Proceedings of the Ninth Hawaii Topical Conference on Particle Physics (1983) 9780824886035

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Proceedings of the NINTH HAWAII TOPICAL CONFERENCE IN PARTICLE PHYSICS (1983)

Edited by R. J. Cence and E. Ma Contributors K. BERKELMAN J. ELLIS H. GEORGI C. RUBBIA

UNIVERSITY OF HAWAII AT MANOA/HONOLULU

© 1984 University of Hawaii Press All rights reserved Manufactured in the United States of America

Library of Congress Cataloging in Publication Data Hawaii Topical Conference in Particle Physics (9th : 1983 : University of Hawaii at Manoa) Proceedings of the Ninth Hawaii Topical Conference in Particle Physics (1983). 1. Particles (Nuclear physics)—Congresses. I. Cence, Robert J., 1930. II. Ma, E „ 1945III. Berkelman, K. (Karl) IV. Title. QC793.H38 1983 539.7'21 83-18126 ISBN 0-8248-0949-1

Distributed by University of Hawaii Press Order Department University of Hawaii Press 2840 Kolowalu Street Honolulu, Hawaii 96822

TABLE OF CONTENTS Page Preface

v TOPICS IN e + e~ PHYSICS K. Berkelman

1. Nonresonant e + e~ Physics

3

2. Quarkonium Physics

16

3.

46

Weak Decays of Heavy Quarks GRAND UNIFICATION AND SUPERSYMMETRY J^ Ellis

1.

Conventional GUTS

95

2.

Globally Supersymmetric GUTS

113

3.

Global and Local SUSY Breaking

130

4.

Supergravity Phenomenology

144

5.

Other Particle Searches

162

BEYOND THE STANDARD MODEL H. Georgi 1.

Effective Field Theories

213

2.

The Chiral Quark Model

228

3.

Higgs vs. Technicolor

243

4.

Flavor

257

5.

A Depressing Speculation

267

PHYSICS RESULTS OF THE UA1 COLLABORATION AT THE CERN PROTON-ANTIPROTON COLLIDER C. Rubbla 1.

The CERN Super Proton Synchrotron (SPS) as a ProtonAntlproton Collider ill

277

2.

Jets

280

3.

Observation of Charged Intermediate Vector Bosons. .

303

4.

Observation of the Neutral Boson Z°

320

5.

Comparing Theory With Experiment

326

iv

Preface

The Ninth Hawaii Topical Conference

in

Particle

Ptiynlr n

was held on the Manoa campus of the University of Hawaii during the two-week period August 11-24, 1983, with support University U.S.

U.S.

the

of Hawaii, the University of Hawaii Foundation, the

Department of Energy (High Energy Physics

the

from

National Science Foundation.

Program),

and

As in previous confer-

ences of this series, which dated back to 1965, the

focus

was

on selected topics at the frontier of particle physics, as presented by four principal lecturers. was

again

fortunate

to

outstanding individuals: Dr. John Ellis and

Conference

have as its principal lecturers four Professor Karl

Berkelman

(Cornell),

(SLAC/CERN), Professor Howard Georgi (Harvard),

Professor Carlo Rubbia

75-minute

This year, the

lectures

to

(CERN/Harvard).

Each

gave

five

an audience of some sixty participants

who were mostly younger members of the active research community, consisting of both experimentalists and theorists. The principal lectures consisted e+e

physics

(Berkelman),

selected

unification

(Ellis), physics beyond the standard physics (Rubbia).

of

and

model

in

supersymmetry

(Georgi),

and

pp

They represented the most up-to-date results

in experiment as well as in theory, and generated discussions

topics

among all the participants.

many

lively

These Proceedings are

a record of those remarkable two weeks and should

be

valuable

to anyone interested in a more comprehensive exposition of current research in high-energy physics than able.

The

is

normally

avail-

contributed seminars given by many of the partici-

v

pants were also an important part of the Conference.

They

are

published separately as a University of Hawaii High Energy Physics Group report, UH-511-518-83. charge

by

Group,

University

Copies are available free of

contacting the Group Secretary (High Energy Physics of

Hawaii,

2505

Correa

Road,

Honolulu,

Hawaii, 96822). Gratitude for the successful conclusion of the goes

first

to

the

four

principal

lecturers.

thanked are the seminar speakers, and personnel, cretary. and

especially

the

Conference

Others to be

Conference

support

Mrs. Caroline Chong, the Conference Se-

Acknowledgment is due also to President Fujio Matsuda

Vice-President

Albert J. Simone

of

the

University

of

Hawaii, and President J.W.A. Buyers of the University of Hawaii Foundation for their interest in and support of the Conference. Lastly, I thank Professor David E. Yount for helping as Conference Associate Director and Professor Robert J. Cence for serving as Editor of these Proceedings.

Ernest Ma Conference Director

vi

TOPICS IN e+e~ PHYSICS

K. Berkelman Cornell University Ithaca, New York

1.0 1.1 The

NONRESONANT e*e~ PHYSICS

THE EARLIEST STORAGE RINGS: TESTING QED

first

storage

electrodynamics.

In

ring

the

theory at small distances

uas

built

to

test

quantum

1950*s tests of the validity of the uere

being

made

uith

wide

angle

electron and muon pair production, and someuhat later uith uide angle bremsstrahlung by electron beams. experiments nucleus.

uas

a

virtual

photon

The sensitivity of

limited

by

unknown

therefore

in

these

tests, such

hadronic

however,

as

uas

form factors,

interactions.

It

uas

important to look for processes in uhich no strongly

interacting particles uere elastic

QED

effects

breakup, and background from

target

in the coulomb field of a

these

nuclear

The

scattering

involved.

Electron

and

positron

on atomic electron targets had been tried,

but the maximum momentum transfer q ^ 2 - -2Em s uas

too

small

to be of interest. As ue all electron

knou,

beam

this

collide

can

be

head-on

remedied

uith

by

another

having

electron

instead of uith stationary electrons, thus allouing momentum uith a

transfer of q ^ sacrifice

in

2

2

- -4E .

maximum

This is obtained, however,

luminosity,

interaction cross section.

a

the beam

the

event

rate

per

unit

Luminosity can be expressed as

jC » njngf/A, and is measured in cm^sec"1.

For a

fixed

target

experiment

n t /A is the number of target particles per unit area, typically at least 10 23 /cm 2 ; and n z f is the beam flux second,

say

10

12

electrons/sec.

might therefore have aC - 10 experiment say 10 10 , A typically

35

For 1

cm^sec" .

in

particles

per

such an experiment ue In a

colliding

beam

ni and n 2 are the number of particles in each beam, is

the

10~2

cm 2 ,

cross

say 10 7 /sec for a 10 m expect around 10

29

sectional

area

of

intersection,

and f is the beam circulation frequency, diameter

ring.

1

cm^sec' .

3

One

uould

therefore

In the late 1950'e Gerry O'Neill and first

to

build

a

coworkers

colliding bean facility.

Here

the

It was a pair of

tangent 550 MeV storage rings [Barber 711 filled uith electrons from

the

Stanford

HEPL linac.

To detect scattered electrons

arrays of lead, scintillators, and spark chambers above

and

belou

the

became the model for all the storage next

decade.

matched

the

houever,

The in

placed

ring

detectors

This

for

the

measured angular distribution (Fig. 1.1.2)

prediction

was

were

intersection region (Fig. 1.1.1).

of

QED.

pioneering

a

The new

real

accomplishment,

nay of doing high energy

physics experiments. The first electron-positron Frascati

[Bernardini

accumulazione). principle

of

It the

601 uas

and an

uas

ring

called

important

single-ring

enough luminosity to test QED. such

storage

uas

AdA

built

at

(annello

di

demonstration

machine,

but

of

the

did not produce

That came uith later

machines,

as ACO at Orsay and VEPP-2 at Novosibirsk, and eventually

at all e*e~ rings (Fig. 1.1.3). There are three reactions usually used to test QED.

They

are of more than just historic interest, so I will discuss them in some detaiI.

a) Elastic e*e~ scattering (Bhabha scattering) In louest order there are tuo diagrams, one exchanged

spacelike

involving

an

photon (as in e'e" scattering), the other

in uhich the e* and e" annihilate into a timelike photon, uhich then creates a pair.

The differential cross section can be uritten as dv/dO - («2W2/2) 4

[(l+cos^J/q" 2 z

+ (2cos 0)/q U

+ (l+cosz0)/2U4],

uhere the first term comes from the exchange diagram, the term

from

the

annihilation,

and

4

the

second

term

is

last the

interference of the tub amplitudes, transfer

squared, q

2

2

q 2 is

the

four

momentum

2

- -4E sin 0/2, 0 19 the scattering angle,

U - 2E is the total energy, and E is the energy of

each

beam.

As you can see from Fig. 1.1.4 this is a large cross section at small

angles—essentially

4«^/E204.

In

fact,

9mall

angle

Bhabha scattering Is used as a monitor of the luminosity in all e*e~ storage ring experiments. b) tluon pair production In

louest

diagram.

The

order

this

involves

only

the

annihilation

cross section is symmetric about 9 - 90° and is

given by d»/dil - (a2/4U2) fy ttl+cos2») + U-fy^sin 2 *]. If the energy U is far enough above

the

threshold

2m^,

the

muon velocity 0 becomes 1 and the cross section simplifies to dv/d8 - (c^MU 2 ) (1 + cos2*). Compared to Bhabha scattering, it is much a much smaller section

(see

Fig. 1.1.4)

and

cross

has no strong forward peaking.

The integral over angle is 9 - 86.9nb/U 2

(U in GeV).

This is the prototype cross section for the production pointlike

of

any

charged fermion pair, and is used theoretically as a

basis of comparison for other cross sections. c) Tuo-photon annihilation In this process the virtual particle in louest order ¡3 an exchanged 2

electron

2

-4E sin 0/2.

uith

space I ike four momentum squared q 2 -

The cross section is strongly foruard peaked

and

of course symmetric about 90°: dff/dQ - a2U23in2e/8q'1 + same uith 0 replaced by ir-8. The early e*e~ storage rings, listed out

these

belou,

all

carried

experiments, each verifying that the measured cross

sections folloued the QED formulas over a of

uide

range

of

the

four

momentum

811.

The testing of QED is no longer considered as exciting as

the virtual particle [Branson 81, Hoilebeck

it uas in the 50's; ue tend to take it for granted as a correct theory.

Strictly speaking, houever, the 5

QED

predictions

for

the

Bhabha

scattering and muon pair production processes have

already proven to be invalid. of

electron

As ue shall see later, the yield

or muon pairs is suddenly enhanced uhen the total

energy matches the mass of a vector decay

branching

ratio

Also, at sensitive

to

into

high

meson

lepton

energies

with

a

detectable

pairs (such as p", w,

the

cross

sections

become

the interference uith the amplitude in which the

neutral ueak vector boson Z° replaces the time I ike photon. interpret

Ue

these as signs that QED is an incomplete theory, not

an incorrect one.

So even if QED uere never really

urong,

it

Mould still be Morthuhile to insist on checking its predictions as higher energies become available.

1.2

EARLY HADRON PRODUCTION

Electron-positron collisions charged

hadrons,

jrV",

K*K",

can

pp,

also

create

pairs

of

through the same virtual

photon diagram responsible for muon pair production.

Hoiiever,

since hadrons are not elementary point I ike particles, the cross section is depressed by a form factor.

That is, for a pair

of

spin-0 charged hadrons uith form factor F ue have d

1012

1.31 ± 0.04

"

*

3097

4.7 ± 0.6

T

9460

1.22 ± 0.06 [Artamonov 821

The energy dependence of the e4e" -» V

•* F

" cross

section

is

given by the Breit-Uigner resonance formula: , a, and 4> mesons decay strongly into pions or kaons. The

quark

final state

and antiquark of the vector meson continue into the as

constituents

of

the

separate

mesons.

For

example, in the dominant 4> decay into KK the original s and s end up each in one of the kaons along uith a new light quark or antiquark partner, for example, allowed

suppressed

fl In the case of the ^ and

T

this

mechanism

^

tl

is

energetically

forbidden, because the lightest mesons (D and B) containing one heavy quark (c and b) have masses uhich are more than half masses

of the vector mesons.

the

As a consequence, a heavy vector

18

meson has to decay by annihilating the heavy quarks, either a

higher

order QCD process or electromagnet i cally.

case the annihilation rate is considerably suppressed to

the

decay

without

annihilation.

This

in

In either relative

qualitative

observation, called the Zweig Rule COkubo 631, explains uhy the pir mode

of ¿ decay has a smaller branching ratio (14.8%) than

the KK mode (83.7%) in space:

M^ - (nip -

spite

of

having

the

most

excitement.

narrou

its

It

uas

Zweig-suppressed annihilation of a bound cc quark

c

with

a

interactions. peaks

are

larger

phase

- 116 MeV and fl^ - 2m K - 33 lleV.

Uhen the ^ uas discovered it uas caused

much

quantum

number,

width

that

explained

as

state

a

of

the neu

charm, conserved in strong

The observed widths of the

^

and

T

resonance

a consequence of the energy spread in the colliding

e* and e" beams, caused by synchrotron radiation.

