Single-Photon Signals at LEP in Supersymmetric Models with a Light

Home , Photino

DOE/ER/40717{36

CTP-TAMU-58/96

ACT-17/96

hep-ph/9611437

Single-photon signals at LEP in sup ersymmetric

mo dels with a light gravitino

1 2;3 4

Jorge L. Lop ez , D.V. Nanop oulos , and A. Zichichi

Bonner Nuclear Lab, Department of Physics, Rice University

6100 Main Street, Houston, TX 77005, USA

Center for Theoretical Physics, Department of Physics, Texas A&M University

College Station, TX 77843{4242, USA

Astroparticle Physics Group, Houston Advanced Research Center (HARC)

The Mitchell Campus, The Wo o dlands, TX 77381, USA

University and INFN{Bologna, Italy and CERN, 1211 Geneva 23, Switzerland

Abstract

We study the single-photon signals exp ected at LEP in mo dels with a

very light gravitino. The dominant pro cess is neutralino-gravitino pro duction

e e

(e e ! G) with subsequent neutralino decay via ! G, giving a +E

miss

signal. We rst calculate the cross section at arbitrary center-of-mass energies

and provide new analytic expressions for the di erential cross section valid for

general neutralino comp ositions. We then consider the constraints on the grav-

itino mass from LEP 1 and LEP161 single-photon searches, and p ossible such

searches at the Tevatron. We show that it is p ossible to evade the stringent

LEP 1 limits and still obtain an observable rate at LEP 2, in particular in the

region of parameter space that may explain the CDF ee +E event. As

T;miss

diphoton events from neutralino pair-pro duction would not be kinematically

accessible in this scenario, the observation of whichever photonic signal will

discriminate among the various light-gravitino scenarios in the literature. We

also p erform a Monte Carlo simulation of the exp ected energy and angular dis-

tributions of the emitted photon, and of the missing invariant mass exp ected in

the events. Finally we sp ecialize the results to the case of a recently prop osed

one-parameter no-scale sup ergravity mo del.

November 1996

1 Intro duction

Sup ersymmetric mo dels with a light gravitino (G) have b een considered for some

time [1, 2, 3, 4, 5], but interest on them has recently surged [6, 7] b ecause of their

ability to explain naturally the puzzling e e +E event observed by the CDF

T;miss

Collab oration [8]. If the gravitino is the lightest sup ersymmetric particle (LSP),

the next-to-lightest sup ersymmetric particle (NLSP, typically the lightest neutralino

( ), as we will assume here) b ecomes unstable and eventually decays into a photon

plus a gravitino ( ! G) [2]. This decay b ecomes of exp erimental interest when

it happ ens quickly enough for the photon to be observed in the detector. Because

the interaction of the gravitino with matter is inversely prop ortional to the gravitino

mass, the neutralino lifetime will be short enough for a suciently light gravitino:

m 250 eV [6]. On the other hand, the gravitino may not b e to o light, as otherwise

it would be copiously pro duced leading to distinctive signals at colliders that have

not b een observed [3, 5] or cosmological [10] and astrophysical [11] embarrassments:

6 3

m > 10 eV . LEP 1 searches strengthen this limit to m 10 eV in large regions

~ ~

G G

of parameter space, when m

Theoretically, light gravitinos are exp ected in gauge-mediated mo dels of low-

energy sup ersymmetry [6], where the gravitino mass is related to the scale of su-

5 2

p ersymmetry breaking via m 6 10 eV ( =500 GeV) . Sp ecial cases of

SUSY

gravity-mediated mo dels may also yield light gravitinos, when the scale of lo cal and

global breaking of sup ersymmetry are decoupled, as in the context of no-scale su-

p ergravity [2, 7], in which case m (m =M ) M , with m the gaugino mass

1=2 Pl Pl 1=2

scale and p 2a mo del-dep endent constant.

Exp erimental searches for sup ersymmetry are considerably more sensitive in

this typ e of neutralino-unstable sup ersymmetric mo dels. First of all, the lightest

observable sup ersymmetric channel is no longer a pair of charginos ( ), but in-

stead a pair of (the usually lighter) neutralinos ( ), or if the gravitino is light

enough (m 10 eV ) the neutralino-gravitino channel ( G). These new chan-

nels allow a deep er exploration into parameter space. Furthermore, b ecause of the

photonic signature in all sup ersymmetric pro cesses, it b ecomes p ossible to over-

come the loss of exp erimental sensitivity that o ccurs when the daughter leptons

b ecome to o soft (as in chargino pair pro duction when m m < 10 GeV or

m >m >m 3 GeV ), and therefore absolute lower b ounds on sparticle masses

b ecome exp erimentally attainable in this class of mo dels. Indeed, diphoton searches

at LEP161 (i.e., s = 161 GeV) [13] have b een recently shown [14] to exclude a

signi cant fraction of the parameter space that is preferred by the sup ersymmetric

interpretations of the CDF event. Ongoing runs at LEP 2 should be able to prob e

even deep er into the remaining preferred region of parameter space.

1 +

We assume that R-parity is conserved, as otherwise the decay p ! GK may o ccur at an

unsuppressed rate [9].

We should emphasize that even though we are encouraged by the natural interpretation of the

CDF event within certain light gravitino scenarios, we b elieve that such scenarios are interesting in

their own right and should b e fully explored irresp ective of the status of the CDF event. 1

Our purp ose here is to consider in detail a complementary signal in light-

gravitino mo dels, namely the asso ciated pro duction of gravitinos with neutralinos.

