The T-Odd R and D Correlations in Beta Decay

We review and discuss the time-reversal-odd R and D correlations in neutron and nuclear beta decay.


Introduction
CP-violation (CPV) 1 has been seen in the mixing of the neutral kaons, and recently also in the K°→ 2π amplitudes [2] and in the decays of the neutral B-mesons [3]. At present there is no unambiguous direct evidence for time-reversal (T) violation. 2 We know however that T-invariance is violated, since the parameter ε in K L → 2π decays is dominated by a CPT-invariant interaction. 2 In the models which we shall consider in the following all the interactions are CPT invariant, and we shall use therefore the terms "T-violation" and "CP-violation" interchangeably.
To date there is no firm evidence against the possibility that the observed CPV effects are due to the Kobayashi-Maskawa phase δ KM in the Standard Model (SM). 3,4 A major question in the field of CPV is whether there are sources of CPV other than δ KM , independently of their relevance or lack of it for the observed CPV. New sources of CPV are present in many extensions of the SM. It is relevant to mention in this connection that δ KM is not sufficient to generate the baryon asymmetry of the universe. 5 The most suitable observables to probe the existence of new CPV interactions are those for which the contribution from δ KM is small. Examples of observables of this kind are the electric dipole moments of the neutron and atoms, and T-odd correlations in leptonic and semileptonic decays.
In this talk we shall review and discuss the status of T-odd correlations in beta decay. In the next section we review the expressions for the coefficients of D and R correlations for a general d → ueνe interaction. In Section 3 we summarize the limits on the CPV beta decay coupling constants implied by beta decay experiments. In Section 4 we consider D and R in extensions of the SM. Section 5 contains a summary of our conclusions.

General Considerations
Time-reversal (T) violating components in the d → ueνe interaction manifest themselves in beta decay through contributions to T-odd correlations in the decay probability [9]. Sensitive experimental information is available on the coefficients D and R of the correlations 〈J〉 ⋅ p e × p ν / J E e E ν and σ σ ⋅ 〈J〉 × p e / J E e (σ σ ≡ electron spin, J ≡ nuclear spin, p e ≡ electron momentum, p ν ≡ neutrino momentum, E e ≡ electron energy, E ν ≡ neutrino energy), respectively. The T-odd correlations are present even in the absence of T-violation, induced by final state interactions. The latter are dominated by contributions from the electromagnetic interaction. We shall write D and R as D = D t + D f and R = R t + R f , where D t , R t represent the T-violating contribution, and D f , R f are the T-invariant contributions due to the final state interactions.
In the SM the d → ueνe transition arises from W-exchange, and has a V-A form: 6 (1) of the Kobayashi-Maskawa matrix. The neutrino state accompanies the left-handed electron in a doublet of SU(2) L . It is a linear combination of the left-handed components of the mass eigenstates: The interaction (1) is CP-(and T-) invariant. In the quark and gluon sector of the SM there are two sources of CP-violation: the Kobayashi-Maskawa phase δ KM in the quark mixing matrix, and the θ-term in the QCD Lagrangian. D t and R t from these sources are extremely small, of the order of 10 -12 a [11], where a is defined in Eq. (10) below. The reason is that δ KM contributes only in second order in the weak interaction, and the θ-term is constrained by the stringent bound |θ | < ∼ 4 × 10 -10 from the experimental limit on the electric dipole moment of the neutron. In the SM with massive neutrinos CP-violation can be present also in the mixing of leptons. The effect of this in beta decay would not show up in first order in the weak interaction either. Thus D t and R t probe sources of CP-violation beyond those present in the SM.
To first order in new d → ueνe interactions D t and R t arise from interference between the SM amplitude and the amplitude from the new interactions. We shall neglect in D t and R t terms proportional to neutrino masses. All the remaining terms must come from interactions involving left-handed neutrinos. The most general d → ueνe (L) interaction involving the neutrino state (2) 7 can be written as 8 (3) where (4) The fields e, u, and d in Eqs. (4) -(7) are the mass eigenstates. The coupling constants are in general complex, in which case the Hamiltonians violate T-invariance. The constant a LL in Eq. (4) contains the SM contribution, and can therefore be written as, a LL = (a LL ) SM + a′ LL , where (a LL ) SM = g 2 V ud /8M 2 W and a′ LL represents contributions from new interactions.
The contribution of the Hamiltonian, Eq. (3) to D t and R t in allowed beta decays is given by [9] (8) (9) where the upper (lower) sign in the first term in Eq. (9) is for decays with e -(e + )in the final state. In Eqs.
Our metric, γ matrices and σ λµ are the same as in Ref. [10].
Couplings involving neutrino states other than ν e (L) are possible, but for those in most cases additional constraints apply. Also, the choice ν e (L) in Eq. (3) guarantees for D t and R t maximal overlap in the interference with the SM amplitude. 8 For a recent review of possible new interactions in beta decay see Ref. [12].
In Eq. (11) λ J′J is an angular momentum factor, defined in Ref. [9]. The quantities g k ≡ g k (0) (k = V, A, S, T) are defined by CVC predicts g V = 1, and (neglecting the effects of the possible new interactions) the experimental value of [13]. The constants g S and g T were calculated in Ref. [14] in connection with a study of neutral current interactions of a general Lorentz structure. Employing a quark model with spherically symmetric wave functions, g S and g T are given by g S = -1 / 2 + 9 /10g A 0.6, g T = 5 /3 ( 1 /2 + 3 /10g A ) 1.46. The uncertainty in these predictions has been estimated to be about 30 % to 60 % [14]. Including an uncertainty of this size, one has
The limit in Eq. (21) on the tensor interaction comes from the result (R Li ) expt = (1.6 ± 2.2) × 10 -3 [21] of a measurement of R in 8 Li → 8 Be + e -+ ν e decay. For this case one has Subtracting from (R Li ) expt the final state interaction contribution, which for this case is yields R t = (0.9 ± 2.2) × 10 -3 [21].
Finally, the limit, Eq. (22), follows from a measurement of the eν correlation in 32 Ar beta decay [22]. A limit, which is weaker than (22), is implied by a measurement of R in 19 Ne decay [23]. An experiment to measure R in neutron decay to an accuracy of 5 × 10 -3 is being developed at PSI [21]. In neutron decay As seen from Eq. (19), such a result, combined with the bound in Eq. (21) will set an upper bound of about 2 × 10 -2 on |g S Ima -LS |.

D t and R t in Extensions of the Standard Model
In this section we shall discuss briefly D t and R t in extensions of the SM. We shall restrict our attention only to models where the required interactions can arise at the tree level, since loop-induced interactions are expected to be weak.

D t
An a LR -type interaction can arise at the tree level in models containing a new charged gauge boson with right-handed couplings to the quarks (as in left-right symmetric models), in the SM model if it is extended to contain new heavy "exotic" quarks which have righthanded couplings to the W and which mix with the  9 In all these cases the a LR -interaction can be represented for beta decay by contact nonderivative four-fermion interactions. Contact a LR -interactions can arise also in composite models, from the exchange of constituents. 10 Since the a LR -interaction is not invariant under the standard electroweak gauge group, it must be proportional to an SU(2) L × U(1) breaking parameter. In leftright symmetric models this is the nondiagonal element of the W L -W R mixing matrix, and in exotic fermion models the light-heavy quark mixing angles. In leptoquark models the a LR -interaction arises from mixing of leptoquarks of different SM quantum numbers. In composite models an a LR -interaction must contain the factor υ/Λ relative to the SU(2) L × U(1) invariant interactions, where υ is the vacuum expectation value of the SM Higgs boson and Λ is the compositness scale.
In left-right and exotic fermion models an a LR -type d → ueνe interaction is accompanied by a strangeness conserving quark-quark interaction of strength a LR , which has a P,T-violating component of the form [26], [12] (23), can be written as (24) where the constant k is expected to be of the order of in view of the left-right structure of the operator, Eq. (23). The EDMs set stringent limits on Ima -LR . The best one is (25) implied by the experimental upper limit (|d( 199 Hg)| < 2.1 × 10 -28 e cm (90 % c.l.) [27]) on the EDM of the mercury atom. 11 An estimate of k [30] using factori-zation and QCD sum rules yielded k 10, implying (26) The neutron EDM, estimated in Ref. [30], leads to the limit |Ima -LR | < ∼ 10 -5 , nearly the same as Eq. (26).
For D t /a from leptoquark exchange the constraints are weaker [31]. The P,T-violating strangeness conserving quark-quark interaction, which is generated at oneloop level from diagrams involving W-exchange and containing a leptoquark propagator in one of the vertices, is suppressed by m 2 u or m 2 d .. The electron EDM and the quark electric and chromoelectric dipole moments do not arise at the one loop level. Based on dimensional estimates of the dipole moments, the conclusion is that they allow D t /a to be as large as the present experimental limit on D t /a.

R t
Scalar d → ueνe interactions can arise at the tree level from the exchange of Higgs bosons, spin-zero or spin-one leptoquarks, and in supersymmetric models with R-parity violation from the exchange of sleptons. Tensor type d → ueνe interactions can arise from the exchange of spin-zero leptoquarks. Scalar and tensor d → ueνe interactions can appear also in composite models, generated by the exchange of constituents.
Let us consider R t in the minimal supersymmetric standard model with R-parity violation [32].
In the minimal supersymmetric standard model (MSSM), unlike in the SM, the conservation of lepton number (L) and of baryon number (B) is not automatic: the superpotential can contain renormalizable and gauge invariant L-and B-violating terms. If both the L-and the B-violating terms are present, some of the products of the corresponding coupling constants would have to be extremely small to prevent too rapid proton decay. One way to deal with this problem is to demand invariance under R-parity [R = (-1) 3B + L + 2s , where s is the spin of the particle; thus R = +1 for particles of the SM, and R = -1 for their superpartners]. This would eliminate both the B-and the L-violating terms. Alternatively, one can demand invariance under "baryon parity" (under baryon parity the quark fields change sign, and the lepton and Higgs fields remain unchanged), which eliminates only the B-violating terms. The model we shall consider in the following is the R-parity violating minimal supersymmetric standard model (R / MSSM), defined as the MSSM with the lepton-number violating terms W L / included in the superpotential. 12 Volume 110, Number 4, July-August 2005 Journal of Research of the National Institute of Standards and Technology 456 9 Ref. [24]. See also Ref. [12]. 10 Ref. [25]. Contact beta decay interactions have been discussed in Ref. [12]. 11 For the Schiff moment and the EDM of mercury we used the results obtained in Refs. [28] and [29], respectively.
The general form of W L / is given by (27) where i, j, k = 1, 2, 3 are family indices, and summations over i, j, k are implied. In Eq. (27), L′ i , Q′ i are the SU(2)-doublet lepton and quark superfields, (2)-singlet charged lepton and up-and down-type quark superfields; H u is the Higgs superfield which generates the masses of the up-type quarks. The primes on the fields indicate that they are the weak eigenstates.
The presence of R-parity violating couplings has rich phenomenological implications. One of these is that they can contribute to SM processes through the exchange of single squarks or sleptons.
There are two classes of contributions to beta decay. One of them is governed by |λ′ 11k | 2 and mediated by thẽ d k R (k = 1, 2, 3) [36]. These d → ueνe interactions have a V -A form [36], and therefore do not contribute to T-odd correlations. The other class, which involves both λ ijk and λ′ ijk , has scalar and pseudoscalar components. There are two such contributions, given by (28) In the Hamiltonian, Eq. (28), the fields are the mass eigenstates; in the sum ν 1 −term for simplicity. The quantity ω B contains the product of the elements of the mixing matrices involved. From (28) we have (29) CP-violation can arise in (28) from complex λ 1j1 and λ′ j11 , and also from complex ω B .
In the following we shall assume for simplicity that and that mixing for the righthanded fields and for u L -type quarks can be neglected. Then a mixing angle in V ei (ν) , and e iφB is a CPV phase. In deriving limits on Ima -LS we shall assume (to preclude additional constraints to apply and the possibility of a cancellation in Ima -LS ) that only one of the products significant size at a time. The limits on |Ima -LS | in Eq. (29) implied by limits on the individual coupling constants λ 1j1 and λ′ j11 , derived from various processes [35], are not better than a few times 10 -2 . A stringent limit (30) on Ima -LS comes from the ratio R π = Γ(π → eν e )/ Γ(π → µν µ ) [37]. This limit arises because the a LP -component of (28) contributes to π → eν e , and a LS = a LP .
Potentially the strongest limits on Ima -LS come from experimental bounds on P,T-violating electron-quark (e -q) interactions. As pointed out in Ref. [38], electroweak radiative corrections to scalar, pseudoscalar, and tensor interactions of any origin induce contributions to P,T-violating e -q interactions. For the Hamiltonian (28) this interaction is of the form 13 (31) with (32) where ρ = (α/4π) ln (Λ 2 /m W 2 ); Λ is a cut-off parameter. Taking conservatively, as in Ref. [38], ln (Λ 2 /m W 2 ) = 1, one has In addition to (k Sd ) r , there is also a tree-level contribution (k Sd ) t to k Sd , governed by the same products in Eq. (28). This is a consequence of gauge invariance of W L / before symmetry breaking. (k Sd ) t is given by (33 ) where ω e contains the product of the appropriate mixing matrix elements. Under our simplifying assumptions that the phases e iφB and e iφe are in general different.
The total contribution to k Sd can be written as (34)  The best limit on k Sd comes from the EDM of the Tl atom. The experimental limit on d(Tl) [40] implies 14 |k Sd | < 4.5 × 10 -8 , so that (35) (k Sd ) r alone would give a limit |Ima -LS | < 2 × 10 -5 . The upper limit on |Ima -LS | could be larger than 2 × 10 -5 if is a cancellation between the two terms in the denominator in Eq. (35). To allow lation would have to occur through 3 orders of magnitude! The bound [Eq. (30)] from R π would however still remain. This bound would become weaker if there is some cancellation between the contributions to π → eν e and π → µν µ . A contribution to π → µν µ is present in the model.
For R t in the other extensions of the SM the situation is similar to the one in the (R / MSSM), provided that the associated P,T-violating e -q interaction involves only the d-quark. If e -u interactions are present, a cancellation between the radiative and tree-level contributions cannot be arranged in more than one atomic EDM. Stringent limits, albeit not as strong as from d(Tl), then persist [32].

Conclusions
In this talk we have discussed tree-level contributions to D t and R t in extensions of the SM. A major question is what experimental sensitivities are required to obtain new information on the new interactions involved.
For D t /a (Eq. 8) in left-right symmetric and exotic fermion models the EDMs of the neutron and of mercury set upper limits about two orders of magnitude below the present direct limits. Since the limits from the EDMs have uncertainties (from the calculation of the hadronic matrix elements and for d(Hg) also from nuclear structure) which are difficult to asses, the possibility that D t /a is larger cannot be ruled out. For D t mediated by leptoquark exchange the conclusion based on dimensional estimates of the electron EDM and the electric and chromoelectric quark dipole moments is that D t /a can be as large as the present experimental limit on D t /a.
For R t in neutron decay (Eq. 19) experimental limits on atomic EDMs set limits which are below the level where R t can be probed. Nevertheless, the possibility that R t is larger, even as large as ∼10 -2 , cannot be completely ruled out. This would require some very fine-tuned cancellations between the contributions to P,T-violating e -q interactions and in the ratio Γ(π → eν e ) / Γ(π → µν µ ).