Decaying dark matter as a probe of unification and TeV spectroscopy

Asimina Arvanitaki,1,2 Savas Dimopoulos,3 Sergei Dubovsky,3,4 Peter W. Graham,3

Roni Harnik,3 and Surjeet Rajendran5,3

1Berkeley Center for Theoretical Physics, University of California, Berkeley, California, 94720, USA2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA

3Department of Physics, Stanford University, Stanford, California 94305, USA4Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow, Russia

5SLAC National Accelerator Laboratory, Stanford University, Menlo Park, California 94025, USA(Received 27 May 2009; published 9 September 2009)

In supersymmetric unified theories the dark matter particle can decay, just like the proton, through

grand unified interactions with a lifetime of order of �1026 sec . Its decay products can be detected by

several experiments—including Fermi, HESS, PAMELA, ATIC, and IceCube—opening our first direct

window to physics at the TeV scale and simultaneously at the unification scale �1016 GeV. We consider

possibilities for explaining the electron/positron spectra observed by HESS, PAMELA, and ATIC, and the

resulting predictions for the gamma-ray, electron/positron, and neutrino spectra as will be measured, for

example, by Fermi and IceCube. The discovery of an isotropic, hard gamma ray spectral feature at Fermi

would be strong evidence for dark matter and would disfavor astrophysical sources such as pulsars.

Substructure in the cosmic ray spectra probes the spectroscopy of new TeV-mass particles. For example, a

preponderance of electrons in the final state can result from the lightness of selectrons relative to squarks.

Decaying dark matter acts as a sparticle injector with an energy reach potentially higher than the LHC.

The resulting cosmic ray flux depends only on the values of the weak and unification scales.

DOI: 10.1103/PhysRevD.80.055011 PACS numbers: 12.10.Dm, 12.60.Jv, 95.35.+d

I. INTRODUCTION

In grand unified theories the proton can decay becausethe global baryon-number symmetry of the low energystandard model is broken by physics at the grand unifica-tion (GUT) scale. Indeed, only local symmetries can guar-antee that a particle remains exactly stable, whereas globalsymmetries are generally broken in fundamental theories.Just as the proton is long-lived but may ultimately decay,other particles, for example, the dark matter, may decaywith long lifetimes. It would seem miraculous that the darkmatter particle’s lifetime is in a range which is longenough to be a good dark matter candidate yet shortenough to be observable today. Nevertheless, this maywell be what happens in grand unified theories. For ex-ample if a TeV mass dark matter (DM) particle decays viaGUT-suppressed dimension 6 operators, its lifetime wouldbe

�� 8�M4

GUT

m5DM

¼ 3� 1027 s

�TeV

mDM

�5�

MGUT

2� 1016 GeV

�4;

(1)

where MGUT � 2� 1016 GeV is the supersymmetric(SUSY) unification scale.

This lifetime is being probed by several current experi-ments, as shown in Table I. This can be understood, at leastfor the satellite and balloon experiments, because these allgenerally have similar acceptances of �ð1 m2Þð1 yrÞ�ð1 srÞ � 3� 1011 cm2 s sr. For comparison, the numberof incident particles from decaying dark matter with a

lifetime of 1027 s is �R10 kpc d3r

r2ð0:3 GeV

mDM cm3Þ�ð10�27 s�1Þ � 10�9 cm�2 s�1 sr�1, where these couldbe photons, positrons or antiprotons, for example,depending on what is produced in the decay. This impliessuch experiments observe �ð3� 1011 cm2 s srÞ �ð10�9 cm�2 s�1 sr�1Þ � 300 events. This coincidencemay allow these experiments to probe physics at theGUT scale, much as the decay of the proton and a studyof its branching ratios would. In fact, HESS, PAMELA,and ATIC may have preliminary indications of dark matterfrom the cosmic ray electron/positron spectrum. The Fermisatellite will test this by measuring both the electron/posi-

TABLE I. A lower limit on the lifetime of a dark matterparticle with mass in the range 100 GeV & mDM & 10 TeV,decaying to the products listed in the left column. The experi-ment and the observed particle being used to set the limit arelisted in the right column. All the limits are only approximate.Generally conservative assumptions were made and there aremany details and caveats as described in [1].

