reasons discussed in the introduction. Therefore, it may be of value to write down analogous expressions to (25) and (26) for the three other parity-violating experiments involving only one polarization. For example, if parity is conserved, one expects no effect on the forwardbackward asymmetry of the scattered proton (i.e., on the ordinary angular distribution ) due to a longitudinally polarized incident beam. If parity breaks down, the following expression must be added to or subtracted from the ordinary cross section, depending on the senseof the initial polarization: Ioag = 34 Tr [MtMd,.fi]
= [+2 Im (C*P)
+ Im ((-B
+ N + H)*Q)
+ Re ( (B + N + H)*R)
- 2 Re (C*X)l cos f
+ [Im ((B - N + H)*X)
+ 2 Im (C*R)
+ Re ((-B
- N + H)*P)
(27)
- 2 Re (C*Q)] sin f ,
which yields I&,
= $4 Re (Mss*Ms~) + X Re( (Moo + MU - Ml-,) *MO,)
+ $$iRe ((Mlo - Mol)*Ml,) + ?d Re ([(Moo - Ml1 + ML-~)* cos /3 - & X ( Mos cos 0 - 4
Mh sin 0) )
(28) (MOM+ n/r,,)* sin 01
80
WOODRUFF
Parity violation would also give rise to polarizations in the scattering plane from initially unpolarized beams and targets. One might have a longitudinal polarization (in the laboratory system) : IOpp = ~~ Tr [Mtdl.PM] = -2
Im (C*P)
+ Re ((B
+ Im ((B
- N - H)*&)
+ N + H)*R)
(29)
- 2 Re (C*S).
giving
+ x Re ((X00*( -
1 + cos 0) + (11111- MM)*(
42
For the transverse I$,
(Mm + Ml,)*
sin 0)
(
M,o cos X -
polarization
in the scattering
= Re ( (B + N - H)*P)
+ 2 Re (C*4?) + 2 Im (C*R)
(30)
1 - cos 0) 4
MS1 sin 5
>>
plane, the expression
+ Im ((B
- N + H)*X)
is (31)
so that IoPK = - k Re
(
2cfs9* Mos sin i + d2 (
- &
Re
- 3; Re ((M,*(
1 -
+ q5
(
(14410- M,,)*
cos
0) + (Mr,
(Mm + Ml,)*
Mls cos i
M,o cos i - l/s
(
- MM)*(
sin 0)
>>
(
1+
cos
M,o sin i + 4
Ml, sin 5
>>
(32)
0) M,r cos i
>>
These expressions may be developed in terms of the phase-shift parameters in the same way as in Section A. Here, however, if T is a good operator, one has two sets of mixing parameters, {i and qj, rather than simply one set (Xi). If both parity and time-reversal invariance break down, then in the linear approximation the four sets of parameters {j, qi , (XX)~ , and (X3) j would appear. We note that modified “P - a” experiments could be performed with the polarization vector in the scattering plane, to test for parity violations. We have
SCATTERING
WITH
P
T
AND
TABLE TRANSFORMATION
III
PROPERTIES
OF VECTORS
P
1:
81
NONINVARIANCE
IN V
T
X
r
-r -P
-r
-P -L -61
L fdl +dz
-P L dz
-dz
dl
(in the linear approximation) yi Tr [lMt[&*K,M]]
= 4 { + Im (C*R)
+ $5 Im ( (B - N + H)*S)
94 Tr [J!J’[dl.P,M]]
= 4 { -1m
+ $5 Im ((B - N - H)*Q)}.
(C*P)
], (33)
Thus the former experiment measures the P-violating T-nonviolating amplitudes and the ‘latter the P- and T-violating amplitudes. However, similar violation effects would take place on the carbon target in these cases, making interpretation of positive results difficult. In fact, with the carbon target, 42’ Tr [U,i[d.K,M,]]
= +4 Im (BO*Co),
$6 Tr [n/r,t[d*P,M,]]
= -4
IV.
