VOLUME 55, NUMBER 26 PHYSICAL REVIEW LETTERS 23 DECEMBER 1985

Could Goldstone Bosons Generate an Observable 1/A Potential~

D. Chang and R. N. MohapatraDepartment ofPhysics and Astronomy, University ofMaryland, College Park, Maryland 20742

and

S. NussinovLaboratory for 1Vuclear Studies, Cornell University, Ithaca, Pew York 14853, and Physics Department,

Sackler Faculty ofScience, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel '(Received 23 October 1985)

We discuss gauge models in which exchange of Goldstone bosons combined with the QCDanomaly can lead to spin-independent I/R and T-nonconserving o. r/R potential. In contrast withaxion-exchange forces which lead to new short-range macroscopic forces (range —10 cm), theseforces could be observable in Eotvos-Dicke-type experiments involving polarized and unpolarizedmaterials.

PACS numbers: 14.80.6t, 04.80.+z

An exactly conserved locally gauged baryon numberwould generate a long-range 1/r force via the exchangeof the corresponding massless vector boson. Thehigh-precision Eotvos-type experiments checking theequivalence principle rule out any appreciable couplingfor such a force. 2

This is not the case for the massless spin-0 Gold-stone bosons corresponding to the spontaneous break-down of a global symmetry. Such particles havederivative couplings3 which vanish in the q =0 limit.Their exchange between nucleons or electrons givesrise to 1/r3 spin-dependent potentials which are veryhard to detect. We would like to point out that thisresult may change when we have a OFF term added tothe QCD Lagrangean leading to interesting phenom-enological and experimental possibilities.

Let us focus on a Goldstone boson P associated withthe spontaneous breaking of some global axial U(1)symmetry, with the corresponding U(1) quantumnumber carried by quarks. Its coupling to quarks is ofa pure y5 type,

~pe gad'lyse

@ exchange is

St2 ——o.t o.2 —3(o.t r) (o., r")

Astrophysics considerations related to the cooling ofred giants imply4 that

F& » 10' GeV,

and similar bounds are obtained if the @ arises from awider context of spontaneous breaking of a globalhorizontal symmetry from limits on K m. + $',p, e+@",etc.3 With such an F&, the interaction (4)is practically undetectable. 5

Consider next this theory as part of a realistic theoryof quarks and leptons, that contains the nonvanishingbut small QCD anomaly term HGG (where G is thecolor gluon field). To study the impact of the anomalyterm on the coupling of the Goldstone boson toquarks, let us review the familiar case of the pseudo-cal r pon-nuceon couping: g NNNySNrr. It has

been shown6 that in the presence of the anomaly term,an induced scalar coupling of type

so that at a low energy it is equivalent to the purederivative coupling

g m„mdNXmF mg+ md

(6)

t)„%ay y 'jtt. (2)

F~ and g& are related via a Goldberger-Treiman rela-tion,

gy = 2 m'/Fy = 2m/Fp, (3)

where m denotes the fermion mass and g~ the axialcoupling (g„=1). The P vertex F~ ' (o- q) vanishesin the q 0 limit and the potential resulting from the

appears. This suggests, by analogy, that if the Gold-stone boson @ of interest couples as (m~/F~) qysq@,the anomaly term will induce a scalar coupling

5Wyiviv = NN$ = p, @NN. (7)u+ md F~

One could think of the induced coupling in terms ofthe diagram in Fig. 1, where the QCD anomaly in-duces the scalar coupling at the NN7r vertex in Eq. (6)and the pion propagator cancels the m2 leading to Eq.

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VOLUME 55, NUMBER 26 PHYSICAL REVIEW LETTERS 23 DECEMBER 1985

2fTlq ITl~

FIG. 1. Typical diagram for the induced scalar coupling inthe presence of the QCD anomaly.

FIG. 2. One-loop graph involving two external Gold-stone-boson lines.

(7). Note that both (6) and (7) vanish when any oneof the quarks has vanishing mass, since in this case theeffect of the 0 term is compensated by an appropriatechiral rotation.

The important observation is that since the Gold-stone boson $ has zero mass (see later), the scalarvertex induces a long-range spin-independent 1/R po-tential

Vww( R)Fp R R

The strength of the new interaction is very weak.From experimental bounds on the electric dipole mo-ment of the neutron, we know7 that

which together with p, = 4 MeV and Eq. (5) implies

2~ 10—41 (10)

and a very weak $ force.However, if the @ force adds up coherently, i.e. , nu-

cleons and leptons possess the U(1) quantum numberwithout exact cancellation as in the case of em charge,then V&(r) could be comparable to the gravitationalpotential

mN2

VNN 1 1G

—= 10 38x-mp( k)

5 = x Vs/ VG = xgs2/10 (12)

from equivalence in EOtvos-type experiments. TheRoll-Krotkov-Dicke experiment limits possible dif-ferent accelerations towards the Sun of different ele-ments to

~(s..)10-". (13)

Furthermore, V&(r) will not respect the inertial-mass-interaction-strength equivalence. Let us denoteby x the degree to which the V&(r)-mass proportional-ity is violated as we vary the isotope (nucleus) con-sidered. x could be as large as 0.1 if @ couples only toup or down quarks. Even if the coupling to u and dquarks is equal, nuclear binding, Coulomb, and n-pmass-difference effects will make x = 0.01.

