Null-test signal for T-invariance violation in pd scattering

# Null-test signal for T-invariance violation in pd scattering

Yu.N. Uzikov, A.A. Temerbayev Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, Dubna, 141980 Russia
Department of Physics, Moscow State University, Moscow, 119991 Russia
L.N. Gumilyov Eurasian National University, Astana, 010008 Kazakhstan
###### Abstract

The integrated proton-deuteron cross section for the case of the incident proton vector polarization and tensor polarization of the deuteron target provides a null test signal for time-reversal invariance violating but P-parity conserving (TVPC) effects. We study the null-test observable within the Glauber theory of the double-polarized scattering. Full spin dependence of the ordinary strong scattering amplitudes and different types of the hypothetical TVPC pN-amplitudes are taken into account. We show that the contribution from the exchange of the lowest-mass meson allowed in the TVPC interaction, i.e. the -meson, to the null-test observable is zero. The axial meson exchange makes a non-zero contribution. We find that inclusion of the Coulomb interaction does not lead to divergence of the cross section and calculate its energy dependence at the proton beam energy 100-1000 MeV.

Time-invariance, polarized proton-deuteron interaction
###### pacs:
24.80.+y, 25.10.+s, 11.30.Er, 13.75.Cs

## I Introduction

CP violation (or T-reversal invariance violation under CPT symmetry) is required to explain the baryon asymmetry of the Universe Sakharov . In baryon systems violation of T-invariance has not been observed yet. CP violation established in physics of kaons and B-mesons leads to simultaneous CP and P-invariance violation. Under the assumption of CPT-invariance this implies existence of T-odd P-odd interactions. These effects are parametrized in the Standard Model by the CP violating phase of the Cabibbo-Kobayashi-Maskawa matrix. Another source for T-odd P-odd effects is the QCD -term, which can be related to electric dipole moments (EDM) of elementary particles and atoms in their ground states.

On the contrary, time-reversal symmetry violating (T-odd) P-parity conserving (P-even) flavor conserving (TVPC) interactions do not arise on the fundamental level within the Standard Model, although they can be generated from the T-odd P-odd interaction by weak radiative P-parity non-conserving corrections. However in this case its intensity is too low Khripl91 ; Gudkov92 to be observed in experiments at present. Thus, observation of the TVPC effects would be considered as indication of physics beyond the standard model.

The existing experimental constraints on the TVPC effects in physics of nuclei are rather weak. So, the test of the detailed balance performed for the reactions and blanke , and complemented by numerous statistical analyses of nuclear energy-level fluctuations leads to the ratio of T-odd to T-even matrix elements as french . Another type of experiment, i.e. polarized neutron transmission through a polarized Ho target gives or huffman . Here is the T-odd P-even coupling constant of the charged -meson with the nucleon introduced in Ref. simonius97 to classify the TVPC interactions in terms of boson exchanges. Charge symmetry breaking determined as difference in scattering of polarized protons off unpolarized neutrons and polarized neutrons off unpolarized protons gives (or simonius97 . One should add that indirect model-dependent estimation based on the existing constraints on EDM gives () haxtonHM94 . However, a more recent analysis showed kurylov that EDM may arise via another scenario which suggests no significant constraints on the TVPC forces.

The integrated cross section will be measured at COSY TRIC in double polarized scattering with a transverse polarized proton beam () and a tensor polarized deuterium target (). This observable provides a real null test of the TVPC forces conzett . This signal is not affected by the initial and final state interaction and therefore its observation would directly indicate time-invariance violation, as in case of the neutron EDM. The experiment TRIC will be performed at a beam energy of 135 MeV. This energy choice was motivated by the theoretical analysis of the integrated cross section performed in Ref. beyer . The aim of this experiment is to diminish the upper bound on the TVPC effects previously obtained in the Ho scattering huffman by one order of magnitude.

The elastic channel and the deuteron breakup were considered in Ref. beyer in the impulse approximation (single scattering mechanism) for estimation of . In the present work we study the null-test observable on the basis of the generalized optical theorem using the forward elastic scattering amplitude calculated within the Glauber theory. Both the single and double scattering mechanisms are considered. The spin-dependent Glauber formalism recently developed in Ref. PK was applied in our previous work TUZyaf to calculate spin observables of the elastic scattering using the strong (time invariance conserving and P-parity conserving) scattering amplitudes as input at 135 MeV. The obtained differential cross section, vector and tensor analyzing powers and spin-correlation parameters were found to be in reasonable agreement with the existing data sekiguchi ; przewoski . Here we generalize this formalism to allow for TVPC scattering amplitudes of several types. This generalized formalism is applied below to derive formulas for the null-test observable and calculate its energy dependence. We show that within the single scattering mechanism this observable is zero in the Glauber theory (for any type of the TVPC interactions considered in the general case in Ref. herczeg ) and, consequently, focus on the double scattering mechanism. We investigate the contribution of several TVPC terms to the scattering amplitudes, in particular, the -meson and axial - meson exchanges. In addition, we investigate the influence of the Coulomb interaction on the cross section not considered in Ref.beyer .

The paper is organized as follows. In Sect. II we consider the spin structure of the forward elastic scattering amplitude including the TVPC term and apply the generalized optical theorem to derive formulas for total spin-dependent cross sections in terms of the forward scattering invariant amplitudes. In Sect. III we construct the Glauber scattering operator taking into account full spin dependence of the elementary -scattering amplitudes for strong and some types of TVPC interactions and S- and D- components of the deuteron wave function. Analytical expressions for the TVPC forward scattering amplitude are derived for the double scattering mechanism with different TVPC terms. The influence of the Coulomb effects on the amplitude is discussed in Sect.IV. Numerical results are shown in Sect.IV.

## Ii Forward transition operator and integrated cross sections

Time-reversal symmetry conserving and P-parity conserving (TCPC or T-even P-even) interactions lead to the following transition amplitude of the elastic scattering at zero degree rekalo

 e′β∗M(0)TCPCαβeα=g1[ee′∗−(me)(me′∗)]+g2(me)(me′∗)+ ig3{\boldmathσ[e×e′∗]−(\boldmathσm)(m⋅[e×e′∗])}+ig4(\boldmathσm)(m⋅[e×e′∗]), (1)

where () is the polarization vector of the initial (final) deuteron, is the unit vector along the beam momentum, is the Pauli matrix, ( are complex amplitudes. To the right-hand side of Eq.(1) one can add the TVPC (T-odd P-even) term in a very general form

 e′β∗M(0)TVPCαβeα=˜g{(\boldmathσ⋅[m×e])(m⋅e′∗)+(\boldmathσ⋅[m×e′∗])(m⋅e)}, (2)

where is the TVPC transition amplitude. To find the total spin dependent cross sections we use the generalized optical theorem phillips

 σti=4√πImTr(ρiM(0))Tr^ρi, (3)

where is the transition operator from Eqs. (1) and (2) for the elastic scattering at zero angle , is the initial spin-density matrix, is the total cross section corresponding to the density matrix . The transition operator is normalized according to the following relation with the differential cross section PK

 dσdt=16TrMM+. (4)

For the sum of Eqs. (1) and (2) one can write (see also uzpepan98 )

 M(0)αβ=g1δαβ+(g2−g1)mαkβ+ig3σiϵαβi+i(g4−g3) σimimjϵαβj+˜gσi(ϵzαimβmz+ϵzβimzmα), (5)

where () are the Pauli spin matrices, is the fully antisymmetric tensor, () are the Cartesian components of the vector .