GENERAL
FORM
OF
THE
(34)
Im (Bo*Do). POTENTIAL
By simply extending the argument of Okubo and Marshak (14), the general form of the proton-proton potential when P and T are not good operators may be written down. The potential V is a function of d1 , dz , r, and p. Making a table similar to that in Section I, we have Table III. The expressions invariant under rotations and exchanges of the particles are: Independent of d: 1, r”, p”, L2, rep + per (the last expression violates T invnriance).8 Linear in d: (&+ 62) .L, (dl - 132).r, (& - 62) .p (the second expression violates both P and T invariance, the third P). Biline:ar in d: dl.de, dl.rd2.r, &.p&.p, c&.Ldz.L + &.Ld,eL, dl.rdn.p + dz-rdl-p + Hermitian conjugate (h.c.), dl.rd2.L - &.r&.L + h.c., dl.pd2.L - dn.pd1.L + h.c. (the last three terms violate, respectively, T, P and T, and P invariance). The terms dl.rdz.L - &.r&.L + h.c. and dl.pdz.L - d2.pdl.L + h.c. are equivalent to terms in dl X &.r and d, X d2.p, and r.p + p.r. Thus the general form 8 See, however, 677 (1954).
G. Mopurgo,
L. A. Radicati,
and B. F. Jouschek,
Nuovo
cimento
Ser. 9,12,
82
WOODRUFF
of the potential V(h
$2
,r,p)
between =
VO +
[(h
two protons +
dd
is
-LVI + h.c.1 + dl.dfV2
+ [dl.rdz.rV3 + h.c.] + [dlapd2.pV4 + h.c.] + [+i(&.Ldp.L
+ 132.Ldl.L)Vg + h.c.]
(35)
+ [(4 - dd .rV6 + h.c.1 + [(dl - d2) .pV? + h.c.1
+ [(dl.rd2.p
+ d2.rdl.p + h.c.)Vs
+ [dl X &.rVg
+ h.c.]
+ h.c.1 + [& X d2-pV10 + h.c.]
where Vi = Vi(r2,p2,L2,r.p
+ p-r).
(36)
ACENOWLEDGMENTS This work was performed wishes to thank Dr. Susumu
under Okubo
the guidance for several
of Professor R. E. Marshak. helpful discussions.
The
author
APPENDIX
Using the phase shifts and ordinary mixing parameters (up to L = 5) given by the Gammel-Thaler potential No. 2 at 220 Mev (10)) and neglecting Coulomb effects, we find in the linear approximation (with Xj as in (5)) lo(P - ~3) = ((6.71 + 22.2 co2 0 - 23.2 cos419)sin 2X2 + ( - 15.5 - 12.0 cos219+ 161cos40 - 115 cos60) sin 2 X4) sin 2 0. 10m2* cm.2
(37)
If X2>> X4, lo(P - c%)is a maximum near 0 = 45” c.m. However, it is not much smaller at 30” c.m., which is the angle actually used: 10(P - a) = (1.2 sin 2x2 + 0.45 sin 2X4) mb/sterad at 45” c.m. = (0.89 sin 2x2 + 1.5 sin 2X4) mb/sterad at 30” c.m.
(38)
The Oxley group (8) obtained P - B = 0.01 f 0.06 at around 230 Mev and a c.m. angle of 54’. If the results of Hillman et al. (8), which were obtained at 180 Mev and 31” cm., are combined with those of Abashian and Hafner (8) (210 Mev and 30” c.m.), assuming that the energy dependence of P - a is not rapid, we obtain (see the paper of Abashian and Hafner) P - a = -0.014 f
0.014.
(39)
Substitution into (38), assuming X4 = 0, yields X2 = -2.0” f 2.0”. Using sin 2X2as an indication of the relative size F of the violating amplitude, we obtain F2 5 5 X 10m3.