The new scalar interaction will therefore cause anapparent deviation

The Braginsky-Panov experiment claims a better ac-curacy for a similar experiment,

~(sun) —Io " (i4)

and the early Eotvos experiment itself'0 claimed5(E,«„)«10 . With x=0.1, we get from (12) and(13) or (14) stronger bounds,

gs2~10-48-10 " (SunI,

gs2~ 10 46 (EarthI

(16)

A. 4,= Fg/8 mq m„. (17)

Note that mq could be as large as the t-quark mass; butbecause of hadronic uncertainties, we cannot be sureof its magnitude and, therefore, we choose the follow-ing plausible range for mq = 10 MeV to 1 GeV. Usingseparate bounds on F& and 0, we find

) ~ & 20-10000 km (i8)Thus, it can be as large as the radius of the earth and

than the one obtained [Eq. (10)] by the bounds on 0and F~ separately.

In the previous discussion we have assumed allalong that the @ is exactly massless. One would ex-pect, however, that the 0 term which leads to violationof the pure derivative coupling will also generate a tinymass for the otherwise exactly massless Goldstone bo-son, which can be seen from the following heuristic ar-gument.

Specifically let us consider a quark-loop diagramwith two external @ legs (Fig. 2). If the interaction atboth vertices is of a pure derivative (8 $) form, thenonly a wave-function renormalization of the kineticterm (B„P) will be contributed. This, in fact, is avery nice reciprocity between the Goldstone theoremand the derivative-coupling theorem. However, thesmall scalar coupling gs may yield a mass correctionroughly estimated to be

8 m„mq gmqm„gs mq

This apparent breakdown of Goldstone's theorem isperhaps related to the nonlocality of instanton effects.(Note that the scalar coupling to fermions of equalmass, unlike the pseudoscalar one, cannot be ex-pressed as a derivative term. ) The correspondingCompton wavelength or range of force is

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VOLUME 55, NUMBER 26 PHYSICAL REVIEW LETTERS 23 DECEMBER 1985

a p= J (g,'/r') pr' dr = Zqg2p (2o)

rather than &~= Rp(Gm~2), and M& will be smallerby a factor of (A.JR)gz2/Gm~2. Thus for X&

——1-10cm, ' as is the case for the axion, M@ will be furtherreduced with respect to the gravitational pull of theearth by 10 9—10 8. A null experiment using a super-conducting gravity gradiometer in Earth orbit has re-cently been proposed by Paik, '4 which can be sensitiveto such deviations from gravitational forces at the levelof 10

From the above discussion it follows that for the "@force" to be significant at large distances we need m~to be smaller than the typical estimate for axion mass'2

m = (f /F )m„= 10 5—10 6 eV.

In the remaining paragraphs, we present two illustra-tive models, where such a long-range spin-inde-pendent potential arises. Consider the class of modelswhere lepton number is spontaneously broken. '5'6This requires an extension of the standard model bythe addition of either'5 (i) a right-handed neutrino anda gauge-singlet Higgs field with L =2 or' (ii) a left-handed SU(2)1-triplet Higgs multiplet with L = 2. Wedo not discuss details of the model except to recall thatin both these models, the massless Goldstone bosonwhich couples to leptons only prior to spontaneouslybreaking acquires a coupling to quarks after symmetrybreaking takes place with equivalent F~ given as fol-lows [q = (u, d)]:

M& —= iF& qy5q@,

where, for case (i),

F~ ' = (GF/167r2)m„;

for case (ii),

F~ ' = GF'i2(uT/vD),

(22)

(23)

(24)

where vT and vD denote the vacuum expectation

will, therefore, affect the gravitational fall towards theearth. For the case of gravitational fall towards theSun, however, since ~& && RE«,hs„„, 6~&«~ will notbe affected by V&. We also wish to point out thatsimilar induced scalar couplings" of the invisible ax-ion' have been discussed in the literature. However,since the invisible axion has mass, ' m, = 10 ' to10 eV, it corresponds to an interaction with range~ 10 cm.

In general the force corresponding to massive @,

(19)

will operate only on distances r ~ A.&. Thus if we havean attracting mass of radius R and uniform density pthe net force due to this on a point object on the sur-face is

values of Higgs triplet and doublet, respectively. Asmentioned, the quark sector is identical to that of thestandard model so that it has an arbitrary value for theanomaly parameter, 0, only restricted by the experi-ments to be less than 10 '0. Using present bounds onm„and u T,

' we find, for case (i),g2 ~ 10—54.

for case (ii),2 ~,0-44 IO-4~

(25a)

(2Sb)

(27)

The strength of this force for the two cases describedhere is, for case (i),

f2~ «10 42 GeV

for case (ii),

fs2n ~10 34 GeV

(28a)

(28b)

The present bounds on these parameters can be in-ferred from the work of Leitner and Okubo'9 to be

fs2n ~ 10 27 GeV (29)

It should be possible in a laboratory experiment to at-tain sensitivity of the level predicted in Eqs. (28).

In conclusion, we find observable strengths for I/rtype potentials (with range = 10 km) in majoronmodels in the presence of the QCD anomaly term, aswell as spin-dependent T-nonconserving I/r2-typeterms. Experimental test of these effects by macro-scopic experiments would throw light on physics atsubmicroscopic distance scales.

One of us (S.N. ) would like to thank the particletheory group at the University of Maryland for hospi-tality when this work was started. We wish to thankC. H. Woo and H. J. Paik for discussions. This workwas supported by a grant from the National ScienceFoundation.

&'~Present and permanent address.

These bounds are well within the observable experi-mental range in Eotvos-type experiments involving theEarth. An upcoming experiment'8 using NASA satel-lites is expected to improve the precision of Eotvos-type experiments by eight orders of magnitude, whichcould therefore test both kinds of models.

The next point we wish to note is that exchange ofthe majoron can also lead to spin-dependent T and P--nonconserving I/r2-type long-range forces. In thiscase only one vertex is spin dependent, leading to thefollowing form of the potential (X4, = 10 km):

Vsn= (fsn/r )(r re (26)where

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VOLUME 55, NUMBER 26 PHYSICAL REVIEW LETTERS 23 DECEMBER 1985

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