New CP violation in neutrino oscillations

New CP violation in neutrino oscillations

M. C. Gonzalez-Garcia,1,2,3,*Y. Grossman,4,†A. Gusso,1,5,‡and Y. Nir6,§

1_{Instituto de Fı´sica Corpuscular, Universitat de Vale`ncia–C.S.I.C, Edificio Institutos de Paterna, Apt 22085, 46071 Vale`ncia, Spain} 2_{Theory Division, CERN CH1211, Geneva 23, Switzerland}

3_{C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3840} 4_{Department of Physics, Technion–Israel Institute of Technology, Technion City, 32000 Haifa, Israel}

5_{Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, Brazil} 6_{Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel}

共Received 22 May 2001; published 9 October 2001兲

Measurements of C P-violating observables in neutrino oscillation experiments have been studied in the literature as a way to determine the C P-violating phase in the mixing matrix for leptons. Here we show that such observables also probe new neutrino interactions in the production or detection processes. Genuine C P violation and fake C P violation due to matter effects are sensitive to the imaginary and real parts of new couplings. The dependence of the C P asymmetry on the source-detector distance is different from the standard one and, in particular, enhanced at short distances. We estimate that future neutrino factories will be able to probe in this way new interactions that are up to four orders of magnitude weaker than the weak interactions. We discuss the possible implications for models of new physics.

DOI: 10.1103/PhysRevD.64.096006 PACS number共s兲: 11.30.Er, 13.15.⫹g, 14.60.Pq

I. NEW CP VIOLATION IN NEUTRINO INTERACTIONS In the future, neutrino oscillation experiments will search for C P-violating effects 关1–22兴. The standard model, ex-tended to include masses for light, active neutrinos, predicts that C P is violated in neutrino oscillations through a single phase in the mixing matrix for leptons. This effect is sup-pressed by small mixing angles and small mass differences. It is not unlikely, however, that the high-energy physics that is responsible for neutrino masses and mixing involves also new neutrino interactions. Such interactions provide new sources of C P violation. In this work we study C P-violating effects due to contributions from new neutrino interactions to the production and/or detection processes in neutrino oscillation experiments. We investigate the follow-ing questions:

共i兲 How would new, CP-violating neutrino interactions

manifest themselves in neutrino oscillations?

共ii兲 Are the effects qualitatively different from the

stan-dard models ones? In particular, can we use the time 共or, equivalently, distance兲 dependence of the transition probabil-ity to distinguish between standard model and new C P vio-lation?

共iii兲 How large can the effects be? In particular, do the

new interactions suffer from suppression factors related to mixing angles and mass differences?

共iv兲 Can the new CP violation be observed in proposed

experiments? What would be the optimal setting for these observations?

共v兲 Which models of new physics can be probed in this

way?

The plan of this paper goes as follows. In Sec. II we present a parametrization of the new physics effects that are of interest to us and explain the counting of independent C P-violating phases in our framework. In Sec. III we evalu-ate the new physics effects on the transition probability in neutrino vacuum oscillation experiments. 共A full expression for the transition probability, without any approximations concerning mixing angles and mass differences, is given in the Appendix.兲 In Sec. IV we investigate the resulting CP asymmetry and compare the new physics contribution to the standard one共that is, the contribution to the asymmetry from lepton mixing兲. In Secs. V and VI we evaluate the new phys-ics effects on, respectively, the transition probability and C P asymmetry, in neutrino matter oscillations. In Sec. VII we study how these effects can be observed in future neutrino factory experiments. In particular, we estimate a lower bound on the strength of the new interactions that can be observed in these experiments. This lower bound is compared to ex-isting model–independent upper bounds in Sec. VIII. We summarize our results and discuss some of the implications that would arise if a signal is experimentally observed in Sec. IX.

II. NOTATION AND FORMALISM

In this section we give a model-independent parametriza-tion of the new physics effects on producparametriza-tion and detecparametriza-tion processes in neutrino oscillation experiments. We put special emphasis on C P-violating phases.

We denote by兩␯i典, i⫽1, 2, and 3, the three neutrino mass eigenstates. We denote by兩␯_␣

典

the weak interaction partners of the charged lepton mass eigenstates␣⫺(␣⫽e,␮,␶):

兩␯␣

典

⫽

兺

i

U_␣i兩␯i

典

. 共2.1兲

*Email address: [email protected]

†_{Email address: [email protected]} ‡_{Email address: [email protected]}

Whenever we use an explicit parametrization of the lepton mixing matrix 关23,24兴, we will use the most conventional one, U⬅U23U13U12⬅

冉

1 0 0 0 c23 s23 0 ⫺s23 c23

冊

⫻

冉

c₁₃ 0 s₁₃ei␦ 0 1 0 ⫺s13e⫺i␦ 0 c13

冊冉

c₁₂ s₁₂ 0 ⫺s12 c12 0 0 0 1

冊

, 共2.2兲

with si j⬅ sin␪ij and ci j⬅ cos␪ij. Alternatively, a

convention-independent definition of the phase ␦ that we will use in our calculations is given by

␦⬅arg

冉

Ue3U␮3*

Ue1U␮1*

冊

. 共2.3兲

We consider new, possibly C P-violating, physics in the production and/or detection process. Such effects were pre-viously studied in Ref.关25兴, and we closely follow the for-malism of that paper. Most of the analysis in Ref.关25兴, how-ever, was carried out assuming C P conservation. We parametrize the new physics interaction in the source and in the detector by two sets of effective four-fermion couplings (G_NPs )_␣␤ and (G_NPd )_␣␤, where ␣,␤⫽e,␮,␶. Here (G_NPs )_␣␤ refers to processes in the source where a ␯_␤ is produced in conjunction with an incoming ␣⫺ or an outgoing ␣⫹ charged lepton, while (G_NPd )_␣␤refers to processes in the de-tector where an incoming␯_␤produces an␣⫺charged lepton. While the SU(2)_L gauge symmetry requires that the four-fermion couplings of the charged current weak interactions be proportional to GF␦␣␤, new interactions allow couplings

with ␣⫽␤. Phenomenological constraints imply that the new interaction is suppressed with respect to the weak inter-action: 兩共GNP s _兲 ␣␤兩ⰆGF, 兩共GNP d _兲 ␣␤兩ⰆGF. 共2.4兲

For the sake of concreteness, we consider the production and detection processes that are relevant to neutrino facto-ries. We therefore study an appearance experiment where neutrinos are produced in the process ␮⫹→e⫹␯_␣␯¯_␣⬘ and detected by the process ␯_␤d→␮⫺u, and antineutrinos are produced and detected by the corresponding charge-conjugate processes. Our results can be modified to any other neutrino oscillation experiment in a straightforward way. The relevant couplings are then (GNP

s

)e␤and (GNP

d

)_␮␤. It is con-venient to define small dimensionless quantities ⑀_␣␤s,d in the following ways: ⑀e␤ s _⬅ 共GNP s _兲e ␤

冑

兩GF⫹共GNP s _兲ee兩2_⫹兩共G NP s _兲e ␮兩2⫹兩共GNP s _兲e ␶兩2 , 共2.5兲 ⑀␮␤d ⬅ 共GNP d _兲 ␮␤

冑

兩GF⫹共GNP d _兲 ␮␮兩2⫹兩共GNP d _兲 ␮e兩2⫹兩共GNP d _兲 ␮␶兩2 .