SCATTERING
WITH
P AND
T
83
NONINVARIANCE
The four triple-scattering experiments, A, A’, R, and R’ are being performed at Rochester.4 Proceeding as above, we find (A + R’) cos;
-
1
8 (A’ - R) sin 2
= {[O.OlO - 0.228 cos 19+ 0.09 cos’ 13+ 0.28 cos3 0 - 0.22 cos4 e
c0s5e] sin 2X2 + [0.143 + 0.491 cos e - 0.79 ~02 e - 1.19 ~0s~e + 1.13 ~0s~e + 0.28 ~0s~e - 0.14 COS” e
-
0.41
(40)
- 0.39 cos’ e] sin 2X4] sin 28 mb/sterad. The X2 term is largest in magnitude expression is (-0.13
sin 2x2 - 0.05 sin 2X4) mb/sterad
at 45’ c.m.,
(-0.21
sin 2hz - 0.17 sin 2X4) mb/sterad
at 34” c.m.,
(-0.24
sin 2Xz - 0.22 sin 2X4) mb/sterad
at 25’ c.m.
January
RECEIVED:
at about 25” c.m. Numerically,
the above
(41)
15, 1959 REFERENCES
1. C. I,. (~XLEY, W. F. CARTWRIGHT, J. ROUVINA, E. BASKIR, D. W. SKILLMAX, Phys. Rev. 91, 419 (1953); 0. CHAMBERLAIN, E. WIEGANI), AND T. J. YPSILANTIS, Phys. Rev. 93,143O (1951);E. AND J. TINLOT, Phys. Rev. 111, 645 (1958). 2. N. TANNER, Phys. Rev. 107, 1203 (1957); D. H. WILKINSON, Phys.
Rev. 109, 1603 (1958) succeeding articles. V. G. SOLOVIEV, J. Exptl. Theoret. Phys. (C!.S.S.R.) 33, 537 (1957); Suelear Physics 6, 618 (1958); S. N. GUPTA Canadian J. Phys. 36, 1309 (1957). L. WOLFENSTEIN AND J. ASHKIN, Phys. fZev. 86, 947 (1952); R. H. DALITZ, Proc. Phys. Sot. A65, 175 (1952). R. J. X. PHILLIPS, Nuovo ciwrato Ser. 10, 8, 265 (1958). J. M. BLATT AND I,. C. BIEDESHARX, Phys. Rev. 86, 399 (1952); Revs. Modern Phys. 24, 258 ((1952). H.P. STSI'P, T.J. YPSILANTIS, AND X.RIETROPOLIS, Phys.Rev. 105,302 (1957);see also S. C. WRIGHT, Phys. Rev. 99, 996 (1955). P.HILLMAS, A. JOHANNSON,.~XI G. TIBELL, Phys. IZev. 110,1218 (1958);A. ABMHIAN ANL) E. 31. HAFSER, Phys. Rev. Letters 1, 255 (1958); see also OXLEY et al., Ref. 1. This experiment was suggested by J. S. BELL AND F. MANDL, Proc. Phys. Sot. (London) ‘71, 272 (1958). P. SIGNELL AND R. E. MARSHAK, Ph,/s. Reu. 106, 832 (1957); Phys. Rev. 109, 1229 (1958). J. L. GAMMEL ANU R. M. THALER, Phys. Rev. 107,291 (1957). E. M. HENLEY AND B. A. JACOBSOHN, Phys. Rev. 108, 502 (1957). L. WOLFEXSTEIN, Phys. Rev. 96, 1654 (1954); Snn. Rev. Nwd. Sci. 6, -13 (1956). H. P. STAPP, University of California thesis UCRL 3098 (1955). S.OKUBO BND R.E. ?rl~~s~~q~lnnal.s ofPhysics 4,166 (1958). and
3.
4. 5. 6. 7. 8.
9. 10. 11.
12. 13.
14.
KLEIN, J. RING AND SEGR&., R. TRIPP, C. HEER, A. ROBERTS,
the