Decay Channel � Lower Limit Experiment

q �q 1027 s PAMELA antiprotons

eþe� or �þ�� 2� 1025 sðTeVmDMÞ PAMELA positrons

�þ�� 1025 sð1þ TeVmDM

Þ EGRETþ PAMELA

WW 3� 1026 s PAMELA antiprotons

�� 2� 1025 s PAMELA antiprotons

� �� 1025 sðmDM

TeVÞ AMANDA, Super-K

PHYSICAL REVIEW D 80, 055011 (2009)

1550-7998=2009=80(5)=055011(7) 055011-1 � 2009 The American Physical Society

tron and gamma-ray spectra with significantly improvedprecision.

Previously [1], we discussed the general framework ofdark matter decays induced by GUT scale physics and itssignals at PAMELA, ATIC and Fermi. In this paper, weconsider the implications of the recently published HESSdata [2] on this framework and discuss models that fit theoverall electron/positron spectrum observed by PAMELA,ATIC and HESS. We focus on the correlated photon andneutrino signals that could be observed at Fermi andIceCube, respectively.

II. THEORETICAL SETUP

To study the observational consequences of decayingdark matter in SUSY GUTs one may follow an effectivefield theory approach and consider an extended MSSMwith higher dimensional operators parametrizing GUTeffects and leading to dark matter decay. A detailed analy-sis of possible higher dimensional operators and the waysto generate them from concrete microscopic SUSY GUTswas presented in [1]. Here, for definiteness, wewill work inthe context of the SOð10Þ models described in [1]. As anexample, in addition to the standard MSSM interactions,we introduce an additional vectorlike ð16m; �16mÞ multipletat the TeV scale and 10GUT multiplet at the GUT scale. Therelevant superpotential interactions involving these fieldsare

W 0 ¼ �16m16f10GUT þm16m �16m þMGUT10GUT10GUT:

(2)

We will assume that the singlet Sm is the lightest compo-nent of the 16m and will therefore be dark matter. AfterGUT scale matter and gauge fields are integrated out oneobtains the dimension 5 operator 16m16m16f16f in the

superpotential and dimension 5 and 6 Kahler terms

16m16m10y, 16ym16m16

yf16f involving m-fields. Of all

these, the only operator that involves two singlet Sm com-ponents of 16m is the dimension 6 Kahler term yielding

SymSm16yf16f (assuming right-handed neutrinos are heavy).

Consequently, in this model a thermal relic abundance ofsinglet fields is produced through dimension 5 decays ofthe charged components of 16m close to the BBN epoch.These decays are interesting in their own right, as they mayexplain the observed Lithium abundances [3]. On the otherhand, dimension 6 decays between different components ofthe singlet supermultiplet may lead to observable astro-physical signals that we discuss in the rest of the paper.

Note that these decays may go through operators gen-erated by integrating out the heavy Uð1ÞB�L gauge boson,or by integrating out heavy 10GUT fields. In the formercase, decays are flavor universal, while the latter generi-cally lead to flavor nonuniversal decays. In the case offlavor nonuniversal decays, since the decay rate scales asthe fourth power of the coupling, it is easy to have decays

to one flavor dominate over the rest. The branching frac-tions of the decay also depend upon the fourth power of theheavy scale. Oð1Þ mass splittings between the doublet andtriplet components of 10GUT can easily alter the GUTinvariant branching relations. For example, if the tripletcomponent is heavier by a factor of �2, the hadronicbranching fraction of the decay is smaller than �0:1,consistent with the antiproton constraints from PAMELA[1]. Depending on the relative strength of gauge and super-potential couplings and the masses of the heavy fields, bothpossibilities can be realized.One may worry that this picture could be spoiled by

lower dimension operators, such as Kahler kinetic mixings

10yGUT10h and 16ym16f. However, these are forbidden by

R-parity (under which 16m is even, and 10GUT is odd), andm-parity under which both 16m and 10GUT are odd. Thesuperpotential (2) also permits a PQ symmetry that issoftly broken by the mass term for the 16m and the �term for the standard model Higgs fields 10h. This PQsymmetry forbids other lower dimensional operators,such as 16m16m10h, that can cause faster dark matterdecay. Other ways to avoid these operators are discussedin [1].For simplicity, in this paper we will focus on the case in

which the scalar ~s receives a TeV scale vev. In this casedimension 6 operators lead to two body decays of thesinglet fields to the MSSM fields. We are thus lead totwo interesting observations:(i) In this case dark matter decay products necessarily

contain MSSM superpartners, because direct decaysof a scalar into two light fermions are suppressed byhelicity.

(ii) The production of superpartners, combined with thegeneric expectation that sleptons are lighter thansquarks leads to decays dominantly into leptonicchannels due to kinematics.

These lead to a possible connection between the branchingfraction of dark matter and the spectrum of its decayproducts on the one hand, and the supersymmetric spec-trum and the decay cascades of superpartners on the other.

III. ASTROPHYSICAL SIGNALS

A. Electrons and positrons

GUT induced dark matter decays lead to several genericexpectations for electron/positron spectra. As discussed inthe previous section, the dark matter is a combination ofthe scalar (~s) and fermion (s) components of the Sm super-field. The two body decay of dark matter will involve

sleptons (~l, the superpartner of a lepton) in the final state.The slepton then further decays to its partner lepton and the

lightest supersymmetric particle (LSP), ~l ! lþ LSP, lead-ing to an injection spectrum of the lepton that is flatbetween a lower and upper edge [1]. When dark matter is

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much heavier than the superpartners the lower edge istypically a few GeV and the upper edge is roughly at

Eedge ¼�1�m2

LSP

m2~l

�mDM

2: (3)

The sensitivity of the upper edge in the injection spectrumto the masses of superpartners may lead to interesting crosschecks at the LHC.

Current data for the positron fraction from PAMELA [4]and the e� þ eþ spectrum from the ATIC [5] and HESS [2]experiments include interesting hints of a possible excessabove background. Because of systematic uncertainties itis premature to interpret spectral shapes observed by ATICand HESS, particularly above 100 GeV. However, this fluxwill be measured in the very near future by both Fermi andPAMELA with improved statistics and systematics. TheHESS experiment will also extend its measurement tolower energies, overlapping with the excess observed byATIC. It is thus interesting to consider several scenarios forthe outcome of upcoming experiments and their implica-tions in the context of our framework.

For concreteness we will focus on two scenarios ofpossible electronþ positron spectra which are roughlyconsistent with current data, given the systematic uncer-tainties, but will be probed further soon:

Scenario 1-A smooth HESS signal—The low energydata from ATIC (below �75 GeV) is used to estimatethe astrophysical background. We then find decay channelsto explain the higher energy HESS excess. Two simpleexamples that realize this are shown in Fig. 1.

One possibility is that a heavy scalar (of order 6 TeVormore) is decaying to a pair of smuons which each furtherdecay to a muon and a neutralino. The muons decay furtherto produce electrons and positrons. For example, a 8 TeVscalar decaying to 200 GeV smuons which then decay to100 GeV neutralinos is shown in Fig. 1. The spectral shape(see Eq. (3)) produced by such a cascade fits the spectrumobserved by HESS quite well.

A second possibility shown in Fig. 1 is that a heavy6 TeV fermion decays to tauþ stau, producing a similarspectral shape. A decay into the third family may bemotivated by minimal flavor violation. In this case theobserved flux is dominated by the electrons and positronsproduced in the subsequent decay of the hard (� 3 TeV)tau [6]. The stau will produce an additional soft componentto the spectrum which is subdominant.

In this scenario, we take the lifetimes of the dark matterto be 1026 seconds for the smuon case and 6�1025 seconds for the tau case. Though both of these possi-bilities produce a similar electron plus positron spectrumthey may be distinguishable by gamma ray and neutrinoobservations, as discussed later in this paper. It is interest-ing to note that both of these possibilities require flavornonuniversal decays of dark matter which may naturally beproduced in the model discussed above. Furthermore, since

the decay rate scales as the fourth power of the coupling,these flavor nonuniversal decays can be easily dominatedby decays to one flavor over the others.Scenario 2-Multiple features—We now focus on some

interesting spectra that consist of multiple features and thatoccur in simple scenarios of GUT induced dark matterdecays. We will not necessarily assume that either theHESS or ATIC spectra are correct, but rather pick threeexamples to demonstrate some of the generic possibilities.In the first example, dark matter decays into leptonsþ

sleptons via s ! l�~l� and to sleptons via ~s ! ~l�~l�. Thelepton final states lead to a hard high energy feature whilethe slepton final state leads to a smoother one at lowerenergies. The purple short-dashed line of Fig. 2 shows theflavor universal decay of a scalar and a fermion of mass1.5 TeVand equal abundance. Here the slepton masses areuniversal at 130 GeV and the LSP mass is 100 GeV. Asshown in [1] a similar decay may yield a spectrum that is inremarkable agreement with the double feature observed byATIC. Since the spectrum of superpartners and the mass ofdark matter set the scale of both spectral features [seeEq. (3)], a nontrivial cross check may be made once theLHC measures the masses of sleptons and the LSP.Another possible way in which GUT induced dark mat-

ter decays can produce multiple spectral features is byvarious cascades of supersymmetric particles. For ex-

ATIC 08HESS 08ATIC 08HESS 08

s

50 100 500 1000 500010

20

50

100

200

500

Energy GeV

E3dN

dEG

eV2

m2s

1sr

1

FIG. 1 (color online). The electronþ positron spectrum pro-duced by the decay of dark matter (s or ~s) to smuon pairs (solidblack line) and tau-stau pairs (dot-dashed red line) as discussedin the text. The HESS and ATIC data is shown by red squares andblus circles, respectively. The systematic error of HESS is shownas a grey band and its superimposed energy uncertainty is shownas a lighter grey band. The background and smuon signalcomponents of the flux are shown by dotted lines.

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ample, if dark matter decays to two sleptons of different

mass (~lL and ~lR) both of which decay to a neutralino LSP,the upper edges of the injection spectra in the two cascadesmay be sufficiently different [see Eq. (3)], leading tomultiple features. In Fig. 2 we show the spectrum producedby a flavor universal decay of dark matter into 145 GeVandand 230 GeV sleptons which subsequently decay to a120 GeV LSP and a lepton (dotted blue line). The branch-ing fraction into the heavier slepton is 20%. Again, sinceboth spectral features are set by relations of the form (3),this interpretation may be tested at the LHC.

Alternatively, multiple features may be produced by twocascades of the same slepton via different neutralino states.For example, a similar spectrum (the dotted blue line inFig. 2) may be produced by a 5 TeV scalar dark matterdecaying to sleptons with a mass of 200 GeV. The sleptonhas two dominant decay channels into two different neu-tralino states with masses of 170 and 100 GeV [7].

Spectral features can also arise from decays of twodifferent dark matter particles with significantly differentmasses. Despite their different masses, the electron fluxfrom the decays of these two particles can be comparable iftheir relic abundance is generated from the decays ofanother particle. For example, in the SOð10Þ model dis-cussed earlier, the relic number density n~s and ns of the

singlets ~s and s are generated through the decays of thecomponents (with mass m) of the 16m that are chargedunder the standard model. With singlet masses m~s and ms

for ~s and s respectively, their relic number densities satisfyn~sns� ðm�m~s

m�msÞ3. In this model, the ~s and s can decay only

when ~s develops a vev. Their respective dimension 6 decay

rates �~s and �s scale as �~s

�s� ðm~s

msÞ3. The relative electron

flux from the two decays is n~s�~s

ns�s� ðm~s

msÞðm�m~s

m�msÞ3. This ratio is

Oð1Þ for m~s þ 0:4ms & m & m~s þ 2:2ms whenm~s � ms.For TeV scale ms and m~s, this results in comparableelectron fluxes for m within a TeV of m~s þms. The ob-servation of these features in the electron spectrum canthus lead to measurement of SUSY parameters and GUTphysics. In Fig. 2, we show the spectrum from the decaysof a heavy 5 TeV scalar, ~s, and a 660 GeV fermion, s, into~s ! ~�� ~�� and s ! ~�� ~��. The decay rates are �~s ¼8� 10�27 s�1 and �s ¼ 3� 10�27 s�1. The smuon wastaken to be at 200 GeVand the LSP at 90 GeV. In this case,the heavier components of the 16m are around �5:5 TeV.All of the scenarios described above are consistent with

the qualitative shape of the PAMELA excess. The positronfraction for some of these cases is shown in Fig. 3. Beforeproceeding to photon signals we will discuss technicalaspects of the figures above. We used GALPROP [9] forgenerating backgrounds and for propagation of the darkmatter signal. In Fig. 1, we assumed the convective diffu-

ATIC 08HESS 08

s lL lL and s lR lR

2 components to

s l l and s l l

50 100 500 1000 500010

20

50

100

200

500

Energy GeV

E3dN

dEG

eV2

m2s

1sr

1

FIG. 2 (color online). The ‘‘multi feature’’ electronþ positronspectra produced in simple dark matter decays scenarios. Theshort-dashed purple line is an example of multiple featuresarising from decays into leptons and sleptons. The blue dottedline is an example of a flavor universal decay to two sleptons ofdifferent masses. The long-dashed green line is for two DMcomponents decaying to muons and smuons. The experimentaldata shown is similar to that in Fig. 1. The two signal compo-nents and the background in the slepton case, as well as thebackground are shown as dotted lines.

PAMELA 08

10 20 50 100 200

0.100

0.050

0.200

0.030

0.150

0.070

Energy GeV

Posi

tron

Frac

tion

FIG. 3 (color online). The positron fraction produced by someof scenarios discussed in the text—decays to smuons (Scenario1, solid black line), a universal lepton-slepton decay (Scenario 2,short-dashed purple line) and decays to sleptons (Scenario 2,dotted black line). The background positron fraction from twopropagation models is shown in gray, thin dotted lines. ThePAMELA data is shown (black circles).

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sion propagation model (DC) of [10]. In Fig. 2, a harderpropagation model was used throughout (model B of [11]).The slope of the background electron flux, which maysignificantly affect both the positron fraction and the totalflux was chosen to be �3:3 at 20 GeV, in agreement withobservations and the large uncertainties [12]. DarkSUSY[13] was used for producing injection spectra.

B. Diffuse gamma rays

Any decay scenario that produces charged particles mustproduce gamma rays from final state radiation (FSR) offthose charged particles. Figure 4 shows the gamma-rayspectra from some of the models discussed above. Theshape of the FSR spectrum is directly related to the shapeof the primary charged particle injection spectrum fromdark matter decay [1]. We do not include gamma rays frominverse-Compton scattering of starlight off the high energyelectrons and positrons from the dark matter decay. Thisdoes not usually give as hard a spectrum as FSR, andpresumably falls off faster than FSR with galactic latitudeas the density of starlight decreases off the plane of thegalaxy (but see [14]). This contribution is difficult tocalculate off the plane of the galaxy due to the anisotropicflux of starlight [14,15]. If a decay mode contains �’s we doinclude the full spectrum from these, e.g. the photons from�0’s produced in the � decay generated using [13].

This diffuse gamma-ray signal can determine whetherthe observed electron/positron excesses do indeed arisefrom dark matter. Dark matter decays give rise to anisotropic gamma-ray signal with a hard, high-energy spec-tral feature such as an edge coming from FSR. The shape ofthe spectrum is the same everywhere across the sky. Theintensity varies slightly, with a dependence on the angle ofobservation that is determined by the known density ofdark matter in the galactic halo and so is only uncertain at

the galactic center. If the electron/positron excesses arisefrom dark matter decay, the gamma-ray signal is probablystrong enough to be observable at Fermi [1]. Observationof such a signal would be impossible to explain by anyknown astrophysical mechanism other than dark matter.These gamma-ray observations may even help distin-

guish decays from annihilations, since decays produce amore isotropic flux than that from annihilations. Thus, thegamma-ray signal from decaying dark matter can be ob-served cleanly with measurements at high galactic latitude,off the plane of the galaxy, where there is little astrophys-ical background.The gamma-ray spectrum carries important information

about the nature of the DM decay that is obscured by theelectron/positron spectra. For example, the two scenariosin Fig. 1 produce almost identical electron/positron spectraeven though the underlying high-energy physics is differ-ent. From Fig. 4 we see that these two are easily distin-guished by their gamma-ray spectra. Further, the shape ofthe FSR spectrum can provide a measurement of the massof the dark matter particle and potentially the masses of itsdecay products such as superpartners. For example, struc-ture in the electron/positron spectrum in Fig. 2 appears aswell in the gamma-ray spectrum as in Fig. 4. In fact, thegamma-ray spectrum can provide extra information sinceFSR photons can come from any interior line in the decaychain that is on shell. For example, in the dashed (green)curves in Figs. 2 and 4 a 300 GeV muon is produced whichis directly visible as an edge in the gamma-ray spectrum.Thus, the gamma-ray spectrum can provide a probe of TeVscale physics complementary to the electron/positronspectra.

C. Neutrinos

When the dark matter particle decays to charged leptons,neutrinos will also be produced due to either SUð2Þ invari-ance or subsequent decays of the produced muons or taus.

FIG. 4 (color online). The gamma-ray spectra from final stateradiation and � decay. The solid (black) and dot-dashed (red)lines are as in Fig. 1. The dashed (green) and dotted (blue) linesare as in Fig. 2. Only the signals are plotted, not signal plusbackground. Gray is the expected background from [42]. Theseare shown at a galactic latitude b ¼ 60� and longitude l ¼ 0�.

200 500 1000 2000 5000 10000 20000

E in GeV

1. • 10• 10

2. • 10• 10

5. • 10• 10

1. • 10• 9

2. • 10• 9

5. • 10• 9

1. • 10• 8

Ed• dE

incm

•2

s•1

sr•

1

FIG. 5 (color online). The average �� flux produced from aDark Matter particle decay in different examples. The solid(black) and dot-dashed (red) lines are as in Fig. 1. The dotted(blue) line is as in Fig. 2. AMANDA data is shown in gray [16].The bin size is taken to be 0.2 in Log10ð E

GeVÞ, as it appears in [16].

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In Fig. 5, we present the �� flux in IceCube [16] for most of

the above examples that predict neutrinos by SUð2Þ invari-ance. The solid black (dotted blue) line comes from ~s !~���ðeÞ þ ~��ðeÞ, while the dot-dashed red line comes from

s ! �� þ ~��. Because the neutrinos are produced at galac-tic distances, the flavor ratios on the earth are 1:1:1. We donot include the flux produced by charged lepton decays asthey are subdominant. We take a bin size of 0.2 inLog10ð E

GeVÞ as it appears in [16], but the energy resolution

could be as low as 0.4 in Log10ð EGeVÞ [17]. In the 100 GeV to

several TeV range there is a large atmospheric neutrinobackground that drops rapidly with energy as a power law,E�3. For example, taking into account the effective area ofIceCube [17] and integrating over a bin of size 0.4 inLog10ð E

GeVÞ centered around the neutrino energy, 5� dis-

covery of a 1 TeV neutrino line could be possible withinroughly a year of observation time, when the dark matterlifetime is 1026 sec . In this estimate we have not includedsystematics in the measurement process. Spectral informa-tion or distinguishing between different scenarios is goingto be harder to deduce because of the atmospheric neutrinobackground and IceCube’s energy resolution. It is worthnoting though that both the IceCube energy resolution aswell as the atmospheric background subtraction could begreatly improved in the near future, increasing the potentialof the experiment to observe and study astrophysical sig-nals of TeV scale dark matter.

IV. CONCLUSIONS

The decaying dark matter scenarios explored in thispaper can naturally explain the cosmic ray spectra ob-

served at HESS, PAMELA, and ATIC without the needfor astrophysical boost factors or additional new energyscales [1,18–41]. In particular, the softer electron spectrumobserved by HESS can fit naturally with the observationsof PAMELA. Substructures in the electron spectrum, cor-related with substructures in the photon spectrum, are alsopossible in these theories. Such substructure may also becorrelated with the spectrum of superpartners observed atthe LHC. Observation of these substructures could open awindow into the spectrum of TeV mass particles. TheHESS observation suggests a heavy ( * TeV) DM mass,too heavy to be produced at the LHC. In this case, astro-physical observations would provide our only probe ofdark matter at such a high energy scale.The same theories typically also contain particles decay-

ing during big bang nucleosynthesis through dimension 5

operators with lifetime �� 8�M2

GUT

m3 � 7 s: Such decays

are recorded by a change in the primordial light elementabundances and may explain the anomalous observed Liabundances, opening another window to unification [1].

ACKNOWLEDGMENTS

We would like to thank Bill Atwood, Patrick Fox,Giorgio Gratta, Francis Halzen, Dan Hooper, GrahamKribs, John March-Russell, Peter Michelson, IgorMoskalenko, Hitoshi Murayama, Roger Romani, SubirSarkar, Bob Wagoner, and Neal Weiner for usefuldiscussions.

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