Since we assume that兩⑀_␣␤s,d兩Ⰶ1, we will only evaluate their effects to leading共linear兲 order. New flavor-conserving inter-actions affect neutrino oscillations only at O(兩⑀兩2), and will be neglected from here on. 关More precisely, the leading ef-fects from flavor-diagonal couplings are proportional to ⑀ (flavor-diagonal)⫻⑀ (flavor-changing) and can therefore be safely neglected.兴

We use an explicit parametrization for only two of the⑀’s, with the following convention:

⑀e␮ s _⬅兩⑀ e␮ s _兩ei␦_{⑀, ⑀} ␮e d_*⬅兩⑀ ␮e d_*兩ei␦_⑀⬘_{. 共2.6兲}

Alternatively, we can define the phases ␦_⑀ and ␦_⑀

⬘

in a convention-independent way: ␦⑀⬅arg

冉

⑀e␮ s Ue1U_␮1*

冊

, ␦_⑀

⬘⬅arg

冉

⑀␮e d_* Ue1U_␮1*

冊

. 共2.7兲

We would like to conclude this section with a comment on the number of independent C P-violating phases in our framework. It is well known that the three-generation mixing matrix for leptons depends, in the case of Majorana neutri-nos, on three phases. Two of these, related to the fact that there is no freedom in redefining the phases of neutrino fields, do not affect neutrino oscillations and are therefore irrelevant to our discussion. The other one is analogous to the Kobayashi-Maskawa phase of the mixing matrix for quarks. The freedom of redefining the phases of charged lep-ton fields is fully used to reduce the number of relevant phases to one. Consequently, it is impossible to remove any phases from the ⑀_␣␤s,d parameters. Each of these parameters introduces a new, independent C P-violating phase.

For example, when we discuss ␯e→␯␮ oscillations, our

results will depend on ⑀_es_␮ and ⑀_␮ed , and the UeiU_␮i* (i ⫽1,2,3) mixing parameters. This set of parameters depends

on three independent phases, one of which is the ␦ of

FIG. 1. The neutrino parameters that dominate Pe_␮ in the

com-plex plane. We show the relevant unitarity triangle, which is the geometrical presentation of the relation Ue1U␮1*⫹Ue2U␮2*

⫹Ue3U_␮3*⫽0, and the two parameters that describe the new

phys-ics in the production, ⑀_es_␮, and in the detector,⑀_␮ed*. The three in-dependent phases defined in the text, ␦, ␦_⑀, and␦_⑀⬘, are shown explicitly. The standard convention puts Ue1U_␮1* on the real axis.

Eq.共2.3兲, while the other two can be chosen to be␦_⑀and␦_⑀

⬘

of Eq.共2.7兲. This situation is illustrated in Fig. 1, where we show in the complex plane the unitarity triangle and the⑀s,d parameters that are most relevant to ␯e→␯␮oscillations.

III. TRANSITION PROBABILITY IN VACUUM In this section we derive the expression for the transition probability in neutrino oscillation experiments as a function of the mixing matrix parameters and the new physics param-eters. We denote by ␯_es the neutrino state that is produced in the source in conjunction with an e⫹, and by␯_␮d the neutrino state that is signaled by␮⫺ production in the detector:

兩␯e s

_典

_⫽

兺

i 关Uei⫹⑀e␮ s _U ␮i⫹⑀e␶ s _U ␶i兴兩␯i典, 共3.1兲 兩␯␮d

典⫽

兺

i 关U␮i⫹⑀␮e d

Uei⫹⑀␮␶ d

U_␶i兴兩␯i典.

关Note that the norm of the states so defined is one up to

effects ofO(兩⑀兩2), which we consistently neglect.兴 We obtain the following expression for the transition probability P_e␮

⫽兩

具

␯␮d兩␯e s

(t)

典兩

2, where ␯_es(t) is the time-evolved state that was purely ␯_es at time t⫽0:

Pe␮⫽

冏

兺

i

e⫺iEit关Uei_U

␮i

*_⫹⑀es_␮兩U

␮i兩2⫹⑀␮ed*兩Uei兩2⫹⑀e␶ s _U

␶iU␮i*⫹⑀␮␶d*U␮iU␶i*兴

冏

2

. 共3.2兲

Our results will be given in terms of ⌬mi j2 , ⌬i j, and x_{i j}, which are defined as

⌬mi j2_⬅m

i

2_⫺m

j

2

, ⌬i j⬅⌬mi j2/共2E兲, xi j⬅⌬i jL/2,

共3.3兲

where E is the neutrino energy and L is the distance between the source and the detector.

Equation 共3.2兲 will be the starting point of our calcula-tions. The full expression for Pe␮ in vacuum is given in the

Appendix, and has been used for our numerical calculations described below. To understand the essential features of our analysis it is, however, more useful to do the following. First we separate Pe␮into a standard model piece Pe␮

SM

and a new physics piece P_eNP_␮. What we mean by P_eSM_␮ is Pe␮(⑀_␣␤s,d⫽0).

This is the contribution to Pe␮ from the standard model

ex-tended to include neutrino masses but no new interactions. In contrast, P_eNP_␮ contains all the⑀_␣␤s,d–dependent terms. Second, since the atmospheric and reactor neutrino data imply that

兩Ue3兩 is small and the solar neutrino data imply that ⌬m12

2

/⌬m₁₃2 is small, we expand P_eSM_␮ to second order and P_eNP_␮to first order in兩Ue3兩 and ⌬m₁₂2 .

For P_eSM_␮ we obtain Pe␮

SM_⫽4x 21

2 _兩Ue2兩2_兩U

␮2兩2⫹4 sin2x31兩Ue3兩2兩U␮3兩2

⫹4x21sin 2x31Re共Ue2Ue3*U␮2* U␮3兲

⫺8x21sin2x31Im共Ue2Ue3*U␮2*U␮3兲. 共3.4兲

The first term is the well known transition probability in the two-generation case. The second term gives the well known transition probability in the approximation that ⌬m₁₂2⫽0. The last term is a manifestation of the standard model C P violation. For P_eNP_␮ we obtain Pe␮ NP_{⫽⫺4 sin}2_x 31Re关Ue3*U␮3„⑀␮e d_*⫹⑀ e␮ s _共1⫺2兩U ␮3兩2兲 ⫺2⑀e␶ s U_␮3*U_␶3…兴 ⫹4x21sin 2x31Re关Ue2*U␮2共⑀e␮ s _兩U ␮3兩2⫹⑀e␶ s _U ␮3 * U_␶3兲兴 ⫺4x21Im关Ue2*U␮2„⑀␮e d_*⫹⑀ e␮ s _共1⫺兩U ␮3兩2兲 ⫺⑀e␶ s _U ␮3

* U_␶3…兴⫺2 sin 2x31Im关Ue3*U␮3共⑀␮e d_*⫹⑀ e␮ s _兲兴 ⫺4x21cos 2x31Im关Ue2*U␮2共⑀e␮ s _兩U ␮3兩2 ⫹⑀e␶ s _U ␮3 * U_␶3兲兴. 共3.5兲

The last three terms in this expression are C P violating and would be the basis for our results.

IV. CP VIOLATION IN VACUUM OSCILLATIONS To measure C P violation, one will need to compare the transition probability Pe␮ evaluated in Sec. III to that of the

C P-conjugate process P¯ ␮¯e . The latter will be measured in

oscillation experiments where antineutrinos are produced in the source in conjunction with e⫺ and detected through␮⫹ production. It is clear that a C P transformation relates the production processes, ␮⫺→e⫺¯␯_␣␯_␣⬘ and ␮⫹→e⫹␯_␣¯␯_␣⬘. As concerns the detection processes, ¯␯_␤u→␮⫹d and ␯_␤d

→␮⫺_{u, the situation is less straightforward. We have G} ␤␮

d ⬀

具

p␮⫺兩␮¯⫺¯u␯_␤d兩␯_␤n

典

and Gd_␤¯␮¯⬀

具

n␮⫹兩␮¯⫹¯d¯␯_␤u兩¯␯_␤p

典

. The relation is through C P and crossing symmetry, but for a four-fermion interaction this is equivalent to a C P transfor-mation.

C P transformation of the Lagrangian takes the elements of the mixing matrix and the ⑀ terms into their complex conjugates. It is then straightforward to obtain the transition probability for antineutrino oscillations. Our interest lies in the C P asymmetry,

AC P⫽

P_⫺

P_⫹, 共4.1兲

where

P_⫾⫽Pe␮⫾Pe¯ ␮¯. 共4.2兲

We quote below the leading contributions for ‘‘short’’ dis-tances, x31Ⰶ1. In some of the observables, we consider two limiting cases for 兩Ue3兩:

the ‘‘large’’ s13 limit:

x21/x31Ⰶ兩共Ue3U␮3兲/共Ue2U␮2兲兩,

共4.3兲

the small s13 limit:

x₂₁/x₃₁Ⰷ兩共U_e3U_␮3兲/共U_e2U_␮2兲兩.

The C P conserving rate P_⫹ is always dominated by the standard model. It is given by

P_⫹⫽

再

8x₃₁2 兩Ue3U_␮3兩2 large s13 8x₂₁2 兩Ue2U_␮2兩2 small s13.

共4.4兲

The C P-violating difference between the transition prob-abilities within the standard model can be obtained from Eq.

共3.4兲:

P_⫺SM⫽⫺16x₂₁x₃₁2 Im共U_e2U_␮2* U_e3*U_␮3兲. 共4.5兲 As is well known, C P violation within the standard model is suppressed by both the small mixing angle 兩Ue3兩 and the small mass-squared difference ⌬m₁₂2 . More generally, it is proportional to the Jarlskog measure of C P violation, J

⫽Im(Ue2U␮2* Ue3*U␮3). For short distances (x21,x31Ⰶ1), the dependence of P_⫺SMon the distance is L3. Since it is C P violating, it should be odd in L. The absence of a term linear in L comes from the fact that the standard model requires for C P to be violated, that all three mass-squared differences do not vanish, that is, P_⫺⬀⌬21⌬31⌬32. In the limit x21/x31

Ⰶ兩(Ue3U␮3)/(Ue2U␮2)兩, we obtain the following standard

model asymmetry:

A_{C P}SM⫽⫺2x21Im

冉

Ue2U_␮2*

Ue3U␮3*

冊

. 共4.6兲

In the small s13limit, the standard C P violation is unobserv-ably small.

The C P-violating difference between the transition prob-abilities that arises from the new physics interactions can be obtained from Eq.共3.5兲:

P_⫺NP⫽

再

⫺8x31

Im关Ue3*U_␮3共⑀_␮ed*_⫹⑀_es_{␮兲兴 large s}

13

⫺8x21Im关Ue2*U␮2共⑀_␮ed*⫹⑀e␮

s _{兲兴 small s}

13.

共4.7兲

We learn that C P violation beyond the weak interactions requires only that either 兩Ue3兩 or ⌬m₂₁2 be different from zero, but not necessarily both. Also the dependence on the

distance is different: for short distances, P_⫺NP⬀L. From Eqs.

共4.4兲 and 共4.7兲 we obtain the following new physics

contri-bution to the C P asymmetry:

A_{C P}NP⫽

冦

⫺ 1 x31 Im

冉

⑀␮e d_*⫹⑀ e␮ s Ue3U_␮3*

冊

large s13 ⫺ 1 x21 Im

冉

⑀␮e d_*⫹⑀ e␮ s Ue2U_␮2*

冊

small s13. 共4.8兲

The apparent divergence of A_{C P}NP for small L is only due to the approximations that we used. Specifically, there is an O(兩⑀兩2_{) contribution to P}

⫹that is constant in L关25兴, namely

P_⫹⫽O(兩⑀兩2) for L→0. In contrast, P_⫺⫽0 in the L→0 limit to all orders in兩⑀兩.

Equations 共4.7兲 and 共4.8兲 lead to several interesting con-clusions:

共i兲 It is possible that, in CP-violating observables, the

new physics contributions compete with or even dominate over the standard model ones in spite of the superweakness of the interactions (兩⑀兩Ⰶ1). Given that for the proposed ex-periments x₃₁ⱗ1, it is sufficient that

max共兩⑀_es_␮兩,兩⑀_␮ed 兩兲⭓min共兩Ue3兩,x₂₁兲 共4.9兲 for the new contribution to the C P-violating difference P_⫺ to be larger than the standard one.

共ii兲 The different distance dependence of P_⫺NP

and P_⫺SM will allow, in principle, an unambiguous distinction to be made between new physics contributions of the type de-scribed here and the contribution from lepton mixing.

共iii兲 The 1/L dependence of AC PNP _{suggests that the optimal} baseline to observe C P violation from new physics is shorter than the one optimized for the standard model.

We carried out a numerical calculation of the probabilities P_⫾ and asymmetry AC P as a function of the distance

be-tween the source and the detector. We use E_␯⫽20 GeV, which is the range of neutrino energy expected in neutrino factories. For the neutrino parameters, we take ⌬m₃₁2⫽3

⫻10⫺3 _eV2_{and tan}2_␪

23⫽1, consistent with the atmospheric neutrino measurements 关26兴, and ⌬m₂₁2⫽10⫺4 eV2 and tan2␪12⫽1, consistent at present with the large mixing angle

共LMA兲 solution of the solar neutrino problem 关26,27兴. As

concerns the third mixing angle and C P-violating phase in the lepton mixing matrix, we consider two cases. First, we take s₁₃⫽0.2, close to the upper bound from CHOOZ

关28,29,26兴, and␦⫽␲/2. This set of parameters is the one that maximizes the standard C P asymmetry. Second, we take s13⫽0, in which case there is no standard CP violation in the lepton mixing. As concerns the effects of new physics, we demonstrate them by taking only兩⑀_es_␮兩⫽” 0. With our first set of mixing parameters 共maximal standard CP violation兲, we take兩⑀_es_␮兩⫽10⫺3and␦_⑀⫽0. With our second set of mix-ing parameters 共zero standard CP violation兲, we take 兩⑀_es_␮兩

⫽10⫺4_and␦

⑀⫽␲/2. Our choice of C P-violating phases can

limit, the C P asymmetry depends on arg关⑀_es_␮/(Ue3U␮3* )兴 ⫽␦⑀⫺␦, while in the small s13 limit it depends on arg关⑀_es_␮/(U_e2U_␮2* )兴⫽␦_⑀. We use the full expression for the transition probabilities that is presented in the Appendix. Consequently, the only approximation that we make is that we omit effects ofO(兩⑀兩2).

The results of this calculation are presented in Fig. 2. The left panels correspond to the first case共maximal standard CP violation兲 and the right ones to the second 共zero standard CP violation兲. For each case we present, as a function of the distance between the source and the detector, P_⫹ 共dotted line兲, P_⫺SMand A_{C P}SM共dashed lines in, respectively, upper and lower panels兲, and P_⫺NPand AC P

NP _{共solid lines in, respectively,} upper and lower panels兲.

We learn a few interesting facts:

共i兲 The new physics contribution to CP violation can

dominate over even the maximal standard C P violation for values of兩⑀兩 as small as 10⫺4. This is particularly valid for distances shorter than 1000 km.