The resulting single-photon signal has b een recently shown to b e observable at LEP 2

in certain range of gravitino masses, but only when the diphoton signal from neu-

tralino pair pro duction is itself not kinematically accessible [12]. Therefore, exp eri-

mental observation of whichever photonic signal will provide very useful information

in sorting out the various light-gravitino scenarios in the literature. The gravitino

mass plays a central role in gravitino-pro duction pro cesses, whose rate is inversely

prop ortional to the gravitino mass squared (1=m ). In contrast, the precise value

of the gravitino mass plays a minor role in the pro duction of the traditional sup er-

symmetric particles, as it determines only the decay length of the neutralino. The

neutralino-gravitino pro cess of interest at LEP is

(e e ! G ! +E ) / 1 ; with = ; (1)

miss

m s

which provides an exp erimental handle on the gravitino mass. This pro cess was

considered originally by Fayet [3] (in the restricted case of a very light photino-like

neutralino) who noted that the threshold b ehavior in Eq. (1) results from sub-

tle cancellations among all contributing amplitudes. Dimensional analysis indicates

4 2 2

that this cross section exceeds electroweak strength when M =(M m ) or

weak

Z Pl

1=2

2 4

m M =M 10 eV . In the context of LEP 1, this pro cess was revisited

e weak

in the restricted case of a neutralino with a non-negligible zino comp onent, where the

resonant Z -exchange diagram dominates [5].

In this pap er we rst calculate the cross section for the e e ! G pro cess

at arbitrary center-of-mass energies and give new analytic expressions for the corre-

sp onding di erential cross section (Sec. 2). Next we reassess the constraints on the

gravitino mass in view of the full LEP 1 data set and imp osing the preliminary lim-

s = 161 GeV), for general neutralino its obtained recently from runs at LEP161 (

comp ositions (Sec. 3). We also comment on the p otential of analogous searches at

the Tevatron. We then p erform a Monte Carlo simulation of the pro duction and de-

cay pro cesses leading to the single-photon signal and obtain energy (E ) and angular

(cos ) distributions for representative p oints in parameter space (Sec. 4), and also

discuss the missing invariant mass distribution exp ected in the events. We show that

one may evade the LEP 1 limits and still obtain observable single-photon signals at

LEP 2, although only when the diphoton signal from neutralino pair pro duction is

kinematically inaccessible. Finally we sp ecialize our results to the case of our pro-

p osed one-parameter no-scale sup ergravity mo del [7, 15] (Sec. 5). Our conclusions

are summarized in Sec. 6.

3 + +

e e e

Gravitino pair pro duction (e e ! G G) exceeds neutralino-gravitino pro duction (e e ! G )

10 eV ). The only for gravitino masses that have already b een excluded exp erimentally (m

single-photon signal is further suppressed by a factor of from the radiated photon. 2

2 The e e ! G pro cess

The Feynman diagrams for neutralino-gravitino asso ciated pro duction at LEP are

shown in Fig. 1, and include s-channel and Z exchange, and t- and u-channel

selectron (~e ) exchange. At the Z p eak one exp ects the s-channel Z exchange

R;L

diagram to dominate, and one may simply calculate the amplitude for Z ! G

decay [5]. This result is accurate as long as the neutralino has a non-negligible

zino comp onent. However, for photino-like neutralinos the other diagrams b ecome

imp ortant. This is also the case for any neutralino comp osition for center-of-mass

energies away from the Z p eak ( s > M ). To deal with all cases at once, we

p erform the complete calculation of all diagrams contributing to e e ! G. We

rst present the general form of the di erential cross section and later sp ecialize the

result for the particular case of a photino-like neutralino in order to exp ound on

certain theoretical issues and generalizations of our results.

In calculating interactions of gravitinos with matter one can pro ceed in one of

twoways. One may calculate with the full couplings in the sup ergravity Lagrangian,

in which case the vertices of interest are given by [1, 10]

e e

e e~ G / P p ; ~ G / p [ ; ] ; (2)

R;L R;L

where the (spin- goldstino comp onent of the) gravitino eld is G / @ =m , and

M =2:410 GeV is the appropriately scaled Planck mass. Alternatively one may

calculate using a set of much-simpli ed e ective goldstino couplings [3]

2 2

m m m

e~ e

p p

P ; ~ / [ ; ] p : (3) e e~ /

R;L R;L

6Mm 6Mm

~ ~

G G

The full and e ective couplings give the same results for the cross sections of pro-

cesses where the typical bad high-energy b ehavior of the gravitational amplitudes is

cancelled completely by the diagrams involving only gravitinos and regular sup er-

symmetric particles. This is the case for the e e ! G pro cess in hand, where we

haveveri ed (in the pure photino limit) that b oth ways of doing the calculation give

identical results. For a derivation and explanation of the meaning of the e ective

couplings see Ref. [10]. The simpli cation of using the e ective couplings is not b ene-

cial in other pro cesses, such as gravitino pair-pro duction, where diagrams including

graviton exchanges must also be included to cancel the bad high-energy b ehavior of

the amplitudes [3, 10]. Also, it is not clear whether this simpli cation may b e used in

the case of broken gauge symmetries, and therefore we have used the full couplings

in the case of neutralino comp ositions other than pure photino, where the s-channel

Z -exchange amplitude must be taken into account. 3

2.1 General case

The di erential cross section for a general neutralino comp osition is given by

(s m )

d F (s; t; u)

= ; (4)

2 2

d cos 32s 6(Mm )

where as usual we de ne

(s m )(1 cos ) ; (5) t =

u = (s m ) (1 + cos ) : (6)

The function F (s; t; u) receives contributions from each amplitude squared and various

interference terms (some of which vanish). In an obvious notation, we nd

F = F + F + F + F + F + F + F + F + F ; (7)

tt uu ZZ t u Zt Zu Z

where

2 2 2

2s(s m )(t + u )

0 2

F = (N e) (8)

2 2 2 2

t (m t)(t) t (m t)(t)

2 2

+(X ) (9) F = (X )

L tt R

2 2

(t m ) (t m )

e~ e~

R L

2 2 2 2

u (m u)(u) u (m u)(u)

2 2

F = (X ) +(X ) (10)

uu R L

2 2

(u m ) (u m )

e~ e~

R L

2 2 2

2 2

2s(s m )(t + u )

g (c + c )

R L

F = N (11)

2 2

cos 2 (s M ) +( M )

W Z Z

2 2

(t)(2st ) (t)(2st )

0 0

F = (N eX ) (N eX ) (12)

t R L

11 11

2 2

s(t m ) s(t m )

e~ e~

R L

2 2

(u)(2su ) (u)(2su )

0 0

F = (N eX ) (N eX ) (13)

u R L

11 11

2 2

s(u m ) s(u m )

e~ e~

R L

2 2

g (t)(2st )(s M )

F = N c X

Zt R R

2 2

cos [(s M ) +( M ) ](t m )

W Z Z

2 2

g (t)(2st )(s M )

+ (14) N c X

L L

2 2

cos [(s M ) +( M ) ](t m )

W Z Z

2 2

(u)(2su )(s M ) g

F = N c X

Zu R R

2 2

cos [(s M ) +( M ) ](u m )

W Z Z

Z e~

2 2

(u)(2su )(s M ) g

+ N c X (15)

L L

2 2

cos [(s M ) +( M ) ](u m )

W Z Z

2 2 2 2

2s(s m )(t + u )(s M )

(c + c ) g

R L

0 0

(16) F = 2 N N e

11 12

2 2

cos 2 s[(s M ) +( M ) ]

W Z Z

Z 4

In these expressions we have used the following e{~e { (X ) and e{e{Z (c )

R;L R;L R;L

couplings [16]

g sin

0 0

; (17) X = N e N

11 12

cos

0 0 2

X = N e N ( sin ) ; (18)

L W

11 12

cos

c = sin ; (19)

R W

c = + sin ; (20)

L W

0 0

where N and N denote the photino and zino comp onents of the neutralino re-

11 12

sp ectively. Indeed, the lightest neutralino may be written in two equivalent ways

[16]:

0 0 0 0

e f f

= N + N Z + N H + N H (21)

13 14

11 12 1 2

0 0

e f f f

= N B + N W + N H + N H (22)

11 12 3 13 14

1 2

related by

0 0

N = N cos + N sin ; N = N sin + N cos (23)

11 W 12 W 11 W 12 W

11 12

2.2 Pure photino case

This sp ecial case is useful in order to exp ose various subtleties in the calculation

that b ecome less apparent (although they are still present) in the case of a general

neutralino comp osition. The e e ! ~ G case is also imp ortant b ecause the result can

be readily taken over to the case of gluino-gravitino pro duction in quark-antiquark

annihilation at hadron colliders (q q ! g~G).

0 0

In this sp ecial case the couplings of the neutralino (photino: N =1,N =0)

11 12

to matter are very simple: X = e = X , F = F = F = F =0, and

R L ZZ Zt Zu Z

~ ~

~ 2 ~

F ! F = e (F + F + F ) ; (24)

R L

where

2 2 2

2s(s m )(t + u )

F = (25)

2 2 2 2

2 2

t (m t)(t) u (m u)(u)

t (2st) u (2su)

F = + + + (26)

2 2 2 2

2 2

(t m ) (u m ) s(t m ) s(u m )

e~ e~ e~ e~

R R R R

2 2 2 2

2 2

u (m u)(u) t (m t)(t)

t (2st) u (2su)

+ + F + (27) =

2 2 2 2

2 2

(t m ) (u m ) s(t m ) s(u m )

e~ e~ e~ e~

L L L L

With this relatively simple expression we can verify certain exp ected b ehaviors

of the cross section. First, in the limit of unbroken sup ersymmetry: m ! m =0,

m ! m 0, s + t + u = m ! 0, one can readily verify from the ab ove equations

e~ e

L;R

that F ! 0. The vanishing of the cross section in this limit is exp ected as the spin-

comp onent of the gravitino (the goldstino) b ecomes an unphysical particle when

sup ersymmetry is unbroken, as it is no longer absorb ed by the gravitino to b ecome

massive.

A related manifestation of this phenomenon can be exp osed by studying the

threshold b ehavior of the cross section. The spin- (goldstino) comp onent of the

gravitino is essentially obtained by taking the derivative of the full gravitino eld,

thus making the goldstino couplings be prop ortional to the goldstino momentum

(k ). At threshold k ! 0 and there is an additional suppression of the cross section

b esides the kinematical one. Threshold corresp onds to the limit s ! m and therefore

2 ~

from Eqs. (5,6) t; u go to zero as (s m ). In the ab ove expression for F , one can

2 3

see that near threshold each term is prop ortional to (s m ) , which combined with

the (s m ) term from the phase space integration [see Eq. (4)] yields a cross section

prop ortional to with = 1 m =s.