The

spread

SU is proportional to UV"1'2, where p is the bending radius in the ring; for example, Fortunately

this

in

energy

CESR

it

ie

3.B

broadening,

MeV

while

at

the

reducing

T. the

resonance peak height, does not affect the area under the peak. The intrinsic T of a narrow indirectly. - T„

resonance

must

be

obtained

If we assume lepton universality, then T n - T ^

and a measurement of one branching ratio, say B w ,

can

be combined with our I"., measurement (see above) to give us T: r - r^/B^. The table below shows the measured

dilepton

branching

ratios

and total widths for the vector mesons. MESON

B^, *

T, keV

0.0043 (ee)

154000 (PDG 821

u

0.0072 (ee)

9900

"

4>

0.031

4200

"

\fr

7.4

63 [Boyarski 75]

T

2.9

45 [Andrews 83, Giles 83]

Note that the ^ and T widths are several tens of keV, much narrow to be observed directly.

19

too

The measurement of B ^ is not easy. uith

and

the

background (Fig. 2.1.2). the

case of B ee .

the

4r

it

It is small to start

signal must be separated from the QED muon pair

i8

The QED background is much

uorse

in

It is uorth pointing out that in the casé of possible

Breit-Uigner

peak

in

interference

uith

the

to

the

measure muon

QED

the

pair

distortion

cross

background

of

section

[Boyarski

the

due to

751,

thus

proving that the ^ has the 1~ quantum numbers of the photon. The Zueig Rule rate for decay quarks

is

predictable

in

by

QCD.

annihilation

of

heavy

I have already mentioned the

vanRoyen-Uei8skopf fromula for the electromagnetic annihilation rate.

Including the next higher order in QCD [Barbieri 751 it

gives r

«> "

T

m " r rr " UGaV/tl 2 ) ||H0) I2 Í1 - 16«s/3jr>.

Electromagnetic annihilation can also states

via

V

•* t

produce

R.M

final

•* qq -» two jets of hadrons, uith a decay

width r q q given by T M multiplied by the (see above).

hadronic

nonresonant

R

value

Thus ue have -

r „

+

T

+

m

r

T T

+

rqq

-

(3 + R)

r...

In QCD the hadronic annihilations must go through a three-gluon intermediate

state:

V

-» ggg

•+ three

jets of hadrons.

A

one-gluon intermediate state is forbidden because one gluon has to

carry

color.

A vector meson cannot decay to tuo mass I ess

gluons for the same reason that it cannot decay to two (the

final

photons

j - 1 photon spin vectors uould have to combine at

an angle to give J - 1, uhich violates the condition

that

the

photon spins have to allign with their antipara!lei motion). The louest order QCD prediction [Appelquist 75al

for

the

three-gluon decay rate is r g w - [160(r2 - 9)/81 II2] «s13 |(MO) I2. Since this uidth is proportional to coupling,

the

cube

of ccg. Ue can remove the dependence on the the

the

strong

uave

function

of

heavy quark-ant i quark system by dividing by the prediction

for r„. been

of

it offers the possibility of an accurate measurement

Although the leading order QCD correction for T m has

knoun

for

some time, it is only recently that Mackenzie

20

arid Lepage [Mackenzie 811 have next

higher

order

succeeded

in

calculating

the

for r ee , which now includes the effects of

four gluons and of gqq.

The prediction for the ratio is

rggg/re C - 3 .

The tensor force, like depends

the

force

betueen

bar

magnets,

on the directions of the spins relative to the line of

separation.

The Breit-Fermi Hamiltonian gives

H T - -(L/12m2) S 1 2 V v "(r), uhere S 1 2 - 4(3 8] 2 n r - 280 MeV.

can

The quarkonium

states cc and bb have no isospin, so if isospin is conserved, the

dipion

state must have T - 0. 0

rate should be twice the irV rate.

This implies that the irV Since a T - 0 dipion state

has even charge conjugation, the states X; and Xf must have the sane C - (-l) u s , for example, % 3

P.

The

transition

- 3 S, ^

-

%

- 'P, 'S

-

•» ^jrjr has been measured both in the

cc system, Br W

•» ifanr) - SO ± 3 * [Abrams 751,

and in bb, Br IT C2S)

T (IS) jrrl - 29 ± 2 X [Giles 83]

Br [T (3S) -» T(lS))rir] - S ± 1 X [Berkelman 83] Br CT (3S) •* T(2S)tnrl - 3 ± 2 X CCUSB 831. Totally reconstructed events, in uhich the dilepton

decay are easily identified.

final

T

undergoes

More accurate branching

ratios are obtained by identifying the final T only as

a

peak

in the missing mass recoiling against jrV" (Fig. 2.7.1). In QCD a hadronic transition uith no change in C mu9t take place through the emission of tuo gluons; one gluon uould carry color and Mould available

is

violate only

a

C

conservation.

Since

751,

however,

expand

the

Gottfried and Van

gluon

radiation

classical mult¡poles, keeping only the louest order, «

1

as

in

energy

feu hundred MeV, this is a soft process

which is not calculable in perturbative QCD. [Gottfried

the

the electromagnetic case.

since

in kr

They cannot calculate

absolute rates, but are able to relate the 2 3 S •* l3Sirr rates in the ^ and T systems through the ratio of Have function spreads: T

firr/Trirr " ( a ).

Ignoring mass dependent phase

space

weights

factors,

the

relative

pairs produced by a U* are 1

for

each

of the various fermion of

the

doublets and 3 (color) for each quark doublet. c* each denote a superposition of d, s, and c the Kobayashi-llaskawa unitary mixing matrix:

SO

or

-» sU*), or a U can produce a

three

lepton

The d*, s*, and parametrized

by

(

-

/Vod

V^

V ^ W d V

( Ved

V*

V* I I s I

Hhere the matrixV elements (In the original K-11 notation) are V Vud-C], U8"81C3» ub"3ls3» S V c d —s 1 c 2 , Vcs-c1c2c3-32s3e' , Veb-CiCjSg+ejCae1®, Vu—s,s2,

Vts-c132c3+c233eiS,

and Sj - sin pj, c, - cos 0j. snail, since

but

they

unmixed

energetically

Vtb-c,3233-c2c3eiS,

The Mixing angles are apparently

are especially important for e and b decays,

decays

to

their

forbidden.

weak boson Z°, but if

the

A

c

and

fermion

mixing

t

partners

are

can couple to a neutral

matrix

is

unitary,

there

cannot be any flavor changing neutral current couplings. There is much to be learned from the ueak heavy quarks.

decays

Is the standard model correct?

Are flavor changing neutral current decays

forbidden?

Uhat

are

the

mixing

angles,

Why are there

the

Are there really

eix quarks?

related to the quark massee?

of

and

really

hou are they

three

generatione

of quarks and leptons?

3.4

MESON DECAY I1ECHANISI1S

For the charged current decay of a meson there possible diagrams:

are

three

annihilation, exchange, and epectator.

annihilation diagram

can

contribute

only

to

charged

The meson

dscay,

and the annihilating quarks must be from the same weak doublet, although

because

of

the

mixing, annihilation of quarks from

different doublets is only suppressed, not forbidden.

This

is

the only mechanism which can give purely leptonic final states, such as

K"

contribute

•* n'y^. to

The

annihilation

eemileptonic

decays, 51

diagram except

does through

not gluon

emission by one of the initial quarks.

The total

annihilation

decay rate (ignoring gluons) is given by Tq, - (GF2m(j2/3ir) |*(0)|2 IVq.,12, or may be written in teras of the decay constant fo,2 - 12 |^(0)|2/m0. Note that ^(0) depends Mainly should

be

140 MeV.

essentially The

V

on

reduced

mass

independent of niQ.

factor

is

the

nq)

and

Empirically, f r -

appropriate

Mixing

Matrix

element: S] (suppressed)

Vus ~ D*

V«. ~

-S] (suppressed)

F*

v« ~

1 (favored)

B*

V„b ~

8182 (doubly suppressed)

The exchange diagraM can contribute only to neutral

Meson

decay. A \

W

It can never produce leptons.

The decay rate (ignoring gluons)

is given by T - (GF2MQ2/«r) |^(0) I2 | W | 2 . uhere the V and V vertices.

For

are the Mixing Matrix eleMents for the

doninant

exchange

the

tuo

decays of the neutral

heavy mesons ue have the following: ed-»uu

VygVyj —

S] (suppressed)

D°;

cu-»ed

VegVuj ~

1 (favored)

B°:

bd-*cu

V^V^ "

8

3 (supressed).

In the spectator diagram only the heavy quark participates in the ueak interaction.

Q

\

This results in seMileptonic and hadronic decays, never leptonic.

Since

the

purely

tuo quarks of the original Meson do not 52

need to interact with each other, the rate ^-(0)

and

is

given

by

is

independent

of

the fol lowing expression (for Bay Q •*

fcq) : T - (GF2niQ5/192ir3l IVQJ2 where 0 is the phase space factor appropriate to of

the

emitted

U

the

products

1 for light quarks or leptons).

this rate is proportional

5

to

HQ ,

it

becomes

the

Since

dominant

mechanism of decay for sufficiently massive quarks. If the spectator diagram dominates, the mean lives of charged

and

neutral

mesons

only in the flavor (u or d) quark.

the

will be equal, since they differ

of

the

noninteracting

spectator

The K*, K$°, and K|_° lifetimes are very different, but

it is interesting to note that the simileptonic

rates

for

K*

and KL° (not normally affected by annihilation or exchange) are almost equal: Br/r Msec'1

r, nsec K"

12.4

3.90 ± 0.04

Kl°

e>.X

51.8

3.73 ± 0.05

Ks°

e>,X

0.0892

Apparently

e>,X

annihilation

and

- ?

exchange

-

dominate

charged

and

neutral decays, but the spectator contribution, although minor, is indeed the same for all decays. It is difficult to make reliable for

theoretical

predictions

D decays because of the undertainty in fQ (i.e., ^(0)) and

in me as uell as the expectation

unknown

effect

of

gluons.

The

early

Mas that the D decay would be spectator dominated.

The measurement of D lifetimes is difficult because of the very short flight paths. and hadron production media,

such

The best data come from high energy photoexperiments

in

fine-grained

as smulsions and high rssolution bubble chambers.

Reeults from different experiments IKalmus range

detection

82]

cover

a

wide

of values, but the consensus is that the charged D has a

longer lifetime than the neutral D, by ae much as a two: r, - 9.3 +2.7,-1.8 x lO'13 sec r0 - 4.0 +1.2,-0.9 x 10"13 sec.

53

factor

of

Moreover, the seniIeptonic branching ratios differ

by

perhaps

the sane factor: Br (D* -» e>.X) - 19 +4,-3 X Br(D° -» e>,X) < 6 X. Apparently, the 0° hadronic

decay

exchange

favored),

diagram

(Cabibbo

rate

is

enhanced

uhile

by

the

the annihilation

diagram (suppressed) contributes Much less to D* decays. In B decay the only available neasureiients of lifetine seRileptonic

branching

ratio

apply to the average of charged

and neutral B Mesons; no one has separated charged and decays.

Because about

B

of the •t, factor in the spectator rate, one If it does, then

whatever

Me

meson decay, rates, branching ratios, spectra,

etc., applies directly to b quark decay. assumption,

neutral

5

expects it finally to doninate. learn

or

although

Models

Ue

will

make

that

[Leveille 81] predict as nuch as

•uch as 3OX nonspectator contribution.

3.5

TESTING THE STANDARD MODEL UITH B DECAYS

Assuning that the b quark decays uithout interacting the

spectator

antiquark, one can Make explicit predictions of

branching ratios

using

the

coMpeting

of

quark

model

b

standard decay.

account, the b should decay into following, 1, 1, 3, 3. the

heavy

e>„

(15X). the

c

Model,

or

indeed

any

Juet taking color into

or

u,

plus

one

of

the

¡LV^, T'PV, d'u, s'c, in the ratio 1,

Putting in the appropriate phase space factors for leptone

and quarks, ue get 1, 1, 0.2 (0.4), 3, 0.6

(1.4) for the b -» c(u)U~ branching

uith

ratio

for

B

cases.

Ue

would

then

predict

Uith estimates (Leveille 811 for the gluon effects

nonspectator

a

-» w j i (same as for B -» jo^X) of 17X

contributions

the

prediction

becomes

and 12X

(10X). Both of the CESR experiments have Measured

the

yield

of

inclusivs electrons as a function of beaM energy U, recognizing electrons as showering charged

54

particles

(Bebek

81,

Spencer

811.

In both experiments the rate for electrons above 1 GeV/c

momentum increases at the T(4S) factor

resonance

by

a

much

than does the total hadronic cross section.

Taking the

acceptance and electron identification efficiency into the

two

groups

larger account

report the following branching ratios for B -»

ep,X [Stone 831s 13.2 ± 0.8 ± 1.4 X

CUSB

11.9 ± 0.7 ± 0.4 X

CLEO.

The CLEO [Chaduick

group

811,

has

made

a

similar

search

which

can

penetrate

depending on the direction. similarly

to

muons

identifying then with drift chambers outside a

60 cm thick iron shield surrounding the detector. momentum

for

that

for

The

minimum

the shield is 1.0 to 1.8 GeV/c

The beam energy dependence behaves

the electrons.