共ii兲 The approximations that lead to Eqs. 共4.7兲 and 共4.8兲

are good for Lⱗ5000 km.

共iii兲 As anticipated from our approximate expressions,

for short enough distances, P_⫺NPgrows linearly with distances and A_{C P}NP is strongly enhanced at short distances.

共iv兲 In the large s13 case, the new C P violation is sensi-tive mainly to the phase difference ␦⫺␦_⑀ and is almost in-dependent of the solar neutrino parameters.

共v兲 In the very small s13 limit, the new C P violation is

proportional to sin␦_⑀. The rate P_⫺NP is suppressed by the solar neutrino mass difference and mixing angle.

V. TRANSITION PROBABILITY IN MATTER Since long-baseline experiments involve the propagation of neutrinos in the matter of Earth, it is important to under-stand matter effects on our results. For our purposes, it is sufficient to study the case of constant matter density. Then the matter contribution to the effective ␯e mass, A ⫽

冑

2GFNe, is constant.

One obtains the transition probability in matter by replac-ing the mass-squared differences⌬i j and mixing angles U_␣i with their effective values in matter, ⌬i jm and U_␣im. The full expression for Pe␮in matter can then be written in terms of

x_{i j}mand U_␣im by a straightforward modification of the vacuum probability given the Appendix. To understand the matter effects it is, however, more useful to take into account the smallness of 兩Ue3兩 and x₁₂. We will expand the transition probability in these parameters to second order for P_eSM_␮ and to first order for P_eNP_␮.

For the standard model case, we obtain

P_eSM_␮⫽4

冉

⌬21 A

冊

2 sin2

冉

AL 2

冊

兩Ue2U␮2兩 2 ⫹4

冉

⌬31 B

冊

2 sin2

冉

BL 2

冊

兩Ue3U␮3兩 2_⫹8

冉

⌬21 A

冊

⫻

冉

⌬31 B

冊

sin

冉

AL 2

冊

sin

冉

BL 2

冊

⫻兵cos x31Re关Ue3*U␮3Ue2U_␮2* 兴 ⫺sin x31Im关Ue3*U␮3Ue2U␮2* 兴其, 共5.1兲 where B⫽⌬31⫺A. 共5.2兲

Again, the first term is the full result for two generations, and the second is the full result for the case of ⌬₂₁⫽0. The last term violates C P. In the limit A⫽0, Eq. 共3.4兲 is reproduced. Note that our definition of B is such that B changes sign according to whether ⌬₃₁ is larger or smaller than A. This is different from the usual convention where B⫽兩⌬31⫺A兩. The standard model results are an even function of B and either definition can be used. But for the new physics results given below, the choice of convention is important.

For the new physics contribution, we find FIG. 2. Transition probabilities and C P asymmetries in vacuum

as a function of the distance. In the upper panels the curves corre-spond to P_⫹SM共dotted兲, P_⫺SM共dashed兲 and P_⫺NP共solid兲. In the lower panels the curves correspond to AC P

NP _{共solid兲 and A}

C P

SM_{共dashed兲. In}

the left panels, s13⫽0.2, ␦⫽␲/2, 兩⑀e␮ s

兩⫽10⫺3_{, and ␦}

⑀⫽0. In the

right panels, s13⫽0, 兩⑀_es_␮兩⫽10⫺4, and␦_⑀⫽␲/2. In all curves E_␯

⫽20 GeV, ⌬m13

2_⫽3⫻10⫺3 _eV2

, tan2␪23⫽1, ⌬m212⫽10⫺4 eV2, and tan2_␪12⫽1.

P_eNP_␮⫽4

冉

⌬21 A

冊

sin 2

冉

AL 2

冊

Re关Ue2*U␮2„⑀␮e d_*⫺⑀ e␮ s _共1⫺2兩U ␮3兩2兲⫹2⑀e␶ s _U ␮3 * U_␶3…兴⫺4

冉

⌬31 B

冊

sin 2

冉

BL 2

冊

Re关Ue3*U␮3„⑀␮e d_*⫹⑀ e␮ s ⫻共1⫺2兩U␮3兩2兲⫺2⑀e␶ s U_␮3* U_␶3…兴⫺2

冉

⌬21

A

冊

sin共AL兲Im关Ue2*U␮2共⑀␮e*

d_⫹⑀ e␮

s _兲兴⫺2

冉

⌬31

B

冊

sin共BL兲Im关Ue3*U␮3共⑀␮e*

d_⫹⑀ e␮ s _兲兴 ⫺8 sin

冉

AL₂

冊

sin

冉

BL 2

冊

cos x31

再冉

⌬31 B

冊

Re关Ue3*U␮3„⑀e␮ s _共1⫺兩U ␮3兩2兲⫺⑀e␶ s U_␮3* U_␶3…兴⫺

冉

⌬21 A

冊

Re关Ue2*U␮2共⑀e␮ s _兩U ␮3兩2 ⫹⑀e␶ s _U ␮3 * U_␶3兲兴

冎

⫹8 sin

冉

AL 2

冊

sin

冉

BL 2

冊

sin x31

再冉

⌬31 B

冊

Im关Ue3*U␮3„⑀e␮ s _共1⫺兩U ␮3兩2兲⫺⑀e␶ s _U ␮3 * U_␶3…兴 ⫺

冉

⌬_A21

冊

Im关Ue2*U_␮2共⑀e␮ s _兩U ␮3兩2⫹⑀e␶ s U_␮3* U_␶3兲兴

冎

. 共5.3兲

Unlike the case of vacuum oscillation, P_⫺ will receive con-tributions from both C P-violating terms共proportional to the imaginary parts of various combinations of parameters兲 as C P conserving terms共proportional to the real parts兲.

Note that, in addition to the effects of new neutrino inter-actions in the source and in the detector, there could be other, independent effects due to new neutrino interactions with matter during their propagation 关30–32兴. Such effects have been studied in the context of solar and atmospheric neutri-nos共see e.g., Refs. 关33–35兴兲 but we neglect them here.

VI. CP VIOLATION IN MATTER OSCILLATIONS Since matter in Earth is not C P symmetric, there will be contributions to AC P even in the case when there is no C P

violation. It is our purpose in this section to evaluate these contributions and, in particular, the fake asymmetry that is related to the real part of ⑀. We denote the matter-related contribution to P_⫺ by P_⫺m⬅P_⫺(A)⫺P_⫺(A⫽0). Since the leading contributions to P_⫹ are the same as in the vacuum case 关Eq. 共4.4兲兴, we can similarly define the matter-related contribution to AC P: AC P

m _⬅P

⫺

m_{/ P}

⫹. Note that in the

evaluation of P_{¯ ␮¯e} from the expressions that we found for P_e␮ we need not only to replace U_␣i and ⑀_␣␤s,d with their complex conjugates, but also A with ⫺A.

For the standard model, we obtain from Eq. 共5.1兲, in the small x31and large s13 limits,

共P_⫺m_兲SM_⫽16 3 x31

4

冉

A

⌬31

冊

兩Ue3

U_␮3兩2. 共6.1兲 In the small s13 limit (x21/x31Ⰷ兩(Ue3U␮3)/(Ue2U␮2)兩) the

standard model effect is unobservably small, and we do not consider it here. Taking into account that 关see Eq. 共4.4兲兴 P_⫹⬇8x₃₁2兩U_e3U_␮3兩2_{, we obtain} 共AC Pm _兲SM_⫽2 3x31 2

冉

A ⌬31

冊

. 共6.2兲

For the new physics contribution, we obtain from Eq.