The initial state spin density matrix is the product of the spin density matrices for the proton

 ρp=12(1+pp\boldmathσ), (6)

where is the polarization vector of the proton, and for the deuteron

 ρd=13+12Sjpdj+19SjkPjk; (7)

here is the spin-1 operator, and are the vector and tensor polarizations of the deuteron, and is the spin-tensor operator. Using Eqs. (3), (5), (6) and (7), one can find the total cross section of the scattering as

 σtot=σt0+σt1pp⋅pd+σt2(pp⋅m)(pd⋅m)+σt3Pzz+˜σppyPdxz, (8)

where () is the vector polarization of the initial proton (deuteron) and and are the tensor polarizations of the deuteron. The OZ axis is directed along the , the OY axis is directed along the vector polarization of the proton beam and the OX axis is chosen to form the right-hand reference frame. The following equations were found in UJH13 for the TVPC terms

 σt0=43√πIm(2g1+g2),    σt1=−4√πImg3, (9) σt2=−4√πIm(g4−g3),    σt3=4√πIm(g1−g2).

We find that the TVPC term in the forward elastic scattering amplitude (2) leads to the following integrated cross section

 ˜σ=−4√πIm23˜g. (10)

In Eq. (8) the terms with are non-zero only for the TCPC (T-even P-even) interactions and the last term is non-zero if the TVPC (T-odd P-even) interactions occur. Thus, the term constitutes the null-test signal for time-reversal invariance violating but P-parity conserving effects. This term can be measured in the transmission experiment TRIC as a difference of counting rates for the cases with and .

We find the following matrix elements of the TVPC transition operator (2):

 <μ′=12,λ′=0|MTVPC|μ=−12,λ=1>=i√2˜g, (11) <μ′=12,λ′=−1|MTVPC|μ=−12,λ=0>=−i√2˜g, (12)

where () and () are spin projections of the initial (final) proton and deuteron on the beam direction, respectively. The diagonal matrix elements of the operator are zeros. For the operator the corresponding matrix elements in Eqs.(11) and (12) are identical and equal to .

## Iii Spin-dependent Glauber formalism with strong and TVPC interaction

### iii.1 Hadronic and Coulomb pN interaction

Hadronic amplitudes of scattering are taken in a form of PK

 MN(p,q;\boldmathσ,% \boldmathσN)=AN+CN\boldmathσ^n+C′N\boldmathσN^n+BN(\boldmathσ^k)(\boldmathσN^k)+ (13) +(GN+HN)(\boldmathσ^q)(% \boldmathσN^q)+(GN−HN)(\boldmathσ^n)(\boldmathσN^n),

where , and are defined as unit vectors along the vectors , and , respectively. Normalization of the amplitudes , , , , , is the same as in Ref. PK

 dσdt=14TrMNM+N, (14)

where is the differential cross section of the elastic scattering. The Glauber formalism for the elastic scattering accounting full spin dependence of the amplitudes (13) and S-and D-components of the deuteron wave function is given in Ref.PK . The unpolarized differential cross section and analyzing powers of scattering calculated within this formalism are in reasonable agreement with existing data in the forward hemisphere at energies 250-1000 MeV PK , jhuz2012 . Further development of this formalism was done in TUZyaf to allow for calculation of spin correlation parameters and inclusion of the Coulomb interaction that is important at lower energies.

We include the Coulomb interaction by adding to the Glauber hadronic scattering amplitude the following pure Coulomb amplitude of the scattering

 MCpd(q)=√πkppfCpp(q)Sd(q/2), (15)

where is the antisymmetric Coulomb amplitude of the scattering MS :

 fCpp(q)=f(θpp)−12(1+%\boldmath$σ$⋅\boldmathσp)f(π−θpp) (16)

with

 f(θpp)=−α4vkppsin2θpp/2exp{−iαvlnsinθpp2+2iχ0}, (17)

here is the fine structure constant, () is the velocity (momentum) of the proton in the cms system, is the Coulomb phase. The momentum transferred in the process is related to the scattering angle in the cms as .

In Eq. (15) is the elastic form factor of the deuteron which can be presented in the form PK

 Sd(q/2)=S0(q/2)−1√8S2(q/2)S12(^q;\boldmathσp,\boldmathσn). (18)

Here

 S12(^q;\boldmathσp,% \boldmathσn)=3(\boldmathσp⋅^q)(\boldmathσn⋅^q)−\boldmathσp⋅\boldmathσn (19)

is the tensor operator, are the Pauli matrices acting on the spin states of the neutron and proton in the deuteron, is the unit vector directed along the transferred momentum . The form factors and are related to the - and - components of the deuteron wave function PK :

 S0(q)=S(0)0(q)+S(2)0(q),S2(q)=S(1)2(q)+S(2)2(q), (20)

here

 S(0)0(q)=∫∞0dru2(r)j0(qr),S(2)0(q)=∫∞0drw2(r)j0(qr), S(1)2(q)=2∫∞0dru(r)w(r)j2(qr),S(2)2(q)=−1√2∫∞0drw2(r)j2(qr). (21)

### iii.2 Tvpc pN scattering amplitudes

In general case TVPC NN interaction contains 18 different terms herczeg . We consider here only the following terms of the t-matrix of the elastic scattering investigated in Ref. beyer :

 tpN=fN(\boldmathσ⋅\boldmathσN)(q⋅k)/m2p+hN[(\boldmathσ⋅k)(\boldmathσN⋅q)+(%\boldmath$σ$N⋅k)(\boldmathσ⋅q)−23(\boldmathσN⋅\boldmathσ)(k⋅q)]/m2p+ (22) +gN[\boldmathσ×\boldmathσN]⋅[q×k]/m2p+g′N(% \boldmathσ−\boldmathσN)⋅i[q×k][\boldmathτ×\boldmathτN]z/m2p.

Here () is the Pauli matrix acting on the spin state of the proton (nucleon ), () is the isospin matrix acting on the isospin state of the proton (nucleon), . In the framework of the phenomenological meson exchange interaction the term corresponds to the meson exchange, and the -term is caused by the axial meson exchange. As shown in Ref. simonius75 , the contribution of the - and -meson exchanges to TVPC NN interactions is excluded. The TVPC NN interaction potential corresponding to , exchange in r-space has a form lazauskas

 Vh(r)=−Gh¯Ghm2h2πm2NY1(x)[(\boldmathσ1¯p)(% \boldmathσ2^r)+(\boldmathσ2¯p)(\boldmathσ1^r)], (23)

where () is the ordinary (TVPC) coupling constant, is the -meson mass, ,  , , ; and are the spin-operator and momentum of i-th nucleon, respectively, and is its r-coordinate ). Making the Fourier-transformation of Eq. (23) we obtain the interaction potential in p-space and, therefore, find the factor in Eq. (22) within the Born approximation for the amplitude as the following

 hN=−iϕh2G2hm2h+q2FhNN(q2), (24)

where is the strength of the T-invariance violating potential of -meson exchange relative to the T-conserving one, and is the phenomenological monopole formfactor in the vertex. Similarly proceeding from the TVPC -meson exchange NN-potential in r-space lazauskas ; haxton93 ; Engel94 we find for the -term in Eq. (22)

 g′N=−ϕρ12g2ρκm2ρ+q2FρNN(q2), (25)

where is the ratio of the TVPC coupling constant to the strong one , is the mass of the -meson, is the anomalous magnetic moment of the nucleon, is the vertex formfactor.

### iii.3 The Glauber operator

The Glauber operator of the elastic scattering in general case can be written as

 M(q,Q;S,\boldmathσ)=∭eiQrΨ+d(r)OΨd(r)d3r, (26)

where is the deuteron wave function, is the transferred momentum, is the spin-operator of the deuteron nucleons, is the spin-operator of the incoming proton. We use the deuteron wave function generated by the NN interaction which conserves time reversal invariance and P-parity and has the following standard form

 Ψd(r;\boldmathσn,\boldmathσp)=1√4πr(u(r)+1√8w(r)⋅S12(^r;\boldmathσn,\boldmathσp)), (27)

where lower index () refers to the neutron (proton) of the deuteron target; and denote the S- and D- wave of the deuteron, respectively; the tensor operator is defined by Eq. (19).

The operator for the single and double scattering mechanisms of scattering in general case (beyond the collinear kinematics) can be written as an expansion over the Pauli matrices and in notations of Ref. PK takes the form

 O(\boldmathσ,\boldmathσn,\boldmathσp)=U(\boldmathσ)+Vn(\boldmathσ)⋅\boldmathσn+Vp(\boldmathσ)⋅\boldmathσp+Wij(\boldmathσ)⋅(σniσpj+σnjσpi), (28)

here are indices of the projections onto directions of three orthogonal vectors introduced in Eq. (13). The operators , , act only on the spin state of the beam proton and do not depend on the spins and coordinates of the target nucleons. When making the matrix element of the operator (28) over the deuteron wave functions (27) we obtain from Eq. (26)

 M(q,Q;S,\boldmathσ)=∭eiQrΨ+d(r)OΨd(r)d3r =US0+VSS(0)0+[Wij{Si,Sj}−Wii]S(0)0−1√2US12(^Q;S,S)S2 −WiiS12(^Q;S,S)S(2)2−2WiiS(2)0− −√2Wij[{Si,Sj}S12(^Q;S,S)+S12(^Q;S,S){Si,Sj}]S(1)2+ +116πWij∫d3r1r2eiQrw2S12(^r;\boldmathσn,\boldmathσp){Si,Sj}S12(^r;\boldmathσn,\boldmathσp). (29)

Here we use the notations and ; the form factors and are defined in Eqs. (20), (21); the tensor operators and are defined in Eq. (19). In Eq. (III.3) summation is performed over repeating indexes . To make the integration over directions of the vector in Eq. (III.3), we used the following relation fealdt

 ∬dΩrexp(−iQr)Tl(^r)=4πjl(Qr)(−i)lTl(^Q), (30)

where is the spherical Bessel function, , ; , and are unit vectors along , and , respectively.

Eq. (III.3) is a generalization of Eq. (18) from Ref. PK . The difference from Ref. PK consists in two following respects. First, the operators , , contain not only T-even P-even terms, but T-odd P-even terms as well. Second, we present in Eq. (III.3) all terms allowed within the Glauber theory, whereas in Ref.PK small spin-dependent terms (of the order higher than two in definitions of Ref. PK ) were neglected. These terms are small at high energies about 1 GeV, however, may be important at lower energies MeV, corresponding the COSY experiment TRIC .

### iii.4 Differential spin observables

Vector analyzing powers and spin correlation coefficients , of the elastic scattering are calculated as

where is the transition operator given by Eq. (26). Details of these calculations in terms of invariant amplitudes are described in Ref. TUZyaf .

## Iv Null-test signal of TVPC interactions

Eq. (III.3) gives the single scattering amplitude if one put , where is the momentum transferred in the scattering. Within the Glauber theory the amplitude of the single scattering mechanism is proportional to the on-shell amplitude. At zero scattering angle the TVPC amplitude (22) vanishes, therefore, the corresponding -scattering amplitude and total cross section of -scattering are equal to zero in the single scattering Glauber approximation. Furthermore, the -term in Eq. (22) gives zero contribution within the Glauber theory both for the single and double scattering, because for the on-shell scattering involved into multistep scattering (28), one has . For the same reason, the component of the -term proportional to vanishes in Eq. (22) too. The rest , and terms contribute to the double scattering forward -elastic amplitude.

The double scattering amplitude is given by integration of Eq.(III.3) over PK

 M(d)=i2π3/2∬d2q′M(q,q′;S,\boldmathσ). (32)

According to Eq. (11), in order to get the TVPC amplitude one has to calculate the matrix element of the operator given by Eq.(32) at over definite initial and final spin states:

 ˜g=1(2π)3/2∫d2q′<μ′=12,λ′=0|M(q=0,q′;S,\boldmathσ)|μ=−12,λ=1>. (33)

When considering the double scattering mechanism, in addition to three vectors defined after Eq. (13) it is convenient to introduce two more sets of orthonormal unit vectors for the first () and second collision as it was done in Ref. PK . At zero scattering angle we have , where () is the transferred momentum in the first (second) collision; , , . In the eikonal approximation vectors and are orthogonal to and we assume PK . The Cartesian projections for unit vectors can be written in terms of components of the two-dimensional vector (OZ axis is directed along ) as , , .

### iv.1 g′ term

Non-zero matrix elements of the isospin-operator connected with the term in Eq. (22) are

 =−i2,=i2. (34)

Therefore, the term contributes only to the charge exchange transitions. Two allowed double scattering amplitudes with one TVPC and another one T-even P-even -interaction are depicted in Fig. 1. Within the operator formalism these two terms can be evaluated in the following way. For pure T-even P-even (TCPC) interactions the transition operator for the charge-exchange mechanism of the process has a form glauber-franco

 OcTVPC=−12[Mc(q2)Mc(q1)], (35)

where is the transition operator for the strong charge-exchange amplitude that is equal to the amplitude. We note that according to Eq. (34), for the TVPC NN-interaction with the terms the amplitude differs from the amplitude by the sign. In order to get the TVPC operator of the charge-exchange scattering, we make in Eq. (35) the replacement , where the index means either or and is the TVPC charge-exchange NN-scattering operator, normalized as in Eq. (13) and related to the - operator given by Eq. (22) as

 TpN=mN4√πkpNtpN. (36)

Furthermore, we neglect the terms of the second order in as compared to the first order and omit the T-even term . As a result, the TVPC charge-exchange operator takes the form

 OcTVPC=−12[Mnp→pn(q2)Tpn→np(q1)+Tnp→pn(q2)Mpn→np(q1)]. (37)

For further evaluation it is convenient to use . Under the sign of the integral over in Eq. (33) the operator (37) is not changed after the substitution . In order to find the operators , , , and introduced in Eq. (28), it is convenient to add to the right side of Eq. (37) the term and divide the obtained sum by the factor of 2:

 OcTVPC=fI+fII, (38)

where

 fI=−14[(Mn(q2)Tpn→np(q1)+Tnp→pn(q2)Mn(q1))+(q1↔q2)], fII=14[Mp(q2)Tpn→np(q1)+Tnp→pn(q2)Mp(q1)+(q1↔q2)]. (39)

Using Eqs. (13), (34) and symmetry in respect of the replacement , we find that , , , and terms cancel in operators and :

 fI=g′Π[Cn(\boldmathσ⋅^n1)(\boldmathσn−% \boldmathσp)⋅n1−C′n(\boldmathσn⋅^n1)(\boldmathσp)⋅n1)+C′nn1^n1], fII=g′Π[Cp(\boldmathσ⋅^n1)(\boldmathσp−% \boldmathσn)⋅n1−C′p(\boldmathσp⋅^n1)(\boldmathσn)⋅n1)+C′pn1^n1], (40)

where

 Π=4√πmNkpN. (41)

Making the sum we find the operators and in Eq. (28) for the -term as the following

 U=g′Π(C′n+C′p)n1^n1, Vp=(Cp−Cn)(\boldmathσ⋅n1)^n1,Vn=(Cn−Cp)(\boldmathσ⋅n1)^n1, Wij=−g′Π(C′n+C′p)n1i^n1j,Wii=−g′Π(C′n+C′p)n1^n1. (42)

One can see from Eqs. (IV.1), that . Furthermore, taking into account the relation , we find that for the -term the operator Eq. (28) does not depend on the spin of the proton beam . As a result the transition operator given by Eq. (III.3) is diagonal in respect of spins of the proton beam. According to Eq. (11), it means that the contribution of term to the TVPC amplitude is equal to zero. We emphasize that this result is true for the S- and D- components of the deuteron wave function and for all spin terms in the transition amplitude (III.3) allowed in the Glauber formalism. It is easy to find that this result is valid for the scattering too.

### iv.2 h and g-terms

The TVPC interaction corresponding to the and terms in Eq. (22) occurs both in the - and - elastic scattering. Following to Ref. glauber-franco (see Eq. (2.7) in it) and PK we consider the symmetric and antisymmetric parts of the operator :

 Q(d)+=12[(Tpp(q1)+Mp(q1)(Tpn(q2)+Mn(q2))+(Tpn(q2)+