Volume 172, number 1
PHYSICS LETTERS B
8 May 1986
NOVEL TIME-REVERSAL T E S T S IN LOW-ENERGY N E U T R O N P R O P A G A T I O N Leo STODOLSKY Max-Plaock-lnstit~t fftr Physik und Astrophysik - Werner-Heisenberg-lnstitut fftr Physik, D-8000 Munich 40, Fed. Rep. Germany Received 13 January 1986; revised manuscript received 27 February 1986
Interesting time-reversal tests are possible for low-energy neutrons propagating in polarized materials. We present a class of tests, involving comparison Of spin states, which avoid the difficulty of rotations induced by fields in the material. Such transmission experiments would permit the use of sensitive neutron-spin-measuring techniques and thus refined searches for T-violation in nuclear interactions.
In an investigation of low-energy neutron scattering [1] we noted the possibility of a T- and P-violating term in the forward-scattering amplitude, the amplitude relevant for describing neutron-transmission experiments. Looking for such a term, which has the structure (o and p the neutron spin and momentum, S the target polarization) ~-s x p
(1)
would seem an attractive way to search for T-violation in nuclear physics since it could permit use of the sensitive spin.measuring techniques as used in the electric dipole moment (edm) [2] or weak-spin-rotation experiments [3]. Furthermore, since it is a "diagonal" amplitude, that for forward, elastic, scattering, such a term is theoretically unambiguous, and implies T-violation without any assumptions about the phases of f'mal-state interactions and the like. In contrast to most T-tests in nuclear physics, but like the neutral K system, parity as well as time-reversal violation is involved. The probability of finding such a "double" violation might seem small, but on the other hand the discovery in recent years of anomalously great and not entirely understood parity violation in certain low-energy neutron reactions [1] (in one case reaching the 7% level [4] ), suggests an interesting field for exploration. The real or "dispersive" part of the amplitude (1) will cause a rotation of the neutron spin around the 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
axis S × p; the absorptive part corresponds, via the optical theorem, to different cross sections for neutrons with spin up or down with respect to S × p. The difficulty with working with a polarized beam and looking for rotation effects is that the magnetic fields associated with the polarization of the target will lead to precessions of the neutron spin difficult to control. This, along with some remedies, was studied by Bunakov and Gudkov [5]. One might also consider using the absorptive part of the amplitude to create a polarization in an initially unpolarized beam [6], which would then be in the S X p direction, but a similar problem arises: through the parity violation, presumably much stronger than the T-violation, a polarization will develop along the beam direction, p. The field in the target will then retare this polarization, also creating a component in the S X p direction. Although such a component in principle has a different time or distance behavior than the T-violating effect induced by (1), it obviously is very undesirable to have to deal with such difficulties when looking for small effects; one would much prefer a true null experiment. Even with a method for polarizing the nuclei without an applied magnetic field on the sample, the nucle ar interaction between the neutron and the polarized nucleus will have a spin-dependent term which in effect is like a magnetic field in the polarization direction, the "pseudo-magnetic field" [5]. In this note we would like to draw attention to the
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possibility of true null experiments, sensitive to the amplitude (1), and that cannot be simulated by targetpolarization effects. We proceed from the observation that the characteristic feature of a T-violating amplitude is that it is anti-symmetric. In the present case if we adopt the usual conventions with the beam along the z axis and the target polarization along the x axis, we see that the amplitude (1) is proportional to Oy, the anti-symmetric Pauli matrix. Thus in forward scattering an amplitude containing (1) will be anti-symmetric in spin space. We note that such an anti.symmetry cannot be created by propagation effects, the index of refraction describing the neutron in the material is a function of the forward-scattering amplitude, and if the forward-scattering amplitude is symmetric, a function of it is symmetric also. To make these remarks more concrete, consider the typical case where we have the usual relation between the forward-scattering amplitude f and the index of refraction n: n - 1 = (2~r/p2)pf(p is the number density of scatterers). Here n and f are 2 × 2 matrices in the neutron-spin space. If Ui is the Pauli spinor for the neutrons entering the material, and Uf that for the transmitted neutrons after travelling a distance z, we have Uf = c~ Ui , where c5 = exp [i(n - 1)pz] .