共5.3兲, in the small x31limit,

共P⫺m兲NP⫽

再

8x₃₁2 A

⌬31

Re关Ue3*U_␮3共⑀_␮ed*_⫺⑀_es_{␮兲兴 large s}

13

8x₂₁2 A

⌬21

Re关Ue2*U_␮2共⑀_␮ed*_⫺⑀_es_{␮兲兴 small s}

13, 共6.3兲 and 共AC Pm _兲NP_⫽

冦

A ⌬31 Re

冉

⑀␮e d_*⫺⑀ e␮ s U_e3U_␮3*

冊

large s13 A ⌬21 Re

冉

⑀␮e d_*⫺⑀ e␮ s Ue2U_␮2*

冊

small s₁₃. 共6.4兲

We would like to make a few comments regarding our results here:

共i兲 Each of the four contributions has a different

depen-dence on the distance. In the short distance limit, we have

共P_⫺m_兲SM_⬀L4_{, P} ⫺ SM_⬀L3_{, 共P} ⫺ m_兲NP_⬀L2_{, P} ⫺ NP_⬀L, 共6.5兲 and, equivalently, 共AC Pm _兲SM_⬀L2_{, A} C P SM ⬀L, 共AC P m _兲NP_⬀L0_{, A} C P NP_⬀1/L. 共6.6兲

One can then distinguish between the various contributions, at least in principle.

共ii兲 If the phases of the⑀’s are of order 1, then the genuine C P asymmetry will be larger 共at short distances兲 than the fake one.

共iii兲 It is interesting to note that the search for CP

viola-tion in neutrino oscillaviola-tions will allow us to constrain both Re(⑀) and Im(⑀).

We carried out a numerical calculation of the probabilities P_⫾m and asymmetry AC P

m

as a function of the distance be-tween the source and the detector. We again use E_␯

⫽20 GeV, ⌬m31

2_{⫽3⫻10⫺3}

eV2, tan2␪23⫽1, ⌬m21 2

⫽10⫺4 _eV2_{, tan}2_␪

12⫽1, and s13⫽0.2 or 0. For the new physics parameters, we take兩⑀_es_␮兩⫽10⫺3. To isolate the

mat-ter effects we now, however, switch off all genuine C P vio-lation, that is, we take ␦⫽␦_⑀⫽0 in both cases.

The results of this calculation are presented in Fig. 3. The left panels correspond to the first case 共large s13) and the right ones to the second 共vanishing s₁₃). For each case we present, as a function of the distance between the source and the detector, P_⫹共dotted line兲, (P_⫺m)SMand (A_{C P}m )SM共dashed lines in, respectively, upper and lower panels兲, and (P_⫺m)NP and (A_{C P}m )NP 共solid lines in, respectively, upper and lower panels兲.

We learn a few interesting facts:

共i兲 The new physics contribution to the fake CP violation

can dominate over the standard contribution for values of兩⑀兩 as small as 10⫺4. This is particularly valid for distances shorter than 500 km.

共ii兲 As anticipated from our approximate expressions, for

short enough distances ( P_⫺m)NPgrows quadratically with dis-tances and (A_{C P}m )NP is independent of the distance.

共iii兲 Both the standard contribution and the new

contribu-tion to P_⫺m are suppressed by a small s13. The s13 suppres-sion is however stronger for P_⫹ than it is for ( P_⫺m)NP_. Con-sequently, the new physics contribution to (A_{C P}m )NPbecomes very large for vanishing s13.

In reality, the measured P_⫺ and AC P will be affected by

both genuine C P-violating contributions and matter-induced contributions. This situation is illustrated in Fig. 4. We present P_⫹ 共dotted curve兲, P_⫺SM and A_{C P}SM共dashed curves in, respectively, upper and lower panels兲, and P_⫺NP and A_{C P}NP

共solid curves in, respectively, upper and lower panels兲, as a

function of the distance. For the neutrino parameters, we always take⌬m31

2_{⫽3⫻10⫺3}

eV2and tan␪23⫽1, consistent with the atmospheric neutrino data. For the other parameters, we take three cases:共a兲 Left panel: we take the LMA param-eters (⌬m₂₁2⫽10⫺4 eV2 and tan␪12⫽1), ‘‘large’’ s13⫽0.2 and maximal phase␦⫽␲/2. This choice of parameters gives maximal standard C P violation. For the new physics param-eters we take 兩⑀_es_␮兩⫽10⫺3 and ␦_⑀⫽0. 共The reason for the choice of phase is that the dominant contributions depend on ␦⫺␦⑀.) 共b兲 Middle panel: we take the small mixing angle

共SMA兲 parameters (⌬m21 2 _{⫽5.2⫻10⫺6} eV2 关26,27兴, tan2␪12 ⫽7.5⫻10⫺4_{), s} 13⫽0.2, ␦⫽␲/2, 兩⑀e␮ s _{兩⫽10⫺3, and ␦} ⑀⫽0.

Here the standard C P violation is unobservably small, but the standard matter effects are still large.共c兲 Right panel: we take the LMA parameters and s13⫽0. With a vanishing s13, the total transition probability is highly suppressed as is the standard matter effect, and standard C P violation vanishes. For the new physics parameters we take 兩⑀_es_␮兩⫽10⫺4 and ␦⑀⫽␲/2. We take a smaller兩⑀e␮

s _{兩 so that our approximation}

will not break down.

We would like to emphasize the following points:

共i兲 Similar three cases will be the basis, in the next

sec-tion, for our analysis of the sensitivity of C P-violating ob-servables measured in neutrino factories to new physics ef-fects 共see Fig. 5兲.

共ii兲 With large s13, the dependence of the new physics effects 共and of the standard matter-induced effects兲 on the solar neutrino parameters is very weak.

共iii兲 A small or even vanishing s13 will suppress all the rates and will introduce a strong dependence on the solar neutrino parameters. The new physics contributions to AC P

will be, however, only slightly affected, because both the standard C P conserving rate and the new physics C P-violating rate are suppressed in the same way.

共iv兲 With large s13, the new physics C P-violating effects are dominated by the combination ␦⫺␦_⑀. With small 共but not vanishing兲 s13, the dependence is on both␦⫺␦⑀and␦⑀.

共v兲 For distances shorter than 800 km, the effects of 兩⑀兩 ⲏ10⫺3 _{are always dominant. For distances shorter than 300}

km, the new physics dominates even for兩⑀兩⬃10⫺4. VII. LONG-BASELINE EXPERIMENTS

We would like to quantify the sensitivity of a neutrino factory to the C P-violating effects from new neutrino inter-actions. For this purpose, we consider the measurement of the following integrated asymmetry关5兴:

AC P ¯_⫽N关␮ ⫺_兴/N 0关e⫺兴兩⫹⫺N关␮⫹兴/N0关e⫹兴兩⫺ N关␮⫺兴/N0关e⫺兴兩⫹⫹N关␮⫹兴/N0关e⫹兴兩⫺ . 共7.1兲

Here N关␮⫺兴/N0关e⫺兴兩⫹ refers to an oscillation experiment that has ␮⫹ decay as its production process: N关␮⫺兴 is the measured number of wrong-sign muons while N0关e⫺兴 is the expected number of ␯e charge current 共CC兲 interaction

events 共in the absence of oscillations兲. Similarly, N关␮⫹兴/N0关e⫹兴兩⫺ refers to an oscillation experiment that has␮⫺decay as its production process: N关␮⫹兴 is the mea-FIG. 3. Transition probabilities and fake C P asymmetries in

matter as a function of the distance. All C P-violating phases are set to zero. In the upper panels the curves correspond to P_⫹SM共dotted兲, ( P_⫺m)SM共dashed兲 and (P_⫺m)NP共solid兲. In the lower panels the curves correspond to (AC P

m

)NP 共solid兲 and (AC P m

)SM共dashed兲. In the left panels s13⫽0.2, and in the right panels s13⫽0. In all curves E_␯

⫽20 GeV, ⌬m312_⫽3⫻10⫺3 _eV2 , tan2␪23⫽1, ⌬m212⫽10⫺4 eV2, tan2_{␪12⫽1, ␦⫽␦} ⑀⫽0, and 兩⑀e␮ s 兩⫽10⫺3_.

sured number of wrong-sign muons while N0关e⫹兴 is the ex-pected number of¯ CC interaction events␯e 共again, in the

absence of oscillations兲. The measured number of wrong-sign muon events can be expressed as

N关␮⫺兴兩_⫹⫽N␮NT ␲m_␮2

E_␮

L2

冕

dE␯f␯共E␯兲␴CC共E␯兲Pe␮共E␯兲,

共7.2兲

where NT is the number of protons in the target detector, N␮

is the number of useful muon decays, E_␮is the muon energy, and m_␮is the muon mass. The function f_␯(E_␯) is the energy distribution of the produced neutrinos. We assume that the muons are not polarized, in which case f_␯(E_␯)⫽12x2(1

⫺x) with x⫽E_␯/E_␮. Finally, ␴CC(E_␯) is the

neutrino-nucleon interaction cross section which, in the interesting range of energies, can be taken to be proportional to the neutrino energy: ␴CC⫽␴0E␯ with ␴0⫽0.67

⫻10⫺38 _cm2_{/GeV for neutrinos and ␴} 0⫽0.34

⫻10⫺38 _cm2_{/GeV for antineutrinos. The expression for} N关e⫺兴兩_⫹ is obtained by an integral similar to Eq.共7.2兲, ex-cept that Pe␮ is replaced by 1.

We define A_{C P}NP as the contribution from new physics共that is, ⑀-dependent兲 to the integrated CP asymmetry. We take into account both genuine C P-violating and matter-induced contributions. 关In the limit of a real lepton mixing matrix, that is, no standard C P violation, the first contributions are proportional to Im(⑀) and the latter to Re(⑀).兴 We define ⌬A to be the statistical error on A¯ . In order to quantify theC P

significance of the signal due to new physics, we compute the ratio A_{C P}NP/⌬A.

The statistical error⌬A scales with distance and energy as follows: ⌬A⯝ 1

冑

N关␮⫹兴兩_⫺⫹N关␮⫺兴兩_⫹⬀ 1

冑

P_⫹SMNCC ⬀ 1

冑

E_␯ . 共7.3兲

To find this scaling, we took into account that the number of CC interactions scales as NCC⬀E_␯3/L2 while, for L

ⱗ3000 km, P_⫹S M_⬀L2_/E

␯

2

. Consequently, the dependence of ⌬A on the distance is very weak. Given our results for A_{C P}NP, we obtain the following scaling with distance of the signal-to-noise ratio:

A_{C P}NP/⌬A⬀

再

1/L genuine C P-violating effects const共L兲 matter induced effects.

共7.4兲

This behavior is illustrated in Fig. 5 where we display the signal-to-noise ratio A_{C P}NP/⌬A as a function of the distance. For simplicity, we consider only the effect of⑀_es_␮. The stan-dard C P violation is presented only in the upper panel, where it corresponds to maximal A_{C P}SM共LMA parameters are

⌬m21 2_⫽10⫺4

eV2 and tan␪12⫽1, large s13 and ␦⫽␲/2), while the middle panel has unobservably small A_{C P}SM 共SMA parameters are ⌬m₂₁2⫽5.2⫻10⫺6 eV2 and tan2␪12⫽7.5

⫻10⫺4_{兲, and the lower panel has zero AC P}SM

(s13⫽0). As concerns the new C P violation, the dashed line corresponds to the case with maximal C P-violating phase (␦_⑀⫽␲/2) and the solid line corresponds to purely matter-induced asymme-FIG. 4. Transition probabilities and fake C P asymmetries in matter as a function of the distance. In the upper panels the curves correspond to P_⫹SM共dotted兲, P_⫺SM共dashed兲 and P_⫺NP共solid兲. In the lower panels the curves correspond to AC P

NP

共solid兲 and AC P

SM

共dashed兲. In all

curves E_␯⫽20 GeV, ⌬m312⫽3⫻10⫺3 eV2_{, and tan}2_{␪23⫽1. In the left panels ⌬m21}2

⫽10⫺4 _eV2_{, tan␪12⫽1, s13⫽0.2, ␦⫽␲/2, 兩⑀}

e_␮ s _兩

⫽10⫺3_{, and␦}

⑀⫽0. In the middle panels ⌬m212⫽5.2⫻10⫺6 eV2, tan2␪12⫽7.5⫻10⫺4, s13⫽0.2, ␦⫽␲/2, 兩⑀e␮

s _兩⫽10⫺3_{, and␦}

⑀⫽0. In the

right panels⌬m212⫽10⫺4 eV2, tan␪12⫽1, s13⫽0, 兩⑀e␮

s _兩⫽10⫺4_{, and␦} ⑀⫽␲/2.

try (␦_⑀⫽0). In our calculations we have assumed a total of 1021useful␮⫺decays with energy E_␮⫽50 GeV and a 40-kt detector.

It is clear from the figure that the maximal sensitivity to new, C P-violating contributions to the production or detec-tion processes will be achieved with shorter distances, while the sensitivity to C P conserving contributions through mat-ter induced effects is almost independent of distance.

A truly short baseline experiment can potentially probe the O(兩⑀兩2)C P conserving effects. But in this case, due to the small signal, systematic errors will dominate over the statistical ones discussed above. It is unlikely that兩⑀兩 smaller thanO(10⫺3) can be signaled in such a measurement.

We next investigate the sensitivity to the size of the new physics interaction that can be achieved by the measurement of the integrated C P asymmetry. In Fig. 6, we show the regions in the 关Re(⑀_es_␮),Im(⑀_es_␮)兴 plane that will lead to A_{C P}NP/⌬A⫽3 共darker-shadow region兲 and A_{C P}NP/⌬A⫽1

共lighter-shadow regions兲 at L⫽732 km, the shorter baseline

discussed for an oscillation experiment at a neutrino factory. We have assumed a total of 1021 useful ␮⫺ decays with energy E_␮⫽50 GeV and a 40-kt detector. In all panels we have ␦⫽0 共no standard CP violation兲, ⌬m₃₁2⫽3

⫻10⫺3 _eV2 _{and tan␪}

23⫽1, and the LMA parameters

⌬m21

2_{⫽10⫺4 eV}2_{, and tan␪}

12⫽1. In the left panels we have s₁₃⫽0.2 and in the right ones s₁₃⫽0. In the upper panels Im(⑀_es_␮)⬎0, which, for our choice of parameters, results in a constructive interference between the matter-induced and

C P-violating effects, while in the lower panels Im(⑀e␮ s

)⬍0, which results in a destructive interference.

In order to illustrate the expected improvement in sensi-tivity to the new physics when the baseline is better opti-mized for this particular purpose, we plot in Fig. 7 the cor-responding regions when the measurement of the integrated C P asymmetry is performed at a distance of L⫽200 km.

We would like to emphasize the following three points:

共i兲 Figure 6 shows that 兩⑀兩 in the range 3⫻10⫺5_–10⫺4

would lead to a ‘‘3␴’’ effect.

共ii兲 A shorter distance will improve the sensitivity to the

new C P violation. Figure 7 shows that, for ␦⫽0, in which case C P-violating effects are proportional to Im(⑀), an im-provement by a factor of about 3 in the sensitivity to Im(⑀) is expected. In contrast, the sensitivity to Re(⑀) is not af-fected by the choice of baseline since the new physics con-tribution to the matter-induced asymmetry is independent of L.

共iii兲 A nonvanishing standard CP-violating phase, ␦⫽” 0,

together with a ‘‘large’’ s13, will change the interference pattern between the matter-induced and C P-violating contri-butions from new physics. The reason is that now some of the contributions depend on␦_⑀⫺␦, so that Re(⑀) and Im(⑀) do not correspond to matter-induced and C P-violating ef-fects in any simple way.

VIII. PHENOMENOLOGICAL CONSTRAINTS The measurements of Pe␮ and P¯ ␮¯e are sensitive to the

four effective couplings, ⑀_es_␮, ⑀s_e␶, ⑀_␮ed and ⑀_␮␶d . These di-mensionless couplings represent new flavor-changing 共FC兲 neutrino interactions. They are subject to various phenom-enological constraints. In this section, we present these bounds in order to compare them with the experimental sen-sitivity that we estimated in the previous section.

Before we present the bounds, we would like to clarify a subtlety that concerns the ⑀s couplings. Each of these cou-plings stands for several different processes. Specifically,⑀_es_␮ gives the amplitude for ␮⫹→e⫹␯_␮¯␯_␴ decays with ␴

⫽e,␮,␶ and, similarly, ⑀e␶ s

gives the amplitude for ␮⫹ →e⫹_␯

␶¯␯␴ decays with␴⫽e,␮,␶. The index␴ is irrelevant

for the analysis of P_e␮ ( P_{¯ ␮¯e} ), where we are only interested in the neutrino共antineutrino兲 interactions. This is the reason that we did not distinguish between the three possible pro-duction processes for each of the ⑀s’s. Most of the bounds that we discuss below do, however, depend on ␴. It is im-portant to understand that it is the weakest of the bounds which applies model independently.

We consider three types of upper bounds:

共i兲 There is a generic bound of O(0.1) on the purely

lep-tonic couplings⑀_␣␤s from universality in lepton decays and a somewhat weaker bound ofO(0.2) on the semihadronic cou-plings⑀_␣␤d from universality in pion decays关36兴. While uni-versality is experimentally confirmed to high accuracy, these bounds are rather weak because deviations from universality are O(⑀2).

共ii兲 By SU(2)Lsymmetry, the couplings are related to FC charged lepton interactions. The latter have not been ob-FIG. 5. The signal-to-noise ratio AC P

NP

/⌬A as a function of the distance L. We considered the following parameters for the experi-ment: E_␮⫽50 GeV, 1021 ␮⫺ decays and a 40-kt detector, and the neutrino parameters␦⫽0, ⌬m312⫽3⫻10⫺3 eV2_{, tan␪23⫽1. In}

the upper and lower 共middle兲 panels we use the LMA 共SMA兲 pa-rameters. In the upper two共lower兲 panels we use s13⫽0.2(0). For the new physics we take 兩⑀e␮

s _兩⫽10⫺3 _{and ␦}

⑀⫽0 or␲/2. In the

upper panel, the dotted curve gives the SM matter-subtracted asym-metry AC P

SM

(␦⫽␲/2)⫺AC P

SM

served, and there are experimental constraints on their strength. There could be SU(2)Lbreaking effects that would

somewhat enhance the neutrino couplings with respect to the corresponding charged lepton couplings. These effects are discussed in detail in Refs. 关37,36兴 where it is shown that they are constrained 共by electroweak precision data兲 to be small. Since our purpose is only to obtain order-of-magnitude estimates of the bounds, we neglect the possible SU(2)L breaking effects.

共iii兲 For some cases, the ⑀s _{coupling contributes at the}

loop level to the␮→e␥ and␮→3e decays. The question of how to extract reliable bounds from loop processes in an effective theory involves many subtleties. A calculation in the spirit of Ref. 关38兴 yields very weak bounds. Instead, we quote below the bounds in specific full high energy models. We emphasize however that, in contrast to the bounds from SU(2)_Lrelated charged lepton tree-level decays, the bounds that we quote for the loop processes may be violated in mod-els other than the ones that we consider.

The ⑀e␮ s

coupling gives the amplitude for ␮⫺→e⫺¯␯_␮␯_␴ decays with ␴⫽e,␮,␶. For ␴⫽e, there is a bound from muonium-antimuonium oscillations 关37兴:

兩⑀e␮

s _{兩⭐3.0⫻10}⫺3 _{共␴⫽e兲. 共8.1兲}

For␴⫽␶, we derive a bound from the␶⫹→e⫺␮⫹␮⫹decay:

兩⑀e␮

s _{兩⭐2.9⫻10⫺3 共␴⫽␶兲. 共8.2兲}

For␴⫽␮, there is no SU(2)L-related tree-level charged

lep-ton decay. Instead, by closing the neutrino lines into a loop, the four-Fermi coupling contributes to the ␮→e␥ and ␮ →3e decays. We quote here the bound in a specific full high energy model: if the effective␮L¯eL␯␮¯ coupling is induced␯␮

by an intermediate scalar triplet, the constraint from the ␮ →e␥ decay reads共see, for example, Ref. 关39兴兲

兩⑀e␮

s _兩⭐5⫻10⫺5 _{共␴⫽␮兲. 共8.3兲}

We emphasize again that the bound in Eq. 共8.3兲 is model dependent, in contrast to those of Eqs. 共8.1兲 and 共8.2兲.

The ⑀_es_␶ coupling gives the amplitude for ␮⫺→e⫺¯␯_␶␯_␴ decays with ␴⫽e,␮,␶. For ␴⫽e, there is a bound from the ␶⫺_→␮⫹_e⫺_e⫺ _{decay关37兴:}

兩⑀e␶

s _{兩⭐2.9⫻10⫺3 共␴⫽e兲. 共8.4兲}

For ␴⫽␮, we derive a bound from the␶⫺→␮⫹␮⫺e⫺ de-cay:

兩⑀e␶

s _{兩⭐3.1⫻10⫺3 共␴⫽␮兲. 共8.5兲}

For␴⫽␶ there is no SU(2)L-related tree level charged

lep-ton decay, but there is a direct one-loop contribution to ␮ →e␥ and ␮→3e. We again quote the bound in a specific FIG. 6. Regions in the plane of

关Re(⑀e_␮ s

),Im(⑀e_␮ s

)兴 that give

A_{C P}NP/⌬A⫽3 共darker shadow兲 and 1共light shadow兲. For the experi-ment, we take L⫽732 km, E_␮

⫽50 GeV, 1021 _␮⫺ _decays

and a 40-kt detector. For the neu-trino parameters, we take ␦⫽0,

⌬m312

⫽3⫻10⫺3 _eV2

, tan2␪23

⫽1, ⌬m212

⫽10⫺4 _eV2_{, and}

tan2␪12⫽1. In the left 共right兲 pan-els we have s13⫽0.2 0).

full high energy model: if the effective␮L¯eL␯␶¯ coupling is␯␶

induced by an intermediate scalar singlet, the constraint from the ␮→e␥ decay reads 共see, for example, Refs. 关40,41兴兲

兩⑀e␶

s _{兩⭐3.5⫻10⫺4 共}

␴⫽␶兲. 共8.6兲

Note that, within the effective theory, the contributions to ␮→e␥ from⑀_es_␮of Eq.共8.3兲 and⑀_es_␶of Eq.共8.6兲 are equal. The factor of O(7) difference in the respective bounds

关which reflects a ratio of O(50) between the contributions to

the rate兴 demonstrates their model dependence.

The⑀_␮ed coupling gives the amplitude for␯ed→␮⫺u. It is

constrained by muon conversion关37兴:

兩⑀␮ed 兩ⱗ2.1⫻10⫺6. 共8.7兲

The ⑀_␮␶d coupling gives the amplitude for␯_␶d→␮⫺u. It is constrained by the ␶⫺→␮⫺␳ decay关36兴:

兩⑀␮␶d 兩ⱗ10⫺2. 共8.8兲

The bound on兩⑀_␮␶d 兩 is the weakest that we obtain. Moreover, it is not unlikely that it is indeed the largest of the couplings since it is the only one not to involve a first-generation lep-ton. For precisely the same reason, however, its contribution

to Pe␮is suppressed by an additional power of兩Ue3兩, which

is the reason that it is omitted in our approximate expres-sions.

To summarize, we expect that all the⑀’s that play a role in the transition probabilities of interest are of O(10⫺3) or smaller. In Sec. VII, we learned that proposed experiments might probe these couplings down to values as small as O(10⫺4_{). This means that the possibility to measure new}

neutrino interactions through C P violation in neutrino oscil-lation experiments is open. Conversely, such future experi-ments can improve the existing bounds on FC neutrino inter-actions which, at present, come from rare charged lepton decays.

IX. CONCLUSIONS AND DISCUSSION We summarize the main points of our study:

共i兲 CP-violating observables are particularly sensitive to

new physics. The reason is that the standard C P violation that comes from the lepton mixing matrix gives effects that are particularly suppressed by small mass differences and mixing angles. Some of these suppression factors do not ap-ply to new contributions.

共ii兲 The fact that matter effects contribute to CP-violating

observables means that these observables are sensitive to both the C P conserving and C P-violating contributions from new physics.

FIG. 7. Regions in the plane of

关Re(⑀e␮ s

),Im(⑀_es_␮)兴 that give

AC P

NP

/⌬A⫽3 共darker shadow兲 and 1共light shadow兲. For the experi-ment, we take L⫽200 km, E_␮

⫽50 GeV, 1021 _␮⫺ _decays

and a 40-kt detector. For the neu-trino parameters, we take ␦⫽0,

⌬m312_⫽3⫻10⫺3 _eV2

, tan2␪23

⫽1, ⌬m212

⫽10⫺4 _eV2

, and tan2␪12⫽1. In the left 共right兲 pan-els we have s13⫽0.2 (0). Note that the scales in the right panels are different from the left panels and from Fig. 6.

共iii兲 The effects of new physics in the production and

detection processes depend on the source-detector distance in a way that is different from the standard one. One conse-quence of this situation is that, at least in principle, it is possible to disentangle standard and new effects. Another consequence is that in short distance experiments the new effects are enhanced.

共iv兲 Our rough estimate is that future neutrino

factories will be able to probe, through C P-violating observables, effects from new interactions that are up to about four orders of magnitude weaker than the weak inter-actions.

共v兲 The sensitivity to new physics effects is better than

most of the existing model-independent bounds.

We would like to mention that a similar共and, for specific models, even stronger兲 level of sensitivity may be achieved by other experiments that search for lepton flavor violation. Particularly promising are those involving muon decay and conversion共for a recent review, see Ref. 关46兴兲: for example, a future experiment at PSI will be sensitive to B(␮→e␥) at the 10⫺14level关47兴, and the MECO Collaboration has pro-posed an experiment to probe ␮⫺e conversion down to 5

⫻10⫺17_{, four orders of magnitude beyond present}

sensitivi-ties关48兴. If these experiments observe a signal, the search for related C P violation will become of particular importance.

What type of new physics will be implied in case that a signal is observed? The ⑀ couplings represent effective four-fermion interactions coming from the exchange of heavy particles related to new physics. If the new physics takes place at some high scale ⌳_NP, then one can set an upper bound:

⑀␣␤s,dⱗ

m_Z2

⌳NP

2 . 共9.1兲

The source of this bound is in the definition of ⑀, which is the ratio of the four-fermion operator to GF, and the fact that

it is maximal when the new physics contribution comes at tree level and the couplings are of order 1. Since the ex-pected experimental sensitivity is to兩⑀兩⭓O(10⫺4), we learn that we can probe models with

⌳NPⱗ10 TeV. 共9.2兲

If the new physics contributes to the relevant processes only at the loop level, there is another suppression factor in兩⑀兩 of order 1/16␲2. This would mean that such models can be probed only if ⌳NPⱗ1 TeV. Finally, if the flavor changing nature of the interaction introduces a suppression factor, e.g.,

兩⑀e␮ s _兩⬃m

␮/⌳NP, that by itself would be enough to make it unobservable in near future experiments. We thus learn that C P violation in neutrino oscillation experiments will explore models with a scale that is, at most, 1–2 orders of magnitude above the electroweak breaking scale, and where the flavor structure is different from the standard model.

Another point concerns the Dirac structure of the four-Fermi interaction. We did not present this explicitly in our discussion of the G_NPs,d couplings. However, it is implicitly assumed in our discussion that the Dirac structure is the same as that of the weak interactions, i.e., a (V-A)(V-A) structure. The reason for that is that the effects that we dis-cuss are a consequence of interference between weak and new interactions. A different Dirac structure would give strong suppression factors related to the charged lepton masses. While our formalism would still apply, these sup-pression factors would make the related effects practically unobservable.

We conclude that a signal is likely to imply new physics at a relatively low scale共up to 1–10 TeV兲 with new sources of flavor共and, perhaps, CP) violation. We know of several well motivated extensions of the standard model that can, in principle, induce large enough couplings. In particular, we have in mind loop contributions involving sleptons and gauginos in supersymmetric models, tree contributions in-volving charged singlet sleptons in supersymmetric models without R parity, and tree contributions involving a triplet scalar in left-right symmetric models. In another class of relevant models, such as the model of Ref. 关42兴, active neutrinos mix with singlet neutrinos. 共Here there can be Z-mediated contributions to the non-standard couplings, and the phenomenological constraints are different关43,44兴.兲 A detailed analysis of new neutrino interactions within relevant extensions of the standard model is beyond the scope of this paper, but preliminary results show that large enough couplings are allowed and in some cases even predicted关45兴.

ACKNOWLEDGMENTS

M.C.G.-G. and Y.N. thank the school of natural sciences in the Institute for Advanced Study 共Princeton兲, where part of this work was carried out, for the warm hospitality. M.C.G.-G. is supported by the European Union through Contract No. HPMF-CT-2000-00516. A.G. is supported by Fundac¸a˜o Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior 共CAPES兲. Y.N. is supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, by the United States–Israel Binational Science Foundation 共BSF兲 and by the Minerva Foundation 共Munich兲. This work was also supported by the Spanish DGICYT under grants PB98-0693 and PB97-1261, by the Generalitat Valenciana under grant GV99-3-1-01 and by the TMR network grant ERBFMRXCT960090 of the European Union and ESF network 86. This work was also supported by the fund for the promotion of research at the Technion.

APPENDIX: TRANSITION PROBABILITY IN VACUUM Neglecting terms of O(⑀2_{) and with no other} approximations, we obtain the following expression for Pe␮: