7 September 2000

Ž .Physics Letters B 488 2000 398–401www.elsevier.nlrlocaternpe

Confronting dilaton-exchange gravity with experiments

H.V. Klapdor-Kleingrothaus a, H. Pas a, U. Sarkar b¨a Max-Planck-Institut fur Kernphysik, P.O. Box 103980, D-69029 Heidelberg, Germany¨

b Physical Research Laboratory, Ahmedabad 380 009, India

Received 14 April 2000; received in revised form 18 May 2000; accepted 3 August 2000Editor: P.V. Landshoff

Abstract

We study the experimental constraints on theories, where the equivalence principle is violated by dilaton-exchangecontributions to the usual graviton-exchange gravity. We point out that in this case it is not possible to have any CPTviolation and hence there is no constraint from the CPT violating measurements in the K-system. The most stringent boundis obtained from the K yK mass difference. In contrast, neither neutrino oscillation experiments nor neutrinoless doubleL S

beta decay imply significant constraints. q 2000 Published by Elsevier Science B.V.

At present we have no indication for the theviolation of gravitational laws. But some theorieslike string theory suggest deviations from the usualgraviton-exchange theories of gravity. Thus it be-comes necessary to find out the extent of applica-bility of the general theory of relativity. Severalexperiments were performed to test the equivalence

w xprinciple 1 for ordinary matter and to test localw xLorentz invariance 2,3 . Attempts were also made to

w xtest these laws in the neutrino sector 4–6 , but theseworks included only tensorial interactions. In theK-system both tensorial and vectorial interactions

w xwere studied by many authors 7 .w xRecently it has been suggested 8 that string

theory may lead to a different kind of violation ofŽ .the equivalence principle VEP via interactions of

the dilaton field, which gives an additional contribu-tion to the usual graviton exchange gravity. The

Žresulting theory is of scalar-tensor type in contrast

.to purely tensorial VEP discussed previously withthe two particle static gravitational energy

V r syG m m 1qa a rr , 1Ž . Ž . Ž .N A B A B

where G is Newton’s gravitational constant and aN j

are the couplings of the dilaton field f to the matterfield of type j, c . This additional contribution mayj

result from a gravitational interaction

Lsm a c c f . 2Ž .j j j j

The distinct feature of this new contribution arespecific couplings of the dilaton field to differentmatter fields, which violates the equivalence princi-ple. It has been discussed recently whether thisfeature can be tested in neutrino oscillation experi-

w xments 9,10 .Unlike the violation of the equivalence principle

through tensorial interactions, in the dilaton-ex-change gravity the gravitational basis is always the

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0370-2693 00 00882-0

( )H.V. Klapdor-Kleingrothaus et al.rPhysics Letters B 488 2000 398–401 399

same as the mass basis, since the additional term dueto dilaton exchange is directly proportional to themass. For this reason, a dilaton-exchange gravitycannot explain the neutrino mixing phenomenon by

w xitself – the discussion in ref 9 seems to overlookthis point. However, it is possible that a mass differ-ence between degenerate neutrinos is implied, what

w xhas been considered in 9 . In this article we discussconstraints on this additional contribution from theK yK mass difference, the non-observation ofL S

neutrinoless double beta decay as well as neutrinooscillation experiments.

In a linearized classical theory one may replacethe effective mass of a fermion by

m) sm ym a f , 3Ž .i i i i c

where f is the classical value of the dilaton fieldc

and is proportional to the Newtonian potential f ,N

f sa f .c ext N

The first term results from the usual graviton-ex-change gravity, while the second term is the newcontribution coming from the dilaton-exchange grav-ity.

To get the interaction of the usual neutrinos ofdifferent flavour, we can rotate the effective hamilto-nian in the mass and the gravitational bases throughthe same unitary rotations

H sU H qH Uy1 . 4Ž .Ž .w m g

In the presence of the dilaton-exchange gravity, theeffective hamiltonian in the mass basis is

H qHm g

21 m ya m f 01 1 1 cspIq .ž /0 m ya m f2 p 2 2 2 c

5Ž .

Here p denotes the momentum, I represents an unitŽmatrix, and for any quantity X we define d Xs X1

. Ž .yX and Xs X qX r2.2 1 2

Assuming there is no CP violation, the effectivehamiltonian becomes real and symmetric and we canparametrize the mixing matrix by

cosu sinuUs , 6Ž .ž /ysinu cosu

where u is the mixing angle. Then the effectivehamiltonian in the weak basis becomes

21 M Mq 12

H spIq 7Ž .w ž /M M2 p 12 y

with1M smq a m qa m fŽ ." 1 1 2 2 c2

1" d mq a m ya m f cos2u ,Ž .1 1 2 2 c2

1M sy d mq a m ya m f cos2u . 8Ž . Ž .12 1 1 2 2 c2

The mass squared difference between the physicalmasses is given by

Dm) 2 sDm2 y2f a m2 ya m2Ž .c 2 2 1 1

qf 2 a 2 m2 ya 2 m2 , 9Ž .Ž .c 2 2 1 1

while the mass difference is given by

m) ym) sd mq a m ya m f . 10Ž . Ž .2 1 2 2 1 1 c

We shall now use the above formalism to analysethe constraints on dilaton-induced gravity from theK yK mass difference, neutrinoless double betaL S

decay and neutrino oscillation experiments. For theK-system, the physical states are the K and KL S

states, while the weak states are the K8 and K8

states. Thus the K yK mass difference can beL Sw xread off from equation 10 ,

mS) )m ym s m ym ym f a y a .Ž .L S L S L c L Sž /mL

11Ž .

The experimental value m) ym) is dominated byL S

the mass difference m ym . Thus no significantL S

cancellations are expected and a conservative boundfor the contribution from dilaton exchange is

mS) )< <m f a y a -m ym .L c L S L Sž /mL

Using also m ,m this implies a bound on theS L

dilaton-induced gravity coupling of

1 m ym 7=10y15L S

da- s , 12Ž .ž /f m fc L cexpt

m y mL S y15 w xwhere we used ;7=10 11 . While themL

bound depends on the absolute value of f , we canc

( )H.V. Klapdor-Kleingrothaus et al.rPhysics Letters B 488 2000 398–401400

obtain a rough estimate by considering the value ofthe Newtonian gravitational potential to be due to thegreat attractor, which is about 3=10y5 and a ;ext

0.03, so that f ;10y6. Then the bound becomesc

da-7=10y9.Unlike for the tensorial interactions, in this case

the measurement of the CPT violating parameter, themass difference of K8 and K8, does not yield anyconstraint. This can be understood by considering the

Ž .mass difference, which can be read off from Eq. 8to be

M yM sd mcos2uqf cos2u a m ya m .Ž .q y c 2 2 1 1

13Ž .

Here the first contribution is the usual mass contribu-tion and the second contribution is the one comingfrom dilaton exchange. Since there is no CPT viola-tion in the usual gravity, we have cos2us0. Thisimplies that there is no contribution coming from thedilaton exchange gravity.

We now turn over to discuss the constraints com-ing from the neutrino sector. The decay rate forneutrinoless double beta decay is given by,

2My1 q 20nbb < <T s G ME , 14Ž .1r2 012me

where ME denotes the nuclear matrix element MEŽ w x.sM yM , for numerical values see 12 , GF GT 01

w xcorresponds to the phase space factor defined in 13and m is the electron mass. In contrast to thee

w xtensorial VEP 6 for dilaton exchange gravity theobservable has no explicit momentum dependence.The contribution of the dilaton exchange to theobservable for neutrinoless double beta decay isgiven by

m11M s m f a qaqdil 2 c 1 22 ž /m2

m1q a ya cos2u . 15Ž .1 2ž /m2

Since no momenta enter the observable, the decayŽrate is suppressed considerably see also the discus-

w x w x.sion in 14 , which criticizes the treatment in 15 .

The dominant contribution to the neutrino oscilla-tions comes through the change in the mass squareddifference, given by

m222 2² :Dm sy2f m a y a . 16Ž .dil c 2 2 12ž /m1

In the almost degenerate case the bounds from thetwo experiments can be compared if we assume that

Žthere is no mean deviation of usual gravity a q1.a s0, which is assumed in most cases. An alterna-2

tive natural choice is to assume that the masses ofthe neutrinos are hierarchical to explain the atmo-spheric and solar neutrino problems. In the case ofhierarchical neutrino masses, m 4m , a drops2 1 1

out from both expressions and without any furtherassumption all experiments can be compared in termsof the single unknown parameter a . In this case it is2

even more difficult to obtain any bound from neu-trino experiments since for neutrino masses m onlyi

upper bounds exist.w xIf we consider according to Ref. 9 the upper

y3'bound of a ; 10 as its absolute value, and theext

dominant contribution to the local gravitational po-tential to be due to the great attractor, which is aboutf s3P10y5, we can estimate the bounds on theN

dilaton couplings. Although this bound cannot betaken seriously, this allows us to compare our resultwith earlier results. Only in the almost degenerate

Ž . Žcase m ;m ;m , assuming ms2.5 eV as an1 2

upper bound obtained from tritium beta decay exper-w x.iments 16 and only for the case the vacuum oscil-

lation solution of the solar neutrino problem will turnout to be realized in nature, the experimental Dm) 2

may be large enough to imply a significant boundŽsee however the discussion of medium effects inw x.10 . The search for neutrinoless double beta decayw x17 , implying M -0.3 eV suffers from an evenqdil

more severe suppression, since the bound for M isqless stringent than the one for Dm) 2.

In summary, we point out that the dilaton ex-change gravity cannot be constrained substantially inthe neutrino sector, while the bound coming from theK yK mass difference is significant, modulo theL S

uncertainty of classical background potential. Unlikethe new tensorial or vectorial gravitational interac-tions, this scalar interaction cannot introduce anyCPT violation in the K-system.

( )H.V. Klapdor-Kleingrothaus et al.rPhysics Letters B 488 2000 398–401 401

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