(ay) = Tr[c3 +tryc3 ] ~ Im C ' B ,
(4)
and analogous results starting from a polarized beam. This is representative of the precession in the target alluded to above, the rotation of the longitudinal polarization created by the imaginary part of C "rotated" by the real part of B. One of the subtle aspects of the anti-unitary character of the T-operation is that a Teven amplitude can give rise to a non-zero expectation value for a T-odd operator. How can we then bring the term D into evidence in a way which cannot be faked by the other interactions? We look for effects proportional to D. Instead of rotations, we are lead to consider certain pairs of reactions. First of all, the presence of an absorptive D term means there is a spin dependence for the transmission of neutrons polarized in the y direction: erob(+ ~- +) - P r o b ( - -+ - ) = 4 Re AD*
(3)
A' represents the main strong-interaction amplitude, B' the spin-dependent strong amplitude, C' the parityviolating term giving the weak-spin-rotation- and helicity-dependent total cross section, and D' the T- and P-violating term (1). All these amplitudes may have real and imaginary parts, giving refraction and absorption. We do not give an external magnetic field explicitly, this may be thought of as incorporated in B'. Taking the beam in the z direction, and chosing the polarization of the target to be in the x - z plane, the D' term is given by oy, as mentioned above. Aside from this t e r m , f is symmetric. Given a specific f, e5 may be worked out using the formula exp(io.b) = 6
cos b + icr-b (sin b)[b, b = x / ~ for the exponentiation of 2 X 2 matrices, c3 will then have a form like eq. (3), with amplitudes A, B, C, D given as functions o f A ' , B', C', D' and the distance z. I f D ' is zero, however, D is also zero, since the exponentation preserves the symmetric property o f f . Thus we must find an unambiguous manifestation of D. Note, however, that even i f D = 0 and c3 is symmetric, the expectation value of the T-odd quantity oy need not be zero in the final state. We have, in fact, with D = 0, and an unpolarized initial state
(2)
Now f can have the following structure on a polarized target with polarization S:
f = A' + B'o .S + C'o .~ + D ' o . S X ~ .
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(~ in t h e y direction),
(5)
where h refers to the direction of spin quantization and + or - means the neutron spin with respect to that direction. For real spinors, that is for ~ in the x - z plane, the lack of symmetry of ci will mean that the amplitude for a reaction and its symmetric reaction will not be the same,
Ciab =/=~ba
(l~ in the x - z
plane).
(6)
There is, therefore, a difference in rate for a process and its symmetric process. It is evident that the effect must involve states a and b which are different and must furthermore involve the D amplitude. The states a and b need not be orthogonal. However, for simplicity and with an eye to a simple experimental set-up, we imagine a and b to be spin-up and spin-down with
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respect to g. Then using the projection operators ~(1 -+ a.~) to make the two states, we Fred that the probability for spin-up going into spin-down is Prob(+ -+ - ) = Tr[~(1 - a-h)6+½(1 + e.h)c3] , (7) while for spin-down going into spin-up the plus and minus signs should be interchanged. Now in order to have a difference in the two cases, the three different Pauli matrices must be involved in the trace, otherwise a unitary transformation could be introduced into the trace to make the D term symmetric. Therefore an interference term between A and D gives no effect. The next largest term would be the B - D interference, ift~ is chosen along thez (beam) direction (helicity states). Consider, then, a helicity-plus beam propagating through a macroscopic length of polarized material and producing helicity-minus transmitted neutrons, as compared with helicity-minus neutrons producing helicity-plus particles. The target polarization is in the x direction. A non-zero difference in rate now arises Prob(+ -+ - ) - P r o b ( - -+ +) = 4 Im BD* (r~ in the z direction).
(8)
Similarly, there will be an effect in interference with the parity.violating term C, where we choose ti in the x (target polarization) direction. Prob(+ ~ - ) - P r o b ( -* +) = 4 Im CD* (li in the x direction).
(9)
In general, of course, we expect this to be small compared with (8), where D interferes with a strong-interaction amplitude. All these effects (5), (8), and ( 9 ) are proportional to D and so are independent of rotation effects in the target. In practice the errors introduced by misalignment of the spins with respect to the required direction would have to be studied in detail. In this respect it is important that the effect (8) is bilinear in the target polarization and so is unchanged when S is reversed. A result which is then averaged over opposite S directions will cancel any term linear in the neutron-spin misalignment out of the x - z plane, leaving only a quadratic error. It is also possible to see how the tests correspond to a manifest time reversal non-invariance. "Running
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the f ilm backward ,~ will • r~verse the neutron spin, target polarization, and mo~nentum. Rotating around the S X P axis will again reverse P, S and the neutronspin components in the x - z plane. This establishes the conditions for (5), (8) and (9). Note we must assume that the material is homogeneous along its length or that ordering effects do not matter, since after the T operation we are going through the material backwards. The unimportance of such ordering effects is also implicit in the use of (2). which originates in the eikonal approximation. This time-reversal argumentation shows that the tests are quite general - up to problems involving the forwards-backwards homogeneity of the material. The mechanism involved in (5) is obvious. A pictorial understanding of the processes in (8) and (9) may be arrived at by considering the two possible initial spin states as they enter the material: The two spin vectors point in exactly opposite directions. Subsequent rotations, as due to a realD term, cannot change this, so the projections on each other's initial spin orientations are always equal to each other. Now, however, let the absorption [Im B in eq. (8)] be switched on. This removes a common component in both cases, with the result that the two spins are no longer pointing in exactly opposite directions. Now a subsequent rotation can produce unequal projections on the initial spin directions. We see then why it is necessary to have the amplitudes relatively imaginary in (8) and (9). Some absorption is needed to create the effect. We now turn to an important problem in the present method, the effect of the magnetic or "pseudomagnetic" field. Such fields will always be present, corresponding to the B term. As explained above, (5), (8), and (9) cannot be induced by such a term without the presence of the true T-violating amplitude D. However, a strong B term, giving many rotations of the neutron spin, will tend to reduce our effects. This may be seen if we calculate S by exponentiating, as explained after eq. (3), giving A = expO(2r:/p)A'oz) cos b ,
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B C
= exp(i(2~r/p)A'pz)
D
X i(2n/p)pz [(sin b)/b]
C' D'
Since B' is presumably the largest o f B ' , C' and D', we will have b ~ (2~r[p)pzB', which has the interpretation of the number of rotations of the neutron spin in the distance z induced by the magnetic or pseudomagnetic field. Let us focus attention on the factor (sin b)/b. If b is small it gives one. For b large, however, we get l/b, thus loosing a large factor. It is therefore important that the magnetic and pseudo.magnetic fields be kept small (perhaps it can be arranged for them to cancel). In this respect our problem is like the one discussed in ref. [5]. We stress, however, that unlike ref. [5], the difficulty is not that false effects can be produced by the pseudo-magnetic field, but rather that our real effects will tend to be reduced. This may turn out to be the most serious problem for the present proposal, since even a small nuclear-spin-dependent force can give a hundred rotations/cm. According to the threshold rules developed in ref. [ 1 ] the parity-violating amplitudes (1 ]p)C' or (1 ]p)D' entering into the tests via the formula for n should go as constants at low energy; for the imaginary parts this is depending on the existence of exothermic channels. Hence constant effects are expected as p -+ 0, assuming b to be small. These tests may be looked upon as a kind of detailed balancing, as has long been practiced in nuclear physics ,1. Instead of different reaction channels, however, we have different spin states; and furthermore coherent propagation effects through a macroscopic length of material are involved. Since both T and P violation are present, it would seem most profitable, as mentioned in the introduction, to conduct such a test where large parity violations have been seen. Complex nuclei also seem to be indicated, at least for (5), by the fact that the existence of the parity-violating absorption at low energy .1 For a recent detailed balancing test see ref. [7].
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is related to the presence of exothermic channels, as explained in ref. [1]. Thus a suitable target material should (a) permit pol.arization of the nuclei, (b) show large parity violations, (c) have exothermic channels, and, as just discussed, (d) have a small spin-dependent (pseudo-magnetic) interaction, preferably with absorption. Operating at a well-defined beam energy, as in the experiments on resonance might also enchance the definition of a possible effect. In this case the factors A, B, C, D will show a complicated resonant behavior. As for previous work involving neutron-spin effects and T invariance in nuclear physics, there is the old experiment of Shull and Nathans [8], where a possible interference between electric dipole moment and nuclear scattering was used to set limits on the electric dipole moment of the neutron. It is instructive to reconsider this work in the present context. The T-, P.violating matrix element involved was of the form a ' A / A 2 , where A is the momentum transfer threevector. Now we might suppose that this experiment could equally well be interpreted as a search for a T-violating component in the nuclear scattering, where the electric dipole scattering matrix element is induced not by an edm of the neutron, but by some hypothet: ical T and P violation in the strong interaction instead. This might indeed be the case, but with one important difference: the edm matrix element involves the factor 1/A 2 arising from the infinite range of the Coulomb field. In the case of a nuclear interaction, not involving photon exchange, we expect a shortrange effect, say 1/A 2 ~ 1/(A 2 + m2), m ~ 102 MeV. Since the energy and so A is very small for reactor neutrons, the matrix element becomes proportional to the momentum and disappears at low energy. In other words, with a short-range interaction the sought for term becomes a o/c effect, and the ShuU-Nathans experiment is still to be interpreted only as a search for the usual electric dipole moment. This is an illustration of the point that elastic non-forward scattering effects involving P violation are expected to disappear at low energy [1]. On the other hand with coherent propagation effects as in the present idea, we deal with the forward amplitude and so via the optical theorem with the total cross section, and thus are sensitive in principle to
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all channels. It is then possible that the presence o f exothermic reactions induces a finite effect at low en. ergy: hence our stress on the necessity o f exothermic channels. Exothermic reactions usually involve radiative capture, thus the tests are indirectly sensitive to some aspects o f the electromagnetic interaction. Finally, we note that due to the formal parallelism with the p h o t o n and its two spin states, the same arguments should hold for light. However, this case does n o t seem as interesting since it is insensitive to the nuclear interaction and no unusual parity effects are known. These ideas arose in the course o f many discussions with R. Golub. I would like to thank him for these as well as for his mentioning the problem o f the "pseudomagnetic field" and for bringing several of the important references to my attention.
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References [1 ] L. Stodolsky, Nucl. Phys. B197 (1982) 213. [2] V.M. Lobashev and A.P. Serebov, in: Workshop on reactor based fundamental physics, J. Phys. (Paris) C3, Suppl. No. 3 (1984)11; J. Morse, in: Workshop on reactor based fundamental physics, J. Phys. (Paris) C3, Suppl. No. 3 (1984) 13. [3] B. Heckel et al., in: Workshop on reactor based fundamental physics, J. Phys. (Paris) C3, Suppl. No. 3 (1984) 89. [4] Alfimenkov et al., in: Workshop on reactor based fundamental physics, J. Phys. (Paris) C3, Suppl. No. 3 (1984) 93. [5 ] V.E. Bunakov and V.P. Gudkov, in: Workshop on reactor based fundamental physics, J. Phys. (Paris) C3, Suppl. No. 3 (1984) 77. [6] L. StodoL_qky,unpublished. [7] E. Blanke et aL, Phys. Rev. Lett. 51 (1983) 355. [8] C.G. Shull and R. Nathans, Phys. Rev. Lett. 19 (1967) 384.