The ab ove results were originally obtained byFayet [3] based on a calculation

of the cross section using the e ective couplings (Eq. (3)), and have b een obtained

here for the rst time using the full couplings (Eq. (2)). Such equivalent expression

for the cross section makes more evident some further prop erties of the results, and

wethus give it explicitly here to o. The expression for F using the e ective couplings

b ecomes

2 2

2s(s m )+4uts=m

F = m

" # " #

2 2

(m t)(t) (m u)(u)

(2su) (2st)

4 2 2

+ m + + m m +

e~ e~

2 2 2 2

R R

2 2

(t m ) (u m ) s(t m ) s(u m )

e~ e~ e~ e~

R R R R

" # " #

2 2

(m t)(t) (m u)(u)

(2su) (2st)

4 2 2

+ m + + m m +

e~ e~

2 2 2 2

L L

2 2

(t m ) (u m ) s(t m ) s(u m )

e~ e~ e~ e~

L L L L

(28)

Despite the seemingly di erent app earances of F and F , it can be veri ed (at

least numerically) that they give identical results. Using F it is immediately ap-

parent that the cross section vanishes in the unbroken sup ersymmetry limit (i.e.,

m ;m ! 0), as it should. (Note also that for a massless photino (a case of in-

R;L

terest in the early literature) the s-channel diagram do es not contribute.) The

threshold b ehavior is not so apparent this time. One can rst note that near thresh-

old F b ecomes indep endentofm and dep ends only on m . A little algebra then

R;L

2 3 8

shows that indeed, near threshold, F / (s m ) , and thus the same threshold

b ehavior results, although this time as a result of a cancellation among all of the

contributing amplitudes.

The F form is also useful in exhibiting the dep endence of the cross section

on the selectron masses. As is evident from Eq. (28), the cross section increases with 6

increasing selectron masses, eventually saturating for very large values of m . Thus,

the decoupling theorem still holds (i.e., large values of the sparticle masses have no

e ect), although its sp eci c implementation here is rather p eculiar.

Before moving on to numerical evaluations of the cross sections, let us note

that the ab ove expressions for the photino cross section (using either the full or

e ective couplings) can b e adapted very easily to describ e gluino-gravitino pro duction

in quark-antiquark collisions at the Tevatron or LHC (q q ! g~G). In this case the

pro cess is mediated by s-channel gluon exchange and t-channel q~ exchange. One

L;R

needs to replace the e{~e { (X ) couplings in Eqs. (17,18) by those appropriate

R;L R;L

for q {~q { , one needs to replace the e{e{ coupling (e in Eq. (24)) by the strong

R;L

coupling (g ), and one needs to insert the appropriate color factor. Of course the

integration over parton distribution functions also needs to be implemented. (A

realistic calculation would also include the gluon-fusion channel, which b ecomes quite

relevant at LHC energies.)

3 Exp erimental constraints

3.1 LEP 1

The single-photon signal ( +E ) has b een searched for at LEP 1by various LEP

miss

Collab orations [17]. We estimate an upp er b ound of 0.1 pb on this cross section. This

estimate is an amalgamation of individual exp erimental limits with partial LEP 1

luminosities ( 100 pb ) and angular acceptance restrictions (j cos j < 0:7). Note

that the single-photon background at the Z p eak (mostly from e e ) is quite

signi cant, as otherwise one would naively exp ect upp er b ounds of order 3=L <

0:03 pb.

A numerical evaluation of the single-photon cross section at LEP 1 versus

the neutralino mass for m = 10 eV is shown in Fig. 2, for di erent choices of

0 0

neutralino comp osition (`zino': N 1; `bino': N = 1, and `photino': N = 1),

12 11

and where we have assumed the typical result B ( ! G)=1. In the photino case

0 0

the Z -exchange amplitude is absent (N =1)N =0) and one must also sp ecify

11 12

the selectron masses which mediate the t- and u-channel diagrams; we have taken

the representative values m = m =75;150 GeV. Increasing the selectron masses

e~ e~

R L

further leads to only a small increase in the cross section, e.g., at m = 0 one nds

= 1:48; 2:09; 2:36 pb for m = 150; 300; 1000 GeV, signalling the reaching of the

decoupling limit for large selectron masses discussed in Sec. 2.2.

In Fig. 2 we also show (dotted line `LNZ') the results for a well-motivated

one-parameter no-scale sup ergravity mo del [7, 15], which realizes the light gravitino

scenario that we study here. In this mo del the neutralino is mostly gaugino, but has

a small higgsino comp onent at low values of m , which disapp ears with increasing

Note that this implies a non-vanishing (p ossibly small) photino comp onent of the neutralino, as

would b e required in the `zino' case discussed ab ove. 7

neutralino masses; the neutralino approaches a pure bino at high neutralino masses.

The selectron masses also vary (increase) continously with the neutralino mass and

are not degenerate (i.e., m 1:5 m 2m ).

e~ e~

L R

This gure makes apparent the constraint on the gravitino mass that arises

from LEP 1 searches: in some regions of parameter space one must require m 10 eV

if m

on the gravitino mass versus the neutralino mass, that results from the imp osition of

our estimated upp er b ound < 0:1 pb. (The curves that extend b eyond m = M

result from constraints from LEP 2 data, and are discussed b elow.) Note the dep en-

dence on the selectron mass in the pure photino case.

3.2 LEP161

Recent runs of LEP at higher center-of-mass energies have so far yielded no excess

of single photons over Standard Mo del exp ectations. The latest searches at s =

161

161 GeV have pro duced upp er limits on the single-photon cross section 1pb

[18]. We have evaluated the single-photon cross sections for the neutralino comp o-

sitions used in Fig. 2 at s = 161 GeV. This time all cases dep end on the choice

of selectron masses. The numerical results are shown in Fig. 4, with the exp erimen-

tal upp er b ound denoted by the dashed line. Note that this line extends only for

m > M , as for m < M the much stronger limits discussed in the previous sec-

Z Z

tion apply. Moreover, for m = 10 eV (as used in Fig. 4), LEP 1 limits require

m M . As the gure makes evident, for m

Z Z

signals at LEP 2is not comp etitive with that at LEP 1.

As discussed ab ove, the cross sections in Fig. 4 increase with increasing selec-

tron masses (saturating at values somewhat larger than the ones shown), and con-

versely decrease with decreasing selectron masses. The choice of selectron masses also

a ects the near-threshold b ehavior of the cross section, with light selectron masses

\delaying" the onset of the threshold dep endence (see Fig. 4). Note also that the

photino, bino, and zino cross sections b ecome comparable ab ove the Z p ole, when

the Z -exchange diagram b ecomes comparable to the other diagrams. In the case of

the one-parameter mo del (`LNZ') a p eculiar bump app ears. This bump is understo o d

in terms of the selectron masses that vary continously with the neutralino mass: at

low values of m the selectron masses are light and the cross section approaches the

light xed-selectron mass curves (`75'); at larger values of m the selectron masses are

large and the cross section approaches (and exceeds) the heavy xed-selectron mass

curves (`150'). This example brings to light some of the subtle features that might

arise in realistic mo dels of low-energy sup ersymmetry.

In spite of their apparentweakeness, LEP161 limits on the single-photon cross

section are useful in constraining the gravitino mass in a neutralino-mass range in-

accessible at LEP 1. Indeed, decreasing the gravitino mass in Fig. 4 by a factor of 3

For reference, in gauge-mediated mo dels of low-energy sup ersymmetry, such gravitino masses

corresp ond to 3TeV .

SUSY 8

will make the cross sections some ten-times larger. The resulting lower b ounds on the

gravitino mass from LEP161 searches are shown in Fig. 3. This gure shows that, as

M . However, b ecause of the threshold exp ected, LEP 1 limits dominate for m

b ehavior at s = M , LEP161 limits `takeover' for neutralino masses slightly b elow

M , and in the `photino' case, considerably b elow M .

Z Z

3.3 Single photons versus diphotons

M the gravitino mass is con- It has b een made apparent in Fig. 3 that for m

strained to m 10 eV . If this would indeed be an absolute requirement on the

gravitino mass (i.e., for all values of m ) then the cross section for neutralino-gravitino

161

pro duction at LEP 2 would be highly suppressed: Fig. 4 shows that 1pb

for m = 10 eV , and / m . In other words, if the minimum observable

single-photon cross section at LEP 2 is 0:1pb (i.e., for L 100 pb ), then

m 3 10 eV app ears to b e the limit of sensitivity of LEP 2.

On the other hand, the pro cess e e ! ! +E is sensitive to m <

miss

s and is indep endent of the gravitino mass. In light of the single-photon constraints

on the gravitino mass obtained ab ove, the diphoton pro cess may be observable at

s LEP 2 (i.e., m < M ) only if m 10 eV , and therefore single photons

will not be simultaneously observable at LEP 2. Conversely, single photons may be

observable at LEP 2 only if m >M , in which case diphotons may not b e observed

s > 2M 190 GeV). This dichotomy between simultaneously (as they require

single-photon and diphoton signals at LEP was rst presented in Ref. [12].

3.4 Other limits

The ab ove lower limits on m are rather signi cant, and improve considerably on

previous limits from collider exp eriments [3, 4, 5, 17] and astrophysical considera-

tions [11], as long as m

hadron collider limits obtained via asso ciated gluino-gravitino pro duction (pp ! g~G),

and via indirect gluino pair-pro duction (pp ! g~g~) where in addition to the usual su-

p ersymmetry QCD diagrams the gravitino is exchanged in the t and u channels. The

multijet signature of these pro cesses has b een contrasted with exp erimental limits

from the most recentTevatron run to show that if gluino pair pro duction is accessible

at the Tevatron (i.e., m 200 GeV) then a lower limit of m > 3 10 eV re-

sults. This limit and our limits in Fig. 3 may b e compared by relating the gluino and

neutralino masses, as o ccurs in sup ergravity mo dels with universal gaugino masses

at the uni cation scale: m 3m 6m . Therefore, m 200 GeV (as required

g~ g~

for the b ound in Ref. [19] to apply) corresp onds to m 35 GeV. Consulting Fig. 3,

we see that the Tevatron limits are stronger in this neutralino mass range. How-

ever, the LEP 1 (161) limit extends up to m M (2M ), which corresp onds to

Z W

m 550 (960) GeV, which is far from the direct reach of the Tevatron.

By considering the further pro cesses pp ! gS; gP , where S and P are very light 9

scalar and pseudo-scalar particles asso ciated with the gravitino, the lower b ound on

the gravitino mass b ecomes much less dep endent on the gluino mass and can b e taken

to be m > 3 10 eV [19]. This lower b ound is comparable with those obtained

ab ove by considering LEP 1 data. However, this result assumes the existence of

additional light and strongly interacting particles (S and P ), an assumption that

dep ends on the detailed nature of the mechanism that leads to a very light gravitino.

3.5 New channels

Another set of channels of interest at the Tevatron consist of the asso ciated pro duction

of gravitinos with neutralinos or charginos

e e

pp ! G; G ; (29)

which have the advantage over pp ! g~G of much less phase space suppression. The

most basic channel is q q ! G, which leads to a +E signal. The cross

T;miss

section for this pro cess can be readily obtained from the expressions given in Sec. 2

by replacing the initial state electron-p ositron pairs by quark-antiquark pairs, the

exchanged selectrons by squarks, and by integrating the resulting expression over

parton distribution functions. We have estimated this cross section and nd that

it may be quite signi cant: up to 85; 25; 15 pb for m = 50; 75; 100 GeV and m =

10 eV , in favorable regions of parameter space. In the b est case scenario of a

Tevatron upp er limit of 0.1 pb (i.e., 10 events in L = 100 pb ), one may conclude

that m (3; 1:6; 1:2) 10 eV for m =50;75; 100 GeV. Taken at face value, these

limits are quite comp etitive with those obtained in Ref. [19]. At the moment there

are no single-photon limits available from CDF nor D0.

To improve the visibility of the signal, one maywant to consider the q q ! G

channel which, dep ending on the chargino decaychannel, may lead to ` + +E

T;miss

or 2j + +E signals. The leptonic signal app ears particularly promising. For

T;miss

all these pro cesses there are some imp ortant instrumental backgrounds that need to

be overcome. For instance pp ! W ! e , where the electron is misidenti ed as

a photon (i.e., b ecause of limitations in tracking eciency), leads to a very large

\single-photon" signal B (W ! e ) 2:4 10 pb [20], which may be reduced

signi cantly by optimizing the tracking eciency and making suitable kinematical

cuts. The other channels mentioned ab ove face similar, although p erhaps less severe,

instrumental backgrounds (e.g., WW ! e +\ "+E ).

T;miss

4 The single-photon signal

The total cross section for neutralino-gravitino pro duction has b een displayed in Fig. 4

s = 161 GeV ) and for some illustrative choices for a sp eci c center-of-mass energy (

of parameter values. The analytic expressions given in Sec. 2.1 allow one to calculate

these cross sections for arbitrary values of the parameters. In this section we would 10

like to explore some characteristics of the actual signal, i.e., the energy and angular

distributions of the observable photon and the missing invariant mass distribution in

the events.

4.1 Monte Carlo technique

Our simulation pro ceeds in a standard way, making use of a `home-made' Monte

Carlo event generator. We start in the rest frame of the decaying neutralino, where

we generate + G events that are isotropic in this reference frame. Energy-momentum

0 0

conservation requires E = jp~ j = m , which leaves two comp onents of the photon

momentum to b e generated at random (i.e., p ). We then b o ost the photon momen-

tum back to the lab oratory frame using the neutralino 4-momentum (E ;~p ), whose

comp onents are constrained by the kinematics of the e e ! G pro cess:

p p

2 2

m m

s s

p p

E = + ; jp~ j = : (30)

2 2 s 2 2 s

Here wehavetwo comp onents of the neutralino momentum unconstrained (cos ; ).

For xed values of these angles we obtain E and cos distributions, which are

purely kinematical e ects. The observable distributions are obtained by varying

(cos ; ) and weighing these kinematical distributions with the corresp onding dy-

d 1

factors calculable from the expressions given in Sec. 2. namical

d cos

In what follows we fo cus on the case of LEP190 ( s = 190 GeV). First we

display in Fig. 5 the total cross sections for single-photon pro duction at this center-

of-mass energy. These should give an idea of the reach in neutralino masses that may

b e accessible at LEP190. As we exp ect neutralino-gravitino pro duction to b e allowed

only for m >M (to avoid the stringent LEP1lower limits on m ), we concentrate

on the following three neutralino mass choices: m = 100; 125; 150 GeV. To gain some

insight into the nal distributions, we start by displaying the normalized neutralino

d 1

as a function of cos for m = 75; 150; 300 GeV, and angular distributions

d cos

bino (Fig. 6), zino (Fig. 7), and photino (Fig. 8) neutralino comp ositions. The total

cross sections for each of the curves can be read o Fig. 5 and for convenience have

b een tabulated in Table 1. As can be seen from Figs. 6, 7, and 8, the angular

distribution varies quite a bit with neutralino mass, although mostly for light selectron

masses. Note that the angular distributions always remain nite, and generally show

a preference for the central region.

4.2 Energy and angular distributions

The observable photonic energy and angular distributions can b e quite unwieldly once

we allow for the manychoices of parameters that wehave considered ab ove. An exam-

ination of all parameter combinations shows that b oth energy and angular distribu-

tions are largely insensitive to the neutralino comp osition, b eing much more sensitive

to the mass parameters (i.e., m ;m ). This result is p erhaps not surprising as the

e~ 11

Table 1: Total cross sections corresp onding to the di erential cross sections shown in

Figs. 6,7,8 at LEP190. All masses in GeV, all cross sections in pb.

Comp osition m m =75 m = 150 m = 300

e~ e~ e~

bino 100 0.34 0.54 1.61

125 0.32 0.19 0.60

150 0.20 0.04 0.11

zino 100 0.19 0.49 1.12

125 0.10 0.17 0.42

150 0.06 0.03 0.08

photino 100 0.37 0.52 1.64

125 0.37 0.18 0.61

150 0.23 0.04 0.12

observable distributions of relativistic particles are dominated by kinematical e ects

which dep end crucially on the mass parameters. Thus, for brevitywe show only the re-

sult in the bino case which, in anyevent, is representativeoftypical sup ergravitymod-

els. The energy (E ) and angular (cos ) distributions for m = 100; 125; 150 GeV

are shown in Fig. 9 and Fig. 10 resp ectively, for m =75;150; 300 GeV.

The energy distributions (Fig. 9) show a signi cant m and m dep endence.

As the neutralino mass grows, it tends to pro duce harder photons. In fact, it is not

hard to show that in the decay ! G, with a neutralino energy and momentum as

in Eq. (30), the photon energy is restricted to the interval

2 s 2

as faithfully repro duced in the simulations. (Near threshold (m s) the photon

carries away half of the center-of-mass energy.) These distributions show that any

given single-photon candidate energy (E ) implies an upp er b ound on the p ossible

neutralino masses consistent with the candidate event,

2 sE : (32) m <

The photonic angular distributions (Fig. 10) are p eaked in the forward and

backward directions, even more so as the neutralino b ecomes heavier. The selectron

mass has an interesting e ect. In the case of m = 100 GeV , from Fig. 6 we see

that for heavy selectron masses the neutralino angular distribution is fairly at, and

therefore the photonic distributions should re ect only kinematical e ects, as they do

(i.e., p eaked in the forward and backward directions). For light selectron masses, the

neutralino avoids the forward and backward directions, and the kinematical e ect on

the photons is diminished. 12

4.3 Missing mass distribution

The dominant background to the neutralino-gravitino signal is single radiative return

to the Z : e e ! Z ! , where the photon is radiated o the initial state and

the Z b oson tends to b e on shell. The most distinctive signature for this background

2 2

pro cess app ears in the missing invariant mass M = E p distribution,

miss

miss miss

which is strongly p eaked at M M .

miss Z

The missing mass distribution for the signal can be easily determined, as in

this case the missing energy and missing momentum are given by

E = s E ; p = j ~p j = E ; (33)

miss miss

and therefore

p p

M = s ( s 2E ) : (34)

miss

The allowed range of M is obtained by inserting the range of photonic energies in

miss

Eq. (31); we obtain

miss

Histograms showing missing mass distributions fall in the range sp eci ed by Eq. (35),

and otherwise favor the upp er end of the M range (corresp onding to the lower end

miss

of the photonic energy range). For brevity, we display these distributions only in the

one-parameter mo del example discussed in Sec. 5 b elow.

We note in passing that the complementary diphoton events from neutralino

pair-pro duction have a background (e e ! Z ! ) that also p eaks for

M M [6]. In this case the missing mass in the diphoton signal varies from

miss Z

( s + s 4m ), in contrast with the result for zero up to a maximum value of

the single-photon signal in Eq. (35). The M distributions of the single-photon and

miss

diphoton signals di er not only in range, but also in shap e.

5 One-parameter mo del example

It should have b ecome clear from the discussion in Sec. 4, that the signals to be

searched for exp erimentally can have a wide range of characteristics b ecause of the

variations in the underlying parameters describing the neutralino-gravitino pro cess.

In reality, the mo del of sup ersymmetry that describ es nature will have all its mass

parameters correlated in some way, and the actual observations may b e a b ewildering

comp osite of the many curves shown ab ove. To exemplify this situation, in this section

we sp ecialize our results to the case of the one-parameter no-scale sup ergravity mo del

that has b een mentioned at various places in the preceding discussion. We have

already shown the corresp onding single-photon cross sections at LEP 1, LEP161,

and LEP190, as the dashed lines in Figs. 2,4,5 resp ectively. The cross sections for

s>M show a p eculiar bump that, as discussed in Sec. 3.2, can b e traced backto

the fact that the selectron masses vary with the neutralino mass. 13

As a rst step towards obtaining the angular and energy photonic distributions,

1 d

in Fig. 11 we show the normalized neutralino angular distributions s = for

d cos

190 GeV and m =60;80; 100; 120; 140 GeV. The total cross sections in each of these

cases are =1:2;1:3;1:0;0:6;0:2pb. Note how relatively at the angular distributions

are: no more than a 10% variation. This is to be contrasted with the wide range of

variability observed in the generic cases shown in Figs. 6, 7, and 8. In fact, the

results in the one-parameter mo del resemble those in the generic mo dels when the

selectron mass is large (i.e., the dotted lines in those gures, which corresp ond to

m = 300 GeV ). This is to be exp ected as in the one-parameter mo del one has

m 1:5 m 2m , indicating increasingly heavier selectrons.

e~ e~

L R

Following the metho d outlined in Sec. 4, in Figs. 12 and 13 we display the pho-

tonic energy and angular distributions at LEP190 for three representative neutralino

masses (m = 100; 120; 140 GeV). The energy distributions show the same restrictive

photon energy b ehavior as predicted by Eq. (31). The angular distributions are also

p eaked in the forward and backward directions.

Finally we consider the missing mass distributions, which are obtained from

Eq. (34), and are shown in Fig. 14. We note the range of M , as prescrib ed by

miss

Eq. (35), and the tendency to favor missing mass values toward the upp er end of

the allowed interval. For the neutralino mass choices shown, the missing mass shows

a distinct preference to be larger than M . (This is in contrast with the M

Z miss

distribution in diphoton events, which is more evenly distributed.)

6 Conclusions

In this pap er wehave attempted to study in some detail the physics of sup ersymmetric

mo dels with a gravitino light enough that it can be pro duced directly at collider ex-

p eriments. Our discussion has centered mainly around LEP, from where the strongest

constraints can b e obtained at the moment. Wehave nonetheless outlined the corre-

sp onding program to be followed at the Tevatron, where instrumental backgrounds

make identi cation of the single-photon signal a more challenging task.

We have provided new and explicitly analytical expressions for neutralino-

gravitino di erential cross sections at e e colliders and have discussed some of the

theoretical subtleties involved in the calculation and some of the p eculiar parameter

dep endences of the cross section. We have used our expressions to obtain new lower

b ounds from LEP 1 data on the gravitino mass for m < M . Weaker limits from

LEP 2 are obtained at higher neutralino masses. Our study includes a Monte Carlo

simulation of the single-photon signal, which should be helpful in the exp erimental

analyses that are just now getting underway. We have also sp ecialized the results to

our one-parameter no-scale sup ergravity mo del, where the signals can be analyzed

much more simply b ecause of the tight correlations between the mo del parameters. 14

Acknowledgments

J. L. would like to thank G. Eppley, T. Gherghetta, and T. Moroi for useful discus-

sions. The work of J. L. has b een supp orted in part by DOE grant DE-FG05-93-ER-

40717 and that of D.V.N. by DOE grant DE-FG05-91-ER-40633.

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e+ G e+ G

(a) (b)

Figure 1: Feynman diagrams for neutralino-gravitino pro duction at LEP: (a) s-

channel and Z exchange, and (b) t-channel selectron exchange (~e ). Additional

R;L

u-channel diagram not shown. 17 + − ~ σ (e e → χ G → γ + Ε ) [pb] √ miss s = MZ

LNZ zino

bino

photino (150)

photino (75)

LEP 1

mχ (GeV)

Figure 2: Single-photon cross sections (in pb) from neutralino-gravitino pro duction

at LEP 1 versus the neutralino mass (m ) for m =10 eV and various neutralino

comp ositions. The `photino' curves dep end on the selectron mass (75,150). The

cross sections scale like / m . The dashed line represents the estimated LEP 1

upp er limit. Also shown is the result for a one-parameter no-scale sup ergravity mo del

(`LNZ'). 18 Lower limit on m~G (eV)

zino bino

photino LNZ LEP 1

LEP 161

mχ (GeV) mχ (GeV)

Figure 3: Lower b ounds on the gravitino mass (in eV) as a function of the neutralino

mass (m ) that result from single-photon searches ( +E ) at LEP 1 and LEP161.

miss

In the `photino' case at LEP 1 and the `photino', `zino', and `bino' cases at LEP161,

the selectron mass in uences the results. We have chosen m = 75; 150; 300 GeV,

denoted by solid, dashed, and dotted lines resp ectively. Also shown are the b ounds

in a one-parameter no-scale sup ergravity mo del (`LNZ'). 19 σ ( + −→ χ ~ → γ + Ε e e G miss ) [pb]

zino (150) √s = 161 GeV bino (150)

photino (150)

zino (75)

LNZ LEP 161 bino (75)

photino (75)

mχ (GeV)

Figure 4: Single-photon cross sections (in pb) from neutralino-gravitino pro duction at

LEP 161 versus the neutralino mass (m ) for m = 10 eV and various neutralino

comp ositions. The solid curves have a xed value for the selectron mass (75,150),

whereas the dotted curve corresp onds to a one-parameter no-scale sup ergravity mo del

where the selectron masses vary continously with the neutralino mass. The cross

sections scale like / m . The preliminary LEP161 upp er limit is indicated.

G 20 σ ( + −→ χ ~ → γ + Ε e e G miss ) [pb] bino (150) √s = 190 GeV photino (150)

zino (150)

LNZ

zino (75)

bino (75)

photino (75)

mχ (GeV)

Figure 5: Single-photon cross sections (in pb) from neutralino-gravitino pro duction at

LEP 190 versus the neutralino mass (m ) for m = 10 eV and various neutralino

comp ositions. The solid curves have a xed value for the selectron mass (75,150),

whereas the dotted curve corresp onds to a one-parameter no-scale sup ergravity mo del

where the selectron masses vary continously with the neutralino mass. The cross

sections scale like / m .

G 21

Figure 6: Normalized angular distribution of neutralinos of bino comp osition in

neutralino-gravitino pro duction at LEP190 for m = 100; 125; 150 GeV and m =

75 (solid); 150 (dashed); 300 (dots) GeV. 22

Figure 7: Normalized angular distribution of neutralinos of zino comp osition in

neutralino-gravitino pro duction at LEP190 for m = 100; 125; 150 GeV and m =

75 (solid); 150 (dashed); 300 (dots) GeV. 23

Figure 8: Normalized angular distribution of neutralinos of photino comp osition in

neutralino-gravitino pro duction at LEP190 for m = 100; 125; 150 GeV and m =

75 (solid); 150 (dashed); 300 (dots) GeV. 24

Figure 9: Photonic energy distributions in neutralino(bino)-gravitino pro duction at

LEP190 for m = 100; 125; 150 GeV and m = 75 (solid), 150 (dashed), and 300

(dots) GeV. 25

Figure 10: Photonic angular distributions in neutralino(bino)-gravitino pro duction

at LEP190 for m = 100; 125; 150 GeV and m = 75 (solid), 150 (dashed), and 300

(dots) GeV. 26 LNZ √s = 190 GeV

60 80 100 120

140

Figure 11: Normalized angular distribution of neutralinos in neutralino-gravitino pro-

duction at LEP190 for m = 60; 80; 100; 120; 140 GeV in a one-parameter no-scale

sup ergravity mo del. 27 LNZ √s = 190 GeV 140

120 100

Eγ (GeV)

Figure 12: Photonic energy distributions at LEP190 in a one-parameter no-scale

sup ergravity mo del for m = 100; 120; 140 GeV.

28 √s = 190 GeV LNZ

100 120 140

cos θγ

Figure 13: Photonic angular distributions at LEP190 in a one-parameter no-scale

sup ergravity mo del for m = 100; 120; 140 GeV.

29 140 LNZ √s = 190 GeV 120 100

(

Mmiss GeV)

Figure 14: Missing invariant mass distributions at LEP190 in a one-parameter no-

scale sup ergravity mo del for m = 100; 120; 140 GeV.