The derived branching

ratio for B -» jo^X is 10.2 ± 0.5 1 1.0 X

[Stone 83],

consistent uithin errors with the electron branching ratio. Only leptons produced in the first generation of the decay are

included

in these numbers; the leptons from decays of D's

coming from decays of B's are mostly of louer if

the

Also,

charged and neutral B have different branching ratios,

the quoted values apply produced

momenta.

at

the

to

T(4S).

the

average

over

charge

states

The average of the electron and muon

branching ratios is 11.6 ± 0.6 X, in excellent agreeement the

standard

also obtained

model

prediction.

semileptonic

B

with

Groups at PEP and PETRA have

branching

ratios,

using

high

momentum and high p? (uith respect to the jet axis) leptons and assuming the parton model bb production measurements

(Fig.

3.5.1)

are

cross

consistent

section. uith

the

All uorld

average of 11.6 ± 0.5 X [Stone 83]. The standard model uith a unitary mixing matrix implies no flavor

changing

[Glashou 701).

neutral

current

decays

(the

Suppose that the standard model

GIU mechanism is

urong

that the b quark can decay to an s or d by emitting a Z°. among the B decays ue should expect some uith a

55

pair

M-

and Then of

high

energy

leptons.

T(4S) -» BB. sources.

CLEO has seen 153 dllepton events from

Nultilepton events, however, can cone from

There

other

are two B's in every event, each one of which

can contribute a single lepton, and if a D is produced

in the

decay, it can decay leptonically, too. Of the 153 dileptons 66 are estimated to cone fron D's, fron QED tau pairs, backgrounds.

or other

The jie pairs cannot of course cone from a single

B. Of the m and ee pairs, about 20 are consistent with p.

being

Some of the pairs have a net momentum too high to come from

the decay of a single B. pairs,

and

subtract

If ue take the remaining

tm and ee

the ue pairs (appropriately weighted for

acceptance), the result

is consistent

uith

zero.

The 95%

confidence upper linit for the dilepton branching ratio is (B ->

B

/vX) < 3 X.

There are in the literature nany

non-standard

models of

the weak interaction, nost inspired by the lack of evidence for a t quark. 801

or

Suppose that the b is either a weak singlet EGeorgi

is

forbids b stable

in a weak doublet but uith a quantum number which ull" or cU" [Derman 791.

Then

either

b decays

would

contain

a

That

is,

lepton and all decays would

contain either a tau or antibaryon (baryon for b decay). has

measured

the

inclusive

rates

lambdas (A or A) from T(4S) •» BB branching ruled

is

(ruled out by the observation of semileptonic decays),

or the b decays to tfrX or qq/ or qq/ only. all

the b

for protons (p or p) and

(Fig. 3.5.2).

ratios of the order of 3% each.

out

by

measurements

the

measured

CLEO

electron

They

imply

A large tau rate is

and

nuon

rates

of the missing neutral energy [Chen 83al.

and

So the

data completely rule out this class of models. Suppose instead that the b is a weak singlet or one

of

a

doublet of charge -1/3 charge quarks, and can mix with s and/or d [Barger 79].

In this case

there

is no

way

to

suppress

completely flavor changing neutral current decays; at least 25% of b decays will be through the Z°. One can [Peekin 80] put lower

limit

on

a

the dilepton branching ratio of the b in this

mode I: 56

(b -> A/-X)/(b -> /j>X) > 1/8. The CLEO upper limit of 3X excludes this hypothesis. Suppose nou that Me put the b in uith

the c [Peskin 811.

flavor changing

neutral

a

right-handed

doublet

This Model has natural suppression of currents.

Its

predictions

can

be

tailored to be very close to those of the standard node I, so it Mill bs very hard to test for it.

Ue nay have to Mait for

the

discovery of the t quark to knoM for sure. Finally, it is possible uith charged Higgs bosons of

Mass

near 5 GeV to concoct a node I in uhich the b decays via a Higgs instead of a U boson.

This Mould shoM up as a

preference

high mass fernions, r > r or sc, in ths final state.

for

A large

rvT rate is ruled out by the electron and nuon measurements; a large sc rate is ruled out by measurements of the Mean charged energy per event.

Fig. 3.S.3 shous the experimental

as determined by CLEO [Chen 83al.

situation

The best fitting Higgs Model

is ruled out at 99.5% confidence. In short, there are no surprises. •odeI

The standard

six-quark

is adequate and most of the models Mithout a t quark are

untenable.

3.6

THE UEAK MIXING ANGLES, BEFORE B MESONS

The three Meak mixing angles

9 Z , and

define

the

three-dimensional rotation of the doMn quark states from the d, s, b basis defining the mass eigenstates, into the d', basis of the ueak interaction. {.

s*,

b*

In addition, there is one phase

As of nou, these are four fundamental parameters of nature,

not predicted by any accepted theory. The rate of hadronic decays involving (e.g.,

light

quarks

n -> pe> 0 ), relative to leptonic decay rates (e.g., iC

-» B'y^f^, is proportional to changi ng

only

V^2

-

cos20,.

decay rates are proportional to V ^ •

Strangeness cos^0g.

Together they determine D\ [Shrock 78], essentially the Cabibbo 57

angle, and imply an upper I ¡ait for 9e: B, - sin

- 0.23 ± 0.01

3 3 - sin ffg < 0.5. C h a m decay rates are proportional to I V d 2 82830"® | 2

2 s

2c

1

for

Z

s

l^edl * i 2 ~ final

states.

i

|cic2c3

-

the favored charm-to-strange decays, and

2 f

or the suppressed decays to

nonstrange

At the present levels of accuracy the measured

rates are not sensitive to the other angles DPDG 821: Br (D*

K"X) . B4 ± IB X

Br (D° -» K°X) - 77 ± 14 X. The decay nodes with reconstructabIe

low

enough

multiplicity

to

be

easily

are the folloHing, Measured at SPEAR [PDG 821.

There should be many others, not so easily recognized. D* •* K V

1.8 ± 0.5 *

K>V

4.6 ± 1.1 *

K°jrV°

13 ± 8 *

E V i r V 8.4 ± 3.5 X D°

KV

2.4 ± 0.4 X

KV

2.2 ± 1.1 X

KVtr"

9.3 ± 2.8 X

k W "

4.2 ± 0.8 x

K V t r V 4.5 ± 1.3 X Notice that the D (D* or D°) contains a c quark

while

the

K

(K" or K°) contains an s quark, eo that the weak coupling c -* sU*

implies

containing

that a

a

K.

D

normally

The

only

decays example

to

a

final

of

a

state

measured

Cabibbo-suppressed mode is D° -» jrV" 0.079 ± 0.038 X. The measurement of K°K° mixing and CP further

constraints

on

the K-H angles and phase.

order weak transition K° - K intermediate

violation

4

involves all the quarks

provide The second in

the

state, so that the mass difference caused by this

interaction, Am - m K S - m ^ - 0.35x10"® eV, depends on all K-M angles as well as all the quark masses.

58

the

In

fact,

the

heaviest

Unfortunately,

quark,

the

t

should

dominate.

Am also depends on the quark-antiquark binding,

uhich is unknoun but parametrized by the factor B, equal to 0.4 in

the

HIT

bag model and 1.0 in a "vacuum" calculation [Chau

83]. The CP violation in K° decay is defined by the angle -0.0227i

through

uhich

the

decaying

Kobayashi-Haskaua

model

the

phase

Am

and

e

ue

uould

have

tuo

If ue kneu mt

relations

because of the uncertainties in mt and constraints

In

0

or jr.

depend on the mixing angles, the phase, the t

quark mass, and the factor B. 0j),

0).

8 provides a natural

explanation for CP violation; CP is conserved if 8 Then

-

K s , K L eigenstates are

"rotated" from the Kj, K 2 CP eigenstates (taking e' ~ the

e

shoun

and

B

(ue

know

among sg, s3, and 8, but B

ue

have

only

the

in Fig. 3.6.1, that is, an alloued band of

values in s 2 versus s 3 , which depends on the value assumed

for

S.

3.7

THE UEAK MIXING ANGLES FROM B DECAY

To learn any more about the K-M angles ue have to look B

meson

mixing.

decays.

at

The b quark can decay only through the d,s,b

There are tuo

measurable

quantities,

the

ratio

of

rates (b-»uU~)/(b-»cU~), which measures the ratio IV^/V,*!2 of d and 8 mixing in the b, and the overall decay rate, or lifetime. A

number

of

different observations indicate that the b decay

through charm is dominant. The first evidence Mas

from

the

inclusive

kaon

rates,

measured by CLEO IBrody 82] from the T(4S)-»BB: 0.72 ± 0.03 K*/B, 0.73 ± 0.03 K° or RVB. This represents a significant increase on the number (2.9

event) belou the T(4S) resonance. as

of

kaons

per event), relative to the continuum production (2.1 per follows.

The naive

calculation

goes

If b -» uU~, you get tuo kaons when U" goes to sc

59

(which it does 1/3 or the tine by simple

counting),

otherwise

none; so you expect an average of 2/3 kaon per B decay. ell", you always get one kaon from

the

c

decay

If b -»

products

and

again an average of 2/3 fron the W"; so you expect 5/3 kaon per B decay. space,

The naive calculation has to be corrected gluonic

enhancements,

for kaons generated fron the vacuum in the later stages.

and

pure

b

b

does

not

0.9

with

the

Our best

kaons

-» cU" would yield about 1.6.

rate of 1.5 per B decay agrees well pure

phase

hadronization

The predictions become very model dependent.

guess is that pure b -» ul4~ would give about decay

for

nonspectator contributions, and

per

B

The measured

prediction

for

•* elf", but because of the model dependence, the result imply

a

very

restrictive

upper

limit

on

(b-HjU") / (b-*cW"). Another charm indicator is the (Fig. 3.7.1).

The

electrons

and

lepton muons

momentum above

mainly from the first generation b-»ctv or utv. of

spectrum

1 GeV/c come

The end point

the spsctrum depends on the effective mass of the recoiling

products of the c or u quark and the the

recoiling

spectator

quark.

Since

mass is likely to be smaller in the b-*tv case,

we expect the lepton spectrum to have a higher end point. measurement

of

The

the yield beyond the end point for B->D£ is a

direct indicator of the suppressed mode b-»uU~ and does not rely on

a

comparison of the b-»cU" rate and the total b decay rate.

Fig. 3.7.2 compares the CLEO electron and muon momentum spectra with

the

cases. no

predictions EAItarelli 821 for the b-»c£ and b-»u£

Fig. 3.7.3 shows the CUSB electron spectrum.

measurable

can contribute.

signal

There

is

above p - «„

0»

Fig. 1.4.1 Transversa moMentum-squared spectra with respect to jet axis [Brande I i K 81].

10 12 u

IGeV/cl2

Fig. 1.4.2 Scatter plots of events in sphericity and aplanarity [Brande 1 ik 79b)

01

06

S P H E R I C IT*

PLUTO

e + e~——q q g

Fig. 1.4.3 A PLUTO threejet event Ololf 80].

79

9.410

9.150

9.490

Fig. 2.1.2 Hadronic cross section at the T [Plunkett 82].

(0) s »

' 1050 1.090 3.100 3.H0 3.120 3.130 ENERGY Et„,.(GiV)

100

Fig. 2.1.1 Measured cross sections at the ip [Aubert 74]

•

*

|cos0|

I 900

F i g . 2.10.3 Corrected

single

photon spectruB fro..

CIS)

[CIJEO 831.

86

' "00

l

I 1300

I I 1500

F i g . 2 . 1 0 . 4 I n c l u s i v e phocon spectrum f r o a ty [Scharre 811.

F i g - 2 . 1 1 . 1 RoertTccnnto of csgC^/J (cco to:ît).

87

3

3.68

3.72

3.76

3.80

Fig. 3.5.1 Summary of measurements of Br(B -* Xer) [Stone 831.

Fig. 3.5.3 Charged energy fraction versus electron rate i n B decays [Chen 83a].

Fig. 3.5.2 Momentum spectra of protons and lambdas from B decays (A I am 83al. 90

0.4

0.4 S* 0.3

0.3

0.2

0.2

0.1

o.i

0.1

0.2

Ss

0.3

m&mmmmmm

0

0.4

0.1

0.2

0.3

0.4

0.4 S2 0.3

0.2 0.1

0

Fig.

3 . 6 . 1

[Stone

Excluded

0.1

0.2

regions

0.3

Sj

of

s

2

0.4

vs.

2

3

ELECTRON ENERGY (G8V) Fig.

3 . 7 . 1

decays

a

3

for

various

8

831.

e l e c t r o n

[KI o p f e n s t e i n

momentum

spectrum

83b].

91

4

from

cseai I e p t o n i c

B

0.27$ 0.325 0.37$ 0.42$ 0.47$ X- t ' C a c u

Fig. 3.8.1 D° momentum spectrum from B deciys [Green 831.

Fig. 3.8.2 Inclusive charged particle spectrum from TC4S) LCLEO 831.

Fig. 3.8.3 Mass spectrum reconstructed B decays IBehrends 831.

5200

5240

MASS (MeV)

5280

Fig. 3.8.4 Maes spectra of electron and rauon pairs from B decays [Gittelman

82] 2.50 3.00 3.50 4.00 4.50

2.50 3.00 3.50 4.00 4.50

MASS (GeV)

92

GRAND UNIFICATION AND SUPERSÏMMETRY

J. Ellis SLAC, Stanford, California, USA and CERN, Geneva, Switzerland

93

1.0 CONVENTIONAL GUTS 1.1

WHY GRAND UNIFY?

The Standard SU(3)C x SU(2)L x U(l)y Model is clearly satisfactory

in many

respects.

un-

Even if one accepts as given

the inelegant choice of gauge group with 3 independent factors, it contains a distinctly "unmotivated" set of fermion representations. placing

If one works in terms of right-handed

fermions

left-handed

fermions,

re-

f R by their left-handed conju-

gates f£, the content of the first generation (u,d,e,ve) is

(1.1) (3,2) + (3,1) + (3,1) + (1,2)

+(1,1)

where we have exhibited their SU(3) x SU(2) representation contents.

The second (c.s.u.v^) and third (t,b,t,vT) generations

transform similarly to (1.1), whose representations systematic

trend except that of being small numbers!

reveal

no

The U(l)

hypercharge assignments of the fundamental fermions (1.1)' pose another

problem.

They all take rational values which ("happen

to") yield a vectorial electromagnetic current:

Q e m = Ij +

Y,

Q e = -1, Q d = -1/3, Q u = + 2/3 implying the remarkable property of charge quantization:

|Qe|/|Qp| = 1 + 0(10-2")

(1.2)

The individual hypercharges Y must have been adjusted to within

95

The individual hypercharges Y must have been adjusted to within the

indicated

upper limits on deviations from equality.

Many

more parameters appear when one examines Higgs interactions

in

the Standard Model, which contains altogether at least 20 arbitrary parameters as seen below Table Is

Parameters of the Standard Model

3 gauge couplings

: g3, g2,

2 non-perturbative vacuum

: 0 3> 6 2

gi

angles >6 quark masses

: "»u^s^b.t

¿3 generalized Cabibbo angles: CP violating phase

: 5

¿3 charged lepton masses

: B, „ .

¿2 boson masses

: m^± H o

1.2 THE PHILOSOPHY OF GRAND UNIFICATION

We will seek2 a semi-simple unifying non-Abelian (or

product

of

group

G

n>l identical group factors G n ) which is sup-

posed to undergo successive stages of gauge symmetry breaking:

G + ...

SU(3) C x SU(2) l x U(1) Y > SU(3) C x U ( l ) e m

Such a theory has a single gauge coupling g. this

How to

(1.3)

reconcile

with the gross inequalities between the SU(3), SU(2), and

U(l) couplings presently observed?

83 »

g2. 8t

96

(1.4)

The answer 3 i s provided

by

the

renormalization

group

which

us t h a t gauge couplings vary with the e f f e c t i v e e n e r g i e s

tells

(momenta) Q a t which they a r e evaluated ( s e e

fig.

1).

Best

known i s the asymptotic freedom1* of the conventional s t r o n g i n teractions: g^(Q) _ 12ir a,(Q) = — r—5 J 4ir (33-2N^)lnQ / A^

(1.5)

where N^ i s the number of quarks w i t h masses mq«Q and Aj i s strong

interaction

scale

parameter:

Aj

a

= 0 ( 0 . 1 t o 1 GeV).

There a r e analogous l o g a r i t h m i c v a r i a t i o n s i n

the

other

cou-

plings (1.4):

e 3 = (33-2N q )/12ir = (33-4N g )/12ir ^ y

2

= P t ln(Q /A*):

1

"

(1.6)

e 2 = (22-N D )/12it = (22-4N g )/12ir

where Ng i s the number of g e n e r a t i o n s and NQ i s the

number

of

weak d o u b l e t s , and we have omitted Higgs boson c o n t r i b u t i o n s t o f o r reasons of s i m p l i c i t y .

We see from e q u a t i o n ( 1 . 6 )

that

the SU(3) and SU(2) couplings approach each o t h e r :

^TQ) - S^Q) =

(p

3 " e2>

ln

- 75,

ln

a s might have been expected on the b a s i s of asymptotic alone

(1.5).

Notice

(1

-7)

freedom

t h a t i n equation ( 1 . 7 ) we have subsumed

t h e two s c a l e s A 2 > 3 i n t o a s c a l e M^ a t which ^u =

+2/3

which implies Qp = 2 ^ + Q d = +1. The embedding 101

of

(1

'18)

SU(2)L

x

U(l)y

in

a GUT group also enables one to calculate3 s i n ^ .

Since Q Is a generator of SU(5), we know em

Q

em = X 3 +

Y = l

3

+ cI

°

(1.19)

where I Q Is an isosinglet generator of SU(5) normalized in the o o same way

as the isotriplet Ij:

ZI| = EI^. We see from equa-

tion (1.19) that Y = cIQ > g' = (1/c) g i

where g' is the conventional hypercharge normalized

coupling

(1.20)

and

g^

is

to equal g 2 and g 3 in the SU(5) synimetry limit. We

can estimate c in (1.19) by evaluating

I Q 2 = Z I 2 + c 2 L I 2 = (1+c2) I I 2 rep xem rep 3 rep o rep 3

in the reducible SU(5) representation

containing a

(1.21)

complete

fermion generation (1.12):

rip «"s

=

V

"b

=

m

x

( 1

Once again, these p r e d i c t i o n s only apply in the SU(5) limit

and

are

subject

to

renormalization

'35)

symmetry

corrections.®

These can be estimated i n one-loop order by using the formalism of

a

momentum-dependent fermion mass

12

.

The inverse fermion

propagator

S - J ( Q ) = $-m f (Q)

with mj(Q) yt

subject One

renormalization

by

gluons,

W±,

Z°,

can picture these as forming a "cloud" round the

fermion as in f i g . parent

to

(1-36)

"weight"

2 , and v i s u a l i z i n g a v a r i a t i o n in of

this

"cloud"

ap-

a s one v a r i e s the d i s t a n c e

s c a l e x = 0(1/Q) around the fermion a t which 105

the

one

is

probing.

In leading order for SU(3) «normalization one finds**'**

m^Q)

a3(Q)

11-4N 3 8

A full two-loop calculation13 which the

renormalization

(1.37)

= 0(3) at Q = 10 GeV

_a3(mx)

of

treats

gauge

the Higgs-fermion

invariantly

Yukawa

couplings

yields for the physical masses

m b /m T = (2.8 to 2.9)

Another success!

(1.38)

Unfortunately, at least one

of

the

predic-

tions for the light quark masses is wrong, since the renormalization group preserves

m

d /m s

= me'-p e /m

(1.39)

Chiral symmetry suggests that m^/nig = 0(1/20), whereas Kg/m^ 0(1/200).

=

Maybe it is possible to cure this problem by adding

small contributions to the fermion masses which do not

destroy

the successful prediction (1.38).

1.6 BARYON DECAY

The basic mechanism in SU(5) is X and Y change

which

order 1/m^.

gives

an

gauge

boson

ex-

effective four-fermion interaction of

Using the gauge

interactions 106

(1.13) one

finds®

that for first generation particles alone:

G

[(e

GU

ijk "kL

^

d

iL

+

^

d

iR> (1.40)

"

(e

d

ijk ^ L

jt ) r V eL

+

V

( h

->]

where

Ggu/^2= g2/ 8m 2 = g2/8m2

(1.41)

to be compared with the Fermi four-fermion coupling

Gp//2 = g2/8m2

(1.42)

We get from (1.40, 1.41) a decay amplitude A a 1/mj^ and hence a decay rate T a |A|2 a 1/m^, or equivalently a lifetime

Tfi = C(m£/mjp

The denominator of m§ is just based the

physics

is

on

(1-43)

dimensional

analysis:

in the unknown coefficient C that we must now

calculate. Short-distance gluon exchanges at momenta between y and m^ renormalize the operator (1.40) by a factor6

a 3 (y) a

3 ( m X)

107

2 11 - 4N 3 8 it, GU

(1-44)

in one-loop order. calculate

The strong interaction problem

, presumably using some stan-

dard hadronic model and 11= 0(1) GeV. often

is

People in the past

have

bag models and/or non-relativistic SU(6), but even

so the strong interaction matrix elements are difficult to calculate

reliably,

in

particular

because there is no reliable

connection to the short distance calculation (1.44).

Previous

estimates have led1** to

Tb = (0.25 to 10) x 10 30 years x (mx/4 x 1 0 ^ GeV)1*

(1.45)

Including the variation (1.32) in m^ one finally gets

T b = 10 2 9 ± 2 years

While people do not agree on the

lifetime,

(1.46)

they

do

tend

to

agree on the dominant baryon decay modes:

p •»• e+ir° (B. Katio -30%?), n + e+ir" (B. Ratio ~60%?)

The estimates (1.46, 1.47) look disastrous when

compared

(1.47)

with

the 1MB experimental limit15

T (p + e+ir°) > 1 x 10 32 years

(.1.48)

We 1 6 have recently re-evaluated the baryon lifetime in two steps:

1) we use current algebra and PCAC to relate17 the bar-

108

yon decay matrix elements to 3-quark annihilation

matrix

ele-

ments :

Ki, pseudoscalar meson P l^-jj) y |B>

(1.49) 1/fp < 7

+ (baryon pole term)

where

(Qj is the axial charge corresponding to the pseudoscalar meson F

with decay constant fp) and the baryon pole term is directly

proportional to ) is a cubic polynomical

F() - a ± j *!«)>;, +

called the superpotential. from

b ljk4'i4>j4>k

Fermion interactions

(2.18)

are

obtained

F() (2.18) by removing two 's and putting in their spin

1/2 i|> components, while taking the scalar components s of

any

remaining

(•i

»2P/»*JLa*i -

+ 118

b

ljk

8k

C2,19)

The first term on the right-hand side of (2.19) mass

term,

teraction.

is

a

fermion

while the second term is a conventional Yukawa inThe multiscalar interactions obtained from F() are

'

1

= |a b S |2 -s i3V i* A *i ' i

(2 20)

'

s

We easily derive from (2.20) a (mass)2 matrix:

(m 2)

s ik " a ij a jk

= (m

u4 _ m mrr sin . 2. 9„ G_ g„2 m 2/2 Gs « Ag - 2 _ £ J p £ -£[F(m~,m~,mg)+F(m~,m~,mg)] (2«37) 3

In equation (2.37), Ag i s a short-distance enhancement factor 1 * 1

As~0.41

while the mc and mg factors come from Higgs

(2.38)

Yukawa

couplings.

We may naively estimate that F = 0( 1/tilling) in which case equa124

tion (2.37) can be used to estimate

Gg = 0(10~12)/mH-i H m, W

(2.39)

which is of the same order as the G of Lecture 1. since mu = GU "3 0(10)mx

and

my = 0(10~ 13 )m x for mx ~ 10 15 GeV.

might expect the baryon l i f e t i m e in minimal comparable

SUSY

Therefore, we GUTS t o

be

with that estimated in Lecture 1 ( 1 . 4 6 ) , though one

might f e e l a tinge of regret that a SUSY baryon dies in such an ignoble way ( f i g .

4).

What are the SUSY baryon decay modes? SUSY

The .combination

of

and colour applied to the dimension 5 operator (2.35) r e -

quire*1 1>1» 5 the presence of second generation particles such SjU.v^.

These

as

are also favoured by quark mass factors in the

Yukawa couplings of the Hc .

In f a c t ,

the

Hc

with

Qem =1/3

couples to sv^ and cy (sy having the wrong charge) and cy i s of course kinematically forbidden. destroyed

This simple conclusion i s

not

by more careful calculations' 11 including the e f f e c t s

of Cabibbo mixing, and the expected hierarchy of decays in minimal SUSY GUTS is 1 » 1

N -»• vK »

vir »

y+K »

y+ir »

e+K »

e+ir

(2.40)

The best experimental limit relevant to this prediction i s 1 5

x(n + "fc 0 ) > 8 x 1030 years

which suggests, i f we use

the

Lecture

125

1 estimate 1 6

(2.41)

of

the

matrix element, t h a t

m„ > 7 x 1 0 1 7 GeV 3

(2.42)

i f we make a simple-minded estimate of Gg from equation ( 2 . 3 7 ) . The

estimate

(2.42)

i s considerably l a r g e r than our previous

estimate 0(10 1 6 )GeV ( 2 . 3 2 ) for n^ i n a SUSY GUT. situation

is

not

However,

n e c e s s a r i l y c a t a s t r o p h i c , s i n c e the nucleon

l i f e t i m e in a SUSY GUT contains more u n c e r t a i n t i e s conventional

GUT.

gauge boson mass? Perhaps

the

SUSY

than

Have we c o r r e c t l y estimated the m^

and

a

factors?

spectrum of SUSY p a r t i c l e s i s such t h a t the funcl/m^m^r?

I t i s perhaps premature t o abandon

GUTS, though i t i s worth noting t h a t non-minimal

SUSY GUTS can be made t o accomodate r a d i c a l l y d i f f e r e n t lifetimes

in

Perhaps the Higgs mass i s l a r g e r than the X

tion F in ( 2 . 3 7 ) « minimal

the

decay modes.

baryon

For example, by imposing a g l o b a l

symmetry 1 ' 2 t o suppress the dimension5 operator ( 2 . 3 5 ) and using SUSY

to

stabilize

a

l i g h t Hc mass, one can arrange 1 , 6 baryon

decay modes

N + v + K, vK

a t comparable r a t e s .

More e x o t i c baryon

(2.43)

decay

modes

(p~K,** 7

even e + i r ° , ' t 8 ) are a l s o obtainable with ingenuity. 2 . 7 EXCERCISES IN MODEL-BUILDING SUSY enables one t o " s e t and f o r g e t , " because l i g h t masses are s t a b l e a g a i n s t r a d i a t i v e c o r r e c t i o n s .

Higgs

S p l i t t i n g the

_5, 5 of Higgses i n t o heavy t r i p l e t s Hc and l i g h t doublets HD i s 126

technically

sound, and one would like to find more attractive

ways to arrange

c

»

h

D

.

Missing doublet models'*9 introduce more Higgs which

contain additional Higgs

with the triplets in the _5 + extra

doublets Hp

smultiplets

triplets H c that can combine

to acquire large masses, but

no

to mix with the Higgses that one wishes to

keep light. The lowest dimensional suitable representations of SU(5) are

50 + 5£ (called here 6 + 6 ) .

One can couple1*9 them

to 5^ + 5^ by using a 75 of Higgses £ to replace the conventional adjoint

24. In fact, if one wishes to be able to use a global

symmetry to forbid a direct 5^ - 5^ HH coupling one needs another 75^ of Higgses Z'. The corresponding superpotential is then1*9

P = X I6H + X I'6H + (QQCH terms, terms to break SU(5) + SU(3) x SU(2) x U(l))

(2.44)

The mass matrix for the triplets of Higgses then has the structure

When we make the transformations ( 3 . 3 ) the neutral gauge

boson

interactions remain flavour-diagonal:

However, the left-handed charged

currents

acquire

CKM angles because the unitary rotations

non-trival

are in general

different:

W

Consider now what could happen

to

the

1

*

^

corresponding

neutral

gaugino interactions:

* L . R < V R *°L.R>

=

S.R^R

D

2,R)(R

- E

The interaction

2

(3.6)

- ^ :

I . R " "2.R 0 and hence that glo-

bal SUSY is spontaneously broken. incorporated

in

GUTs

The basic idea (3.17) can be

in many different ways:

class of models 65 is illustrated in fig.

6.

one particular

There is

primor-

dial SUSY breaking at a scale m g in a sector of the form (3.17) containing just gauge singlet chiral then

coupled

to

superfields.

in

are

gauge singlet superfields which acquire SUSY

breaking from (3.17) through one-loop non-singlet

These

turn

feed

smultiplet in two-loop order.

diagrams.

These

gauge

SUSY breaking through to the gauge The

gauge

smultiplet

feeds SUSY breaking to the known matter superfields. ture could be complicated by additional

loops

in

in

turn

This picthe

chain:

the end result is 2 rSnijj or 5m~ q

2 x g or 6m~ = — (16iO

P

x

m

S

where the powers of Yukawa couplings X, gauge couplings (l/16ir2) are model-dependent.

(3.19)

g

and

It is clearly possible in such a

scenario to have m g » m y , perhaps as large as m^ or m p 65

Note

that the feed-through of nig to the squarks and sleptons depends

135

only on their gauge representations and hence (3.19))

are

6m2

the

q't

(see

flavour-independent, which avoids (3.10) any fla-

vour-changing neutral interaction catastrophe. An alternative scenario for exploiting the F model is

expressed

in

the

class

geometric66 hierarchy models. =0,

it

of

(3.17)

inverted52

so-called

or

Notice that while V (3.18) fixes

leaves

unconstrained.

The potential is flat in the X direction

at

the

tree

level.

The idea 52 is that radiative corrections may determine /ms = exp(0(l)/g2)

by

causing

a

deviation

from

flatness

(3.20)

as

in

fig.

7.

Renormalization group analyses67 confirm that such a hierarchically related minimum (3.20) can be obtained by a suitable justment

of

parameters.

ad-

However, if mg=0(m^) one finds many

unwanted particles with masses m = 0(m|/mx) To make these unobservably

heavy

(3.21) (>0(mw))

we

therefore

need 66

m g > 0( /mjjm^)

(3.22)

Unfortunately, the low energy spectrum in initial these

models

was

sufficiently

richer

standard model which led to (2.32) that m^

variants

of

than that of the SUSY was

calculated

to

greater than nip, in which case gravity should not have been ignored.

Indeed, there were so many 136

light

particles

that

the

gauge

couplings became 0(1) before reaching m^, so the pertur-

bative

renormalization

reliable68.

group

Furthermore,

equations

if

were

no

longer

there are more than one genera-

tion, some sleptons acquire m£n/mp~*) terms, coming for example from quartic, etc.

terms in the superpotential: 138

3 - 0 ( 0 +

F (+)

o^y /lX 1 *

„ / 1 \ An A + ••• + 0

It has been suggested that such play

an

Non-trivial

non-renormalizable

terms may

role in generating fermion masses 70 » 75 » 76 ,

essential

decay70»1*3,

baryon

(3.25)

* +

the

gauge

contributions

to

the

hierarchy50,

etc.,

etc.

novel chiral function g($)

have a similar form:

4>n + •••

g()|2 term

in

it is about to come in useful.

You remember that in the Higgs mechanism a massless spin 0 Goldstone

boson is eaten by a massless spin 1 gauge boson, be-

coming the helicity state it needs to become massive. cally

SUSY

theories,

In

lo-

the corresponding super-Higgs mechanism

involves a massless spin 1/2 Goldstino fermion being eaten by a massless

spin

3/2

gravitino, becoming the two extra helicity

states it needs to become massive.

The

order

parameter

for

local SUSY breaking is :

m

3/2 =

/m£ = «|/«p

The (-) sign in (3.24) enables us to have and

139

(3-27)

hence

m

3/2* 0 ,

while also

having < o | 3F/34>| 0> * 0 (the order parameter

for global supersymmetry breaking),

=

constant.

0,

corresponding

It

is

and

nevertheless

keeping

to the absence of a cosmological

apparent

from

(3.24)

that

if

=m|mgy2,t0, there are other SUSY breaking terms in the scalar potential. (3.27)

If we

take

the

mp-*00 keeping

limit

m

2,/2

fixed then the F() terms among the positive terms in V

give 7 7 » 7 8

6V = 0(m2 /2 ) |3/2^ decreases.

Notice that m t tends to increase

There is a boundary region in fig.

where gauge symmetry breaking arises from a SUSY analogue 88 the

as

rumoured 30 to 40 GeV are compatible with this radiatively

broken supergravity scenario. as

in-

Coleman-Weinberg92

mechanism,

9 of

light sleptons weighing as

little as 20 GeV are possible, and the lightest

neutral

Higgs

boson weighs less than 20 Gev. To see how this scenario works, let us examine more closely the low energy Higgs potential93:

V=

(g2 + g'2)(|H|2-|H|2)2 + mf |H|2 + m||H|2 - 2 m 2 HH

145

(4.2)

the first term in (4.2) is a D term (2.22), the next two emanate last

from

term

the

is

model-dependent.

super-Higgs

the Higgs

terms

mechanism (3.28, 3.30) and the

mixing

term

whose

magnitude

is

There is breaking of SU(2)T x U(l)v * U(l) l> i em

i f 93

> 2m

n»i +

W

1

m

case

is

positive

when

«», while the condition (4.3b) ensures that the orgin

|h|=|h|= 0 is unstable. ing

(4.3b)

2

Condition (4.3a) ensures that the potential |H|,|H|+

(4.3a)

where

absent, namely m

Let us consider the simplified

limit-

the parameter not required by supergravity is +0.

In this case the conditions

(4.3)

re-

duce to

+ m| > 0

m2m|
0(60 GeV) (less if m * 0)

(4.8)

and

m

H ± ~ "W + ' ""H0' ~ m Z °

(4.9)

and there is a light neutral Higgs boson with

m„o < 20 GeV H

(4.10)

Ultimately, one would like to have a no-scale standard model in which

the

gravitino

mass scale was determined dynamically as

well as my, but that would take us91* beyond the range of

these

lectures. 4.2 SPARTICLE MASS MATRICES

We recall that quarks and leptons are four-component Dirac fermions:

q ; L, R

I

L,R

- but only Vj. ^

terms of left-handed fields only, we their

conjugate

If we wish to work in

can

antiparticle fields q£,

replace

q^,

t^ by

All of the quark

and lepton helicity states have the corresponding spartners.

K++K*

(4.11)

If we prefer, we can work in terms of the antiparticles and

which are just the spartners q^,

of

of q R and Jt^. As

discussed in section 3.1, the squarks and sleptons q, £ can mix

148

in

helicity

(L,R)

space,

a s well as in flavour space.

As a

f i r s t example, l e t us look a t a SUSY world with the conventiona l Higgs-quark Yukawa superpotential term

"Hqq

which y i e l d s the F-terms

v

» g 2H^q „ - [|q L H|2 + |q£ H|2 + . . . ]

(4.13)

When we give the Higgs a vacuum expectation value = m^

v:

= gjj-q v , we get from ( 4 . 1 3 ) the following c o n t r i b u t i o n s t o

the squark mass m a t r i x :

(

V

V

/

\ (4.14)

Notice that as we would expect,

q

-

m

a4

i n t h i s imaginary SUSY world, a d i s a s t e r which we must by

introducing

SUSY breaking.

149

(4.15)

rectify

This conclusion i s not a l t e r e d

by including the D-term contributions to (4.14), which

in

any

case vanish if = . In the presence of spontaneous SUSY breaking of

the

type

induced by the super-Higgs effect, the squark mass matrix may 9 5 be parametrized in the form

(4.16)

where you will recognize A=0(1) (3.28), and L 2 * R 2 in general, thanks

to

the

action

of different radiative corrections re-

flecting the different SU(2)^ x U(1)Y quantum numbers of q L and qR.

The matrix (4.16) has the generic form (3.10), and the ef-

fects of D-terms if * can be subsumed into lumped

parameters L and R, while off-diagonal terms introduced

by a non-zero value of m can be lumped into the cal

A

parameter

of (4.16).

phenomenologi-

Clearly the matrix'(4.16) can be

diagonalized in flavour space by the same Uf1

ly R

ized

the

which

diagonal-

the quark mass matrix (3.3), ensuring the absence of fla-

vour-changing neutral interactions (3.6). onalized in helicity space by a rotation 9

tan 26, LR

It can then be diag:

-2Am q /(L 2 -R 2 )S

150

(4.17)

which is small for light quark masses as long as L 2 * may

be

large for the t squarks.

R2,

but

After diagonalization (4.17)

the matrix (4.16) has diagonal eigenvalues:

(4.18)

Some potentially amusing consequences95 of (4.18) in

the

case

of the possible large mixing (4.17) of t squarks will be mentioned later. Supersymmetric fermions are in spin

general

mixtures

1/2 partners of the gauge and Higgs bosons.

of

the

As discussed

previously, we need 2 Higgs smultiplets with equal and opposite hypercharges:

(4.19)

in order to cancel triangle anomalies and give

masses

charge +2/3 (H) and charge -1/3, -1 fermions (H).

to

the

The possible

SUSY breaking gaugino masses are:

M 3 (g a g a ) +

for

SU(3)C,

model

is

SU(2) l and U(l)y

eventually

M2(WAWA) +

MJ

respectively.

(BB)

If

(4.20)

the

standard

embedded in a grand unifying non-Abelian

151

group, the mass parameters are 96 in the

ratios

of

the

gauge

couplings to leading order:

^

: M 2 : M x = aj: (»2 : | a'

The gaugino/shiggs mass matrix can also

(4.21)

receive

contributions

from a possible H - H mixing term:

e (H H)

(4.22)

where we might expect

M2,e=0(mw)

(4.23)

Using (4.19) to (4.23) we can now construct the general charged fermion mass

m a t r i x

9 7

»

9 0

.

g2v

where = v, v. come

from

g2v

W

-e

H-

The

substituting

off-diagonal

(4.24)

entries

in

(4.24)

these Higgs vacuum expectation values

into the HHW couplings, related by supersymmetry to the familar HHW

couplings.

Interesting results can be extracted from the

matrix (4.24) in the limit

152

Mj.e+O

:

mass eigenstates

: masses (4.25)

(H-, W + ) L

; (ÌT, H + ) : g ^ , g 2 v

where the H and W mix completely:

swiggses or wiggsinos?

Also

of Interest are the limits

mass eigenstates M 2 + », e+0

: w

, W1") ; (H~, H1")

(W

masses g

2'

2 vv 2Z MM" 2

(4.26)

g22 v v-

M 2 + 0, e+®

e

in which the gauginos and the shiggses separate. The neutral fermion mass matrix has a structure to

(4.24),

but

is more complicated97»98»99 because there are

four neutral fermions to mix: shiggses

(H°, H°,).

2

(W3,

gauginos

B)

and

two

Thus each entry in (4.24) becomes a (2 x

2) submatrix: 0

'ill

A

82v\

•2 5

o^

- - M (W3, B, H°, H° )T

analogous

0

3

a2

Zill

Ìli

•2

/2

fz!

ziii

•2

153

ill 'ill /2 0

/2 e

H°

;

e

7/

W

(4.27)

It is of interest to pick out the lightest mass eigenstates

in

certain limits:

M 2 - 0:

s i n 2 ^ M2

Y = s i n ^ W -cos^B, m~ =

(4.28)

where s i n 2 ^ = g'2/(g'2+g2) as usual; and 2 vH + vH \ ): v

2 \n

=

"H

,. '29)

6

(4

where v^ = v 2 + v 2

Typical mass contours for the lightest charged mass eigenx* and for the two lightest neutral mass eigenstates x°>

state

X°' are shown98 in fig.

11.

We see that in much of the param-

eter space in + < m,T± and m o, m o' < m,o. Xin fig.

™

X

The horizontal lines

A

X

11 correspond to approximate H states, while the vert-

ical lines correspond to approximate y states. There are important cosmological constraints32»33 possible

mass parameters (e, M ).

Lecture 3 always involve pairs of that

they

can

stable. this

Thus

Since the SUSY couplings of sparticles,

it

is

every

sparticle

must

contain

evident

another

the lightest SUSY particle (LSP) is probably

(This conclusion can be avoided if

seems

the

only be pair-produced in association, and that

the decay products of sparticle.

on

*

0,

but

to be a bizarre possibility.) Sparticles were pre-

sumably present in abundance in the

154

early

Universe,

and

the

LSP's should be present today as supersymmetric r e l i c s from the Big Bang.

Their properties

are

constrained 32

by

the

upper

l i m i t on the present mass density of the Universe:

p < 2 x 10~29 gm/cc

(4.30)

This may perhaps 33 be strengthened by one order of magnitude to p


20 GeV.

The boundaries of the domain depend somewhat

the masses assumed for the other sparticles q, J:

ing fig.

12 we have assumed the minimum

values

of

in drawabout

20

GeV. 4.3 SLEPTON AND SQUARK SEARCHES Hie most obvious place to look l+l~i

for u and x

156

for

sleptons

is

e + e~

+

, + q(e e

_ 1 «3 + - " 4 3 y y )

+

o(e e

per flavour and helicity, which exhibits a

(4.33)

nasty

3

P-wave

sup-

+

pression factor (5 . The cross-section for e e~ -»• e'ë" could be augmented 1 0 by crossed

channel

y

exchange.

As

seen

from

(3.10), we can expect79 almost thresholds for the three generations

of

sleptons,

degenerate88.

The

though decays

the

and

Jt^ may

not

be

to look for are I* •*• I + 7 » which

give a final state event signature of missing energy E c m > and acoplanar final state leptons.

of

about

Another possible re-

action is 1 0 5 e + e~ • e* e T y:

o(ln x=m^/E^m * 0, nu = 0) x —fZ s îf; m a(e e

- y

*

v

„ )

»/«e

1Z1Î

t2/x +

18

e

(A>3A)

- 54x + 34x2 + 3(3 - 3n - 4x2)lnx - 9xln2x]

which is sensitive to m^ + m^ £ E , whereas the previous reacg y cm tion

(4.33) was

sensitive

cross-section (4.34) comes beam-pipe.

to me

when

the

£ e±

E . cm

Most

of

the

goes

close

to

the

The decaying e then provides a single large angle e

157

as the event signature, with missing

energy-momentum

carried

off by two photinos. Turning now to ep collisions, the cross-section for ciated

production

seen in fig.

of

13a.

an

e and a q can be quite large, 106 as

The corresponding charge exchange

+ vq' may also be important,106 as seen in fig.

ep

asso-

reaction 13b.

HERA

with its centre-of-mass energy of 314 GeV should give access to m + m

< 200 GeV. 5

I

The sneutrino can also be produced in other ways, for ample 1 0 7 » 1 0 8

via

pp + (W 4 -»• e 1 v) + X.

have a branching ratio similar

Z°

+

. 2 2 (4.35)

4 "Hi

which is plotted in fig. for

The decay W • e v may

to that for W + ev:

/ 2 +. "g 2

r(W •*• e v) = 1 T(W + e v) 2

rate

107

14.

ex-

There can also be a

competitive

vv decays. 107 > 109 The total cross-section for

e + e~ •»• vv via t-channel W exchange as well as s-channel Z°

ex-

change is 1 0 7 :

dq dcos9

2 ira s 32 sin

4m~ \3/2 1 -I

(H

sin26

Zsln 2 ^ 1 - sin QJJ /\m§ - t/\(ml - s)£+ ' ( 4 s i n \ - 1) 2 + r v

8(l-sin\)2

;

158

2 V (m z ~

s2

s)

+

. _2 2 r

z m z,

(4.36) T ^

The corresponding ratio to e+e~ +y* • 15.

The

is

shown

in fig.

v has several different decay modes available to it

which may be competitive107:

visible decays

v + Audg, vuug and perhaps eud

(4.37a)

and invisible decay modes

v + vf

(4.37b)

The ratio of these decay modes (4.37) depends on masses of the

the

relative

e and q. The production and subsequent decays

of vv pairs can give (visible (4.37a))-(visible (4.37a)) = C C

combinations, and (invisible (4.37b))-(visible 4.37a)) = N -

C combinations, whose shown107

in fig.

ratio

as a

function

of m^ e

(tO u

is

16. Possible signatures from W + ev are e +

missing Ej, e + hadrons + missing E^. and e+e~ + hadrons + missing E t , while Z° •*• vv can give in addition to these one-sided "zen" events where one v decays invisibly - the N-C events of fig. 16. The cross-section for e+e~ •*• qq pair production is

°;v_" ^ i o(e

per flavour and per helicity:

+

| qJ 3 3

(4.38)

recall (3.11) that we expect the

159

first

two generations (u, c) and 3, s) to be almost degenerate

in m a s s 7 9 .

The principal decay modes of the q, if kinematical-

ly accessible, would be q + q + g, q + q + y:

r(Q + Q + * > r(q + q + Y)

=

i SLL 3 a

(4.39)

Q^

Thus the q + q + g decay mode is favoured if m presumably

be

< m , and would q

g

followed by g + qqy (or perhaps g + g + y).

mg ^ > m q , or else in the few per cent of events where q

+

If q

+

y), we would expect about 50% of the e + e~ centre-of-mass energy to be missing in the form of photinos and the events to be acoplanar

as

in fig.

17.

However, the amount of missing energy

would be much less in the events q + q +

Cross-sections 31

g.

for hadron + hadron ->• (qq") + X are shown in fig.

18:

they may

be detectable at the CERN pp collider for m q^ £ 0(50) GeV. best

signature would be events with acoplanar jets and missing

E t from q + q + y decay, as illustrated in fig. 9

A final remark ® about squark searches and

The

the t squark.

20 GeV.

17.

concerns

toponium

We know that m t > 20 GeV, and also that m >

If they are comparable, there

will

be

large

mixing

(4.17) between the t, and t R , and it could be that | m ^ - m | = 0(m t ) t l 2 and in an extreme case m . < m,.!

If so,

large branching ratio for toponium 0 + t

160

(4.40)

there 1

t : 1

would95

be

a

_2\3/2 r(9

-

4 / M

T(Q + y* * e V ) ~

3

2

(

V« / (l V

1

'

+ m

(4

2 / 2 _ m | / 2) g t t, t '

'41)

which is about 500, multiplied by a fudge factor which could be ¡> 0(1)!

Predominantly,

However, before concluding that 0 +

we should notice that if hl. > m

+ m , it will 1

be

favourable

7

for the t and the t in 0 to decay first into t + y before annihilating via a gluino as in (A.AO): r(0 + t + (t + t.Y) + (herm. conj.) * 300 x mt(Gev) j r(0 + y + e e )

which is probably still larger than (A.41). bizarre

possibilities

can

be

Of

course,

one can conclude that m^ J 4 important decay mode: 0 • gg:

r(0 T(0

Kg)

+ -

which

> m,.. Even so, there may be an

x C :C =

e e

these

excluded if the t quark is ob-

served to decay canonically into b + qq or b + (I v), in case

(A.A2)

L

2

-

2

(2 - m|/m^)(L + R) 2 -A 2

(A.A3)

which could also be 0(1), depending on the model-dependent parameters

L,R and A.

Toponium may yet turn out to be a good la-

boratory for studying SUSY!

161

5.0 OTHER PARTICLE SEARCHES

5.1 GLUINOS

The

obvious

place

to

look

Is

In

hadron-hadron

collisions31, and traditional beam dump experiments110 have already established a limit

0(2)GeV

g

(5.1)

based on hh+gg + X production followed by g^qqy decay, but pending

somewhat

on

the

de-

assumed squark mass as seen in fig.

19, since m^ enters both in the decay lifetime (a

m^)

and

in

q

the

y cross-section (am"1*).

Perturbatlve QCD calculations of

q

(gg) pair-production cross-sections are shown in fig. these

can

be

sensitive to sensitive

used to estimate111 that the CERN ISR should be

< 0(5)GeV, while the CERN pp collider should be g ~

to m^ £ 0(50)GeV. g

One would look for the missing E-, 1

signature discussed previously in connection with Fig.

20, and

q

searches.

21 shows the results of Monte Carlo calculations112 for a

collider with Ecm=800GeV, showing that gluino

pair

production

could be distinguished from conventional light quark jets using either of the variables -P •P •P -Jet, -Jet2

2

P out

162

(5.2)

The number of gluinos present in the proton when

measured

at

any given Q 2 (e.g., in hard scattering processes) can be calculated 113 » 106 » 11 ** using the Altarelli-Parisi equations to

include

sparticles.

Table

modified

2 shows the asymptotic (Q2+°°)

momentum fractions carried by both conventional and supersymmetric partons, assuming either the absence of any sparticles, or the presence of gluinos alone 113 , or the presence of both gluinos

and squarks106 (the figures in parentheses are the percen-

tages if there are 6 quark quite

a

flavors:

It

seems

that

high percentage of hard scattering processes may con-

tain supersymmetric final states. ymptotic

Nq=6).

As seen in fig.

22, the as-

values are approached quite slowly for gluinos alone,

since there is no

direct

squarks.

develop

Things

q«-»g coupling faster

in

the

absence

above the squark threshold:

squarks may also provide an observable signature

in

in

deep inelastic scattering106. TABLE 2:

Asymptotic Momentum Fractions

Pure QCD 3N q +




a(e e

~

_3

x

t{)

at

(5>3)

+ qq)

for E^m/m£=10 to 100 as seen in fig. seen

„

PEP

and

PETRA

at

the

22. level

cross-section, this would mean that m

g

If gluinos were

not

of 10"3 of the total

> 0(5)GeV.

can also be pair-produced in quarkonium decays 113 :

Gluino pairs for example

for the 3 S 1 (QQ) state T(3S, • gggg) r( 3 s 1 • ggg)

> 10

for

m_ i m Q


> E„ cm is

165

°


VV,YY)> the

(5

-6)

exact

g2 cm

2 x x Y\ o 2 . xXY

[

d2o

f/,

(1 - x )(1 Y 2

[Nv(8V

~ X

+

S

P

+

2(8

V

+

g

A

+

+

v(1

4

- x )cos 2 6 y 1

for

\

(5.7)

tt3

2

3

"e

,f o r __ T t

R or L

where N v is the number of light neutrino s p e c i e s , 1 2 0 g v = 1/2 = - g A , and we have assumed that « m ^ or vice versa: the e e _ R L (YY)Y cross-section is doubled if m ^ ~ m ^ . We see from fig. e e R L 25

(solid

line) that an experiment sensitive to a bremsstrah-

lung cross-section of 1 0 ~ 3 8 c m 2 at E c m = 3 0 G e V would be sensitive to

e

L or R

£ 0(50)GeV, m „ £ 0(10)GeV Y

(5.8)

Shown for comparison are the sensitivities in the (m ,m ) plane e_ y achievable line)

105

.

with e + e ~ + e + e ~ (dotted line) and e ^ ' + e ^ Y Although

it

is

more

indirect,

(dashed

the

reaction

e+e~-»-(YY)Y is sensitive to much larger ranges of m^. e

To get to

the level (5.7) one needs a detector without veto

the

QED

holes

which

can

(etc.) backgrounds such as e + e~+YYY or e + e ~ Y by

being sure that no other observable particles caped detection.

could

have

es-

Another possible w a y 1 2 1 of looking for photi-

nos is via the reaction e + e~+e + e~YY, but cross-section 1 2 2 :

166

this

has

a

smaller

+ _ + -39 /50Gev\ 4 2 a(e e • e e y y ) = 0.5 x 10 I j cm

/e

(5.9)

which makes It less suitable for a front-line search. It may also be possible to detect SUSY at the CERN pp collider via W ^ + x o decay. 123.98,124 W e

lightest

r e c a l l that the

charged gaugino/shiggs mixture x* and the lightest neutral mixture x° (most probably the y) are often (fig. than

the W ± ,

accessible98 W±

and

in

the

*X X° » where x° ±+

W^x^X0

the decay shaded

region

is

of fig.

11) both lighter kinematically 12.

[The decay

is the second lightest neutral (most prob-

ably thefl°),is also accessible in the cross-hatched region of fig.

12.] As discussed in section 4.2, we

expect m x ± >m x 0

in

general, so that the decays

X ± * X° + U v ) or (q'q)

are expected to dominate. (5.10) offer

the

(5.10)

If my! » m x ± + m ^ the (q q) decays

"zen" event signature98 of fig.

26: a ha-

dronic jet system on one side of the beam axis, recoiling against missing E t on the other side.

If m w ± is not » m x ± + m x 0 ,

then the x ± will be produced

relativistically

less

and will

decay more isotropically, yielding an acoplanar event signature more like fig. ward-backward

17. The kinematic availability, rates and forasymmetries98

for W* + x* + X° decays depend on

the ratio v/v of Higgs vacuum expectation values, and mass

parameters

on the

e and Mj introduced in section 4.2. Fig. 27

167

shows the r e l a t i v e r a t e s W ± +x ± X°/W ± + e ± v (dotted l i n e s ) and f o r ward-backward

asymmetries

(dashed l i n e s ) in the k i n e m a t i c a l l y

allowed domains f o r d i f f e r e n t v a l u e s of see

that

paramters.

We

the r a t e s may be s i g n i f i c a n t f r a c t i o n s of the W ± +e ± v

decay r a t e . smaller

these

Since gaugino/shiggs masses are o f t e n considerably

than

squark and s l e p t o n masses, the r e a c t i o n W^x^+x"

may o f f e r the b e s t immediate prospect for d e t e c t i n g SUSY a t the pp c o l l i d e r . There may a l s o be " z e n " events in present-day e + e ~ annihil a t i o n due to the r e a c t i o n e + e ~ + x ° + X°* followed by x ° " + (qq or J l + J O d e c a y 9 9 » 1 2 5 . quires

both

x°

and

The presence of t h i s r e a c t i o n

x°"

t 0

an

d X°'«

y

C r o s s - s e c t i o n s are shown in f i g .

fl 28

v^.v,,. "

r a t i o a ( x ° X ° ' ) / ® ( v | 1 v ) > 10 in the shaded r e g i o n s , and > 1

i n the s t i p p l e d r e g i o n s . conservative figs.

re-

and

f o r m^ = 20GeV r e l a t i v e to the cross—section f o r e e e The

X°

r e l a t i v e l y l i g h t , the e to be

be

q u i t e l i g h t ( £ 0(40)GeV) and some mixing between the components of x °

+

12,

These a r e bounded

cosmological 27.

It

is

by

somewhat

more

bounds than were assumed in drawing even

possible

that

the

reaction

e + e ~ + x ° ' x ° ' roay be k i n e m a t i c a l l y a c c e s s i b l e and have an observa b l e r a t e , a s seen in f i g .

29.

This r e a c t i o n would

have

the

signatures

e+e~+

(i

( *

+

l~) +

0

+

(l

+

l~)

+

missing energy,

+ (qq) + missing energy, + ( q i ) + m i s s i n g energy

thanks to the d i f f e r e n t decay modes a v a i l a b l e to the x ° • 168

(5.11)

Because of the great uncertainties in sparticle masses, as witnessed

by the "Ramsey plot" histogram fig.

30, of possible

sparticle masses extracted from a survey of different SUSY dels,

mo-

the search for SUSY must be a broad-band one and we can-

not be sure where it will first turn up. theorists

However, many

of

us

are convinced that SUSY has something to do with re-

ality, and have the following message for our experimental colleagues:

SUSY

He

To encourage the experimentalists among you in let

us

close

your

searches,

this section with the following historical rem-

inder: 1954 - Gauge theory (Yang & Mills)

1973 - SUSY (Wess & Zumino)

1961 ± - Weak interaction models (Glashow, etc.)

1977 ± - SUSY models (Fayet » etc.)

1967-1968 - Weinberg-Salam model

? - Whose model?

1971 - Renormalizability ('t Hooft)

1981 - Hierarchy problem

1972 - Searches for neutral currents

1983 - These lectures

1973 - Neutral currents found in Gargamelle

? - What?

1974 - Charm

? - What?

1983 - Discovery of the W±, Zo

? - What? the SUSY revolution?

the gauge revolution

169

5.3 CONVENTIONAL HIGGSES

The gauge revolution chronicled above

Is

as

yet

Incom-

p l e t e , since no-one has seen any of the Higgs particles associated with the spontaneous breakdown of gauge symmetry. all

our

Higgses.

rigmarole

about

Indeed,

SUSY was motivated by problems with

Therefore i t seems appropriate to f i n i s h

these

lec-

tures with a few words about how to f i l l this lacuna by finding a Higgs b o s o n 1 2 6 » 1 2 7 . Generically, the couplings of Higgses to fermions are proportional

to

m^, while the couplings to gauge bosons are pro-

portional to M^. U(1) Y

In the minimal Standard

SU(3)C

x

SU(2) L

x

model with just one Higgs doublet there i s just one phy-

s i c a l neutral Higgs H°, and no charged

Higgs

boson

H±.

The

couplings of the H° are completely s p e c i f i e d 1 2 6 ' 1 2 7 :

g _ = 0; 4v, e > 0;

ai

(b) v - 2v, e < 0;

to

(c) v -

and (d) v = 4v, e < 0. 10 V* +

U+y"). 136

208

FIG.

34

Cross-section ratios a(pp + W * or Z° + H ° + X)/a(pp

+ W * or Z ° + X ) . 1 3 6

FIG.

35

Cross-sections for hadron + hadron + (gg

X.137

209

+

H°)

+

FIG.

36

Excluded domains

branching ratio for H~ •»•

of

n^i

as

T>T decays.11,1

210

a

function

of

the

BEYOND THE STANDARD MODEL

H. Georgi Harvard University Cambridge, Massachusetts

211

j.

Effective Field Theories I will spend most of my time, in these lectures, talking

about subjects which are beyond conventional grand unification. But just to clear the air, I will begin with some comments on the status of the minimal SU(5) model. esting questions. ible with data?

Can it be right?

There are two inter-

That is, can it be compat-

and How can it be right?

That is, how can

such a theory with its apparent unnaturalness and arbitrary parameters possibly stand theory of particle interactions?

I

will return to the second question near the end of this series of lectures, after we have discussed some of the alternatives. Now, I want to discuss the first question. To kill any possible suspense right away, let me say that I believe that minimal SU(5) is not yet ruled out by any data but only because of our abismal ignorance of quantitative long distance strong interactions.

I am slightly embarrassed by

this conclusion, because I came to it rather late, only in the last year, in the process of teaching a course on weak interactions.

This embarrassment is one of the reasons why I will

spend the first lecturesdiscussing what we do and do not know theoretically about the strong interactions.

The other reason

is that this discussion will serve as a useful introduction to the effective field theory language, without which a real understanding of modern particle theory is almost impossible. But before I get really started, I should probably bore you with a brief review of the minimal SU(5) model and its experimental predictions.

The minimal model consists of an

SU(5) Yang-Mills theory, with a 5 and 24 of scalars and three 213

families of fermions, each a 10 and 5 of left-handed (LH) fields.

The 24 gets a large vacuum expectation value (VEV) of

order M„ * 10 14 GeV which breaks SU(5) down to SD(3) xSU(2) xU(l). G

The 5 contains the usual Higgs doublet whose VEV breaks SU(2) x U(l). The theory depends on about 20

adjustable parameters.

One combination of these (which determine the ratio of the VEVs of the 5 and the 24) must be tuned to be extraordinarily small -24 (^10

).

Several others (i.e., 8 and the Yukawa coupling to

the light families) must also be taken to be very small. On the other hand, the theory makes a number of predictions which are, at least in principle, very definite. these seem to work:

sin2 6,, * .215; w

Two of (1.1)

m^/m^ * 2-3.

(1.2)

m /m, = m /m ; y e s d

(1.3)

Three seem to fail;

m /m -v 3-4; s y —-BB + 0

- 4.5 x 10 29±1 ' 7 yr.

(1.4)

(1.5)

e ir Of the three apparently bad predictions, (3) is probably the least serious.

As John Ellis and Gaillard and Nanopoulos

have emphasized, the d and e masses are so small that physics well above the unification scale, at the Planck scale could modify this relation and bring it into agreement with experiment. 214

On the other hand, (3) also differs from (4) and (5) In that It does not depend on the details of long distance strong interactions (which drop out of the ratio of the current algebra masses). The question is, how seriously does our ignorance of the strong interactions affect the predictions (4) and (5)?

Before

I try to answer that question, I will subject you to a long digression on the subject of effective field theories. The effective field theory idea is important because physics involves particles with very disparate masses, and because we study physics in experiments involving various energies.

If we had to know everything about all the particles,

no matter how heavy, we would never get anywhere.

But we don't.

We can concentrate only on what is important at the energy scale of interest. Quantum electrodynamics, for example, describes the properties of electrons and photons at energies of the order of 1 MeV or less pretty well, even if we ignore the muon, quarks, QCD, the weak interactions, and anything else that may be going on at high energy.

This works because we can write

an effective quantum field theory involving only the electron field and the photon field.

The corresponding particles are

the only things light enough to be produced at energies of MeV.

Thus if we write down a completely general quantum

field theory involving these fields, including arbitrary nonrenormalizable interactions, we can describe the most general possible interactions consistent with relativistic invariance, unitarity of the S matrix and other general properties like

215

TCP symmetry.

So we don't give up any descriptive power by

going to an effective theory. It might seem that we have given up predictive power, because an arbitrary effective theory has an infinite number of nonrenormalizable interactions and thus an infinite number of parameters.

But this is not quite right for two reasons,

one quantitative and one qualitative.

Quantitatively, if we

know the underlying theory at high energy, then we can calculate all the nonrenormalizable interactions.

Indeed, as we

will discuss, there is a straightforward and useful technology for performing these calculations.

Thus quantitative calcu-

lations can be done in the effective theory language. The qualitative message is even more interesting.

As we

will see in detail, so long as the underlying theory makes sense, all of the nonrenormalizable interactions in the effective theory are due to the heavy particles.

Because of this,

the dimensional parameters which appear in the nonrenormalizable interactions in the effective theory are determined by the heavy particle masses.

If these masses are all very large

compared to the electron mass and the photon and electron energies, the effects of the nonrenormalizable interactions will be small, suppressed by powers of the small mass or momenta over the large masses. Thus not only do we not lose any quantitative information by going to the effective field theory language, but we an important qualitative insight.

gain

When the heavy particle

masses are large, the effective theory is approximately renormalizable.

It is this feature that explains the success 216

of renormalizable QED. More generally, the more nonrenormalizable a term is, the less Important it is at low energies.

So we need only a

finite number of NR interaction terms to calculate to a given accuracy. To extract the maximum amount of information from the effective theory with the minimum effort, we will renormalize the theory to minimize the logarithms which appear in perturbation theory.

In practice, we will use a mass independent

renormalization scheme such as the MS scheme, and choose the renormalization scale u appropriately.

If all the momenta in a

process of interest are of order y, there will be no large logarithms in perturbation theory. the theorists.

(A technical aside, for

It is only because the decoupling of heavy

particles is incorporated automatically in the effective QFT idea that we can use an unphysical scheme like MS—the physics of decoupling is

entirely in the effective theory program.)

The standard technology of the renormalization group can be used to change from one u to another. In the extreme version of the effective field theory language, we can associate each particle mass with a boundary between two effective theories.

For momenta less than the

particle mass, the corresponding field is omitted from the effective theory.

For larger momenta, the field is included.

The connection between the parameters in the effective theories on either side of the boundary is now rather obvious.

We must

relate them so that the description of the physics just below the boundary (where no heavy particles can be produced) is the

217

same in Che two effective theories.

In lowest order, this

condition is simply that the coupling constants for the interactions involving the light fields are continuous across the boundary.

Heavy particle exchange and loop effects

introduce corrections, as well as new nonrenormalizable interactions.

The relations between the couplings imposed by

the requirement that the two effective theories describe the same physics are called "matching conditions".

The matching

conditions are evaluated with the renormalization scale y in both theories of the order of the boundary mass, to eliminate large logarithms. If we had a complete renormalizable theory at high energy, we could work our way down to the effective theory appropriate at any lower energy in a totally systematic way. Starting with the mass M of the heaviest particles in the theory, we could set u = M and calculate the matching conditions for the parameters describing the effective theory with the heaviest particles omitted.

Then we could use the

renormalization group to scale y down to the mass M' of the next heaviest particles.

Then we would match onto the next

effective theory with these particles omitted.

Then use the

renormalization group again to scale y down further. on...I

And so

In this way, we obtain a descending sequence of

effective theories, each one with fewer fields and more small nonrenormalizable interactions than the last.

I will discuss

some examples of this procedure in a moment. There is another way to looking at it, however, which

218

corresponds more closely to what we actually do in studying physics.

We can look at this sequence of effective theories

from the bottom up.

In this view, we do not know what the

renormalizable theory at high energy is, or even that it exists at all.

We can replace the requirement of

renormalizability with a condition on the nonrenormalizable terms in the effective theories.

In the effective theory

which describes physics at a scale u> all the nonrenormalizable interactions must have dimensional couplings less than 1/y to D-4 the appropriate power (1/y

for operators of dimension D).

If there are nonrenormalizable interactions with coupling 1/M to a power, for some mass M > y , there must exist heavy particles with a mass m i M which produce them, so that in the effective theory including these particles the nonrenormalizable interactions disappear.

Thus as we go up in energy scale in

the tower of effective field theories the effects of nonrenormalizable interactions grow and become important on the boundaries between theories, at which point they are replaced by renormalizable (or at least less nonrenormalizable) interactions involving heavy particles, in matching conditions. This condition on the effective theories is probably a weaker condition than renormalizability.

One can imagine (I

suppose) that this tower of effective theories goes up to arbitrarily high energies in a kind of infinite regression. This is a peculiar scenario in which there really is no complete theory of physics, just a series of layers without end.

More likely, the series does terminate, either because

219

we eventually come to the final renormalizable theory of the world, or (most plausible) because at some very large energy scale (perhaps the Planck mass?) the laws of relativistic quantum field theory break down and an effective quantum field theory is no longer adequate to describe physics. Whatever bizarre things happen at high energies, they don't effect what we actually do to study the low energy theory.

This is the great beauty of the effective field

theory language. I will return later in these lectures to the question of generalizations of QFT at high energies.

For now I am

interested in low energies and will use the effective field theory idea as a calculational tool.

It is usually by far

the simplest way to do calculations involving very different scales, and often makes it possible to do nontrivial radiative correction calculations in a few lines, and so to really understand them. The most familiar example of such a calculation is 2

probably the G Q W calculation of sin 9 W scale in GUTs.

and the unification

Here the unification scale M^ is the boundary

between the GUT and SU(3) xSU(2) xU(l) effective theory. The lowest order matching condition is just that the gauge couplings are continuous at the boundary.

This plus the one

loop renormalization group (RG) equations for the SU(3), SU(2) and U(l) couplings gives the G Q W result.

It is

straightforward to improve the calculation by calculating the matching conditions to order a

and using the two loop RG.

220

Note that the matching condition is a power series in the coupling constant at the boundary, while the RG gives all the large logarithmic effects. As another example of the utility of effective field theories, consider the W and Z masses, as analyzed by three of my former students, Sally Dawson, John Hagelin and Larry 2 Hall.

In the tree approximation in the SU(2) xU(l) theory,

2 JI e 2 VC. = % ,M * 8G sin 6 Z F We know from QED that

= MW /cose.

(1.6)

a = e2/4ir = 1/137.036.

(1.7)

We can determine G_ from the u decay rate, r i

T -1 = v

G V _JLjl

(1.8)

192ir

which for measured T and m gives jj u G„ = 1.164 x 10"5 GeV~2. F

(1-9)

2

Finally sin 6 as measured in neutral current experiments is sin26 = .23.

(I.10)

Putting these together gives ^

= 77.8 GeV,

= 88.7 GeV.

(1.11)

Suppose we would like to calculate the leading corrections to this result.

By far the easiest way to do this is to

adopt an effective field theory language. effect can be understood as follows.

The dominant

Below M^, the effective

theory involves QCD and QED, with the weak interactions

221

appearing only as nonrenormalizable 4-fermion interactions. In lowest order, the matching condition which determines these 4-fermion interactions comes from single W and Z exchange.

That means that in leading order, the tree level

relations are correct if G_, e r

2

2 and sin 9 are interpreted as

parameters renormalized at n the energies are

describing processes in which

Higher order contributions to the

matching condition will change these expressions.

The correc-

tions will be a power series in the SU(2) and U(l) coupling constants, of order

2 > 4irsin 6

(1.12)

2 * 4ircos 9

These effects are not very important, less than 1%. There are, however, much more important effects.

The

most important is that the a which appears is not given by 1/137.

It should be interpreted as «(M^j) where a (y) is the

running coupling constant in the effective QED theory renormalized at u.

The other effects are similar.

The effec-

tive 4-fermion operators obtained from the matching condition are renormalized at M^, but the experiments which are used to determine the parameters are done at smaller momentum.

The

renormalization group must be used to find the form of 4fermion operators renormalized at the u appropriate for each experiment. Actually, the renormalization group is an affectation in these calculations.

The important point is that these

corrections involve large logarithms,

222

h

(1.13)

and that all flavors of quarks and charged leptons give contributions.

The renorealization group automatically adds

up the higher powers of a In M^/p, but here a is small enough to make the higher order terms negligible.

The terms pro-

portional to (1.13) can be extracted directly from one-loop diagrams. I will first discuss the renormalization of a. The conventional a is renormalized on the electron mass shell. However, the difference between this definition and a reasonable running coupling (like the MS scheme) evaluated with y =m it.

e

is not very large because it has no large logarithm in

Thus to a good approximation we can take a(m ) = 1/137.

Now we can follow a(u) up to y - M ^

(1.14) The y dependence comes

from the vacuum polarization diagram, (1.15)

All charged particles which have masses less than y contribute in the loop because these are the charged particles in the effective QED theory at the scale y. A standard calculation gives a(M„) = 1/129.

(1.16)

This is a 6% increase in a, which corresponds to a, which corresponds to a 3% increase in M.. and M .

223

There are several relevant 4-fermion operators.

G^ is

determined from the y decay rate, so we must look at the renormalization of the 4-fermion operator ^ " ( l - t ^ e - y~ Yu(l-hf5)vy.

(1.17)

For convenience, I will work in Landau gauge where the fermion wave function renormalization from the diagram (1.18)

has no In y dependence.

Then the renormalization comes only

from the diagram

no electric charge.

But this gives no i.n y dependence.

The

diagram is finite. You can see this by direct calculation or note that we could have used Fierz transformations to write the operator in so-called "charge retention form" G

F - u ~ v Yy(l+y )v u y (1+yJe . 3 3 v y

Z2

(1.20)

Here it is clear that the y and e fields appear only in the form of the left chiral current, which is conserved in the limit that the y and e masses vanish.

But a conserved current

does not get multiplicatively renormalized. a charge which is a physical observable.

224

It is related to

Thus there are no corrections of order a i,n M^/u to (1. That means that the G

F

)

determined by the u decay rate is the

same as the G^ that appears in the formula for M y and M^, to a good approximation. The operator which contributes to charged current neutrino (or its h.c. to antineutrino) hadron scattering is G — _ -± v~ y p (i+y,)vUy,,(i+y s )d. u 5 /2 5 y

(1.21)

It gets renormalized because of the following diagram a (1.22)

which gives for the coefficient

c(u) = -4

a % 1 + - An -=•

(X.23)

•2

Note there is an effect only when the photon is exchanged between legs with the same outgoing handedness.

Otherwise, as

discussed above, there is no large logarithmic renormalization. This tends to increase the strength of neutrino induced charged current interactions relative to neutral current interactions at low energies.

This decreases the p parameter

by about 1%. The four fermion operator relevant in neutral current neutrino scattering is - E 7, Yv^ (1+Y s )v • I i Y ^ i T j d + Y ^ ) - 2 sin 8Q)*.

Jl

i

i

225

(1.24)

where the sum runs over all flavors of leptons and quarks. This is like the y decay operator in that there is no multiplicative renormalization.

Here, however, there is a new

effect produced by the diagram V

y One might think (for a fleeting moment) that the photon 2

propagator would produce at pole for momentum transfer q =0, which would give rise to a long range neutral current interaction.

This cannot happen, because the subdiagram 1/ V

where x is the electromagnetic current, must vanish as the momentum transfer goes to zero because the neutrino has zero electric charge.

Thus the Feynman integral produces a factor

2 of q which kills the pole. The result is a four fermion interaction which is a product of the neutrino current times the electromagnetic current. In other words, it simply renormalizes sin2 8. This 2 decreases sin 0 t t L

2 compared to sin 6 I , _ „ by about 5%. W|a few GeV

Dawson, Hagelin and Hall also analyze the corrections to the SLAC experiment on polarized electron-hadron scattering. This is much more complicated, because the relevant operator mixes with four quark operators which gets QCD into the game. The renormalization group must be used to incorporate the QCD

226

corrections.

The result, however, is quite consistent with

the results of the above analysis.

Putting all these together,

they predict MJJ = 82.6 GeV,

M z = 93.4 GeV.

(1.27)

The results of a more conventional analysis of radiative corrections by Marciano and Sirlin = 82.0 GeV,

are (for m t " 25 GeV)

M z = 93.0 GeV.

(1.28)

2 These predictions include the order a/sin 6 effects which we neglected in the effective field theory analysis.

But as you

can see, we can do rather well with very little effort using the effective field theory. In the next lecture, 1 will apply these ideas to the nonrelativistic quark model and attempt to explain what we do and do not understand about the strong interactions. References 1.

For a long list of references on the effective theory idea, see A. Cohen, H. Georgi and B. Grinstein, "An Effective Field Theory Calculation of the p Parameter", Phys. Lett. B, to be published.

2.

S. Dawson, J. Hagelin and L. Hall, Phys. Rev. D 23 (1981) 2666.

Note:

The material in these lectures has been slightly rearranged and grouped according to subject matter.

227

II.

The Chiral Quark Model One of the striking facts about the low lying hadrons is

that they can be described accurately by a simple nonrelativistic quark model in which the spin 1/2 octet and spin 3/2 decuplet of baryons are three quark states and the vector and pseudoscalar nonets are quark-antiquark states.

This picture

explains the success of SU(6) symmetry arguments (which are hard to justify in any other way) and gives a qualitative understanding of the signs of all mass splittings.

Unlike

symmetry arguments, the nonrelativistic quark model predicts not only the ratios but the magnitude of the baryon magnetic moments.

The simplest version of these arguments

is accurat'e

to about 10 percent. The dominant contribution to the hadron masses in the nonrelativistic quark model is the sum of the constituent quark masses.

Thus the u and d quark masses are roughly one

third the nucleon (or delta) mass.

In addition there are

various relativistic corrections, the most important of which is the spin-spin interaction due to color magnetism proportional to 0.«0. Si ~i J

quark pairs

It is this interaction which makes the A heavier than the N, because all quark spins in the A are alligned. dependence explains the X-A splitting.

The mass

The £ is heavier because

the two light quark spins are alligned. The leading contribution to the magnetic moments of the

228

octet and decuplet baryons is the sum of the quark moments cr (II

I ViiT' quarks ^ ~ ~i

-2)

The proton magnetic moment is about three nuclear magnetons because the u and d quarks are about 1/3 the nucleón mass. The lowest order contributions work to about 10 percent, and 2 2 order v /c

corrections appear to be able to account for the

rest. But this simple and appealing picture of the hadrons has many problems. v2/c2 is not very small. This is a quantitative question which I will not say much about. My general view is that 2

2

v /c

corrections are of the order of 10 percent, and that any

much larger discrepancy requires further explanation. One such discrepancy is gA/gy f°r the neutron.

The NRQM

prediction is 5/3 while the experimental value is about 5/4. The isosinglet pseudoscalar mesons, ri and n' are mysterious.

The NRQM would suggest an ideally mixed pair like 0)

and (J), but the ri is primarily octet, and certainly not degenerate with the it (as CO is with p). Mixing with gluon states may account for all this, but detailed prediction is difficult. Perhaps the most confusing feature of the NRQM is the quark masses themselves. give

Standard current algebra arguments

m +m, _u_d

a

m2 _JL

(II.3)

\ which is completely inconsistent with the NRQM picture.

229

Thus

one distinguishes the "constituent masses"

(m^) of the NRQM

from the "current algebra masses" (m^) which are proportional to the mass terms in the QCD Lagrangian.

One often assumes

that these are related as nij = m + m^

(II. 4)

where m is a flavor independent constant which incorporates the effect of confinement. Still the it doesn't make sense. We believe (on the basis of current algebra) that as the u and d current algebra 1/2

masses go to zero the tt mass must go to zero like (m^+m^)

so that for m , =0 the it's can be identified as the Goldstone u,d bosons associated with the spontaneous breakdown of chiral SU(2) x SU(2) symmetry.

But (2,3-4) suggest that m^ and m^ are

very small, of the order of 10 MeV, Thus taking m^ ^ to zero causes very little change anywhere except the pion mass. This is impossible to understand in the NRQM. In this lecture, I will argue that when chiral symmetry is properly included, most of these problems are eliminated, while many of the successes remain.^ When a global internal symmetry is spontaneously broken, the dynamics determines the length of certain vacuum expectation values (VEV's) but leaves the angles (which give the direction in the internal symmetry space) completely undetermined.

The long wavelength fluctuations in these undetermined

angles are the massless Goldstone bosons.

The angles

themselves, suitably normalized, are the Goldstone boson fields.

In the modern effective field theory language, the

230

results of current algebra can be derived from an effective field theory of these angles, ignoring all the rest of the dynamics. In QCD with three massless quarks (corresponding to the three relatively light quarks, u, d and s), there is a chiral SU(3) x SU(3) symmetry under independent SU(3) transformaL K tions U and U on the left- (L) and right- (R) handed quark L K fields.

This symmetry, we believe, is spontaneously broken

down to SU(3) which corresponds to Gell-Mann's SU(3). The LtR object whose VEV breaks this symmetry is the quark field bilinear i|jt ¡L which under the symmetry transforms as L K

- VlVR-

(ii

-5)

The dynamics fixes the VEV of this object tp be nonzero, but its direction is not determined.

Thus in the effective theory

which describes the Goldstone boson fields, we replace 21?r/f V r "

1 = e

(II

~

*6)

where ir = ~

it A , a • 1 to 8

(II.7)

2 a a

is a hermitian 3x3 matrix of Goldstone boson fields,

f is a

constant with the'dimensions of mass which we have introduced in order to normalize the T 's conventionally.

When we write

an effective field theory of the 7r's, we throw out most of the dynamics which goes into fixing the VEV of (2.5) and keep only the angles which must exist because of the symmetry. This is a sensible thing to do if we are interested in energies 231

and momenta so small that only the Goldstone particles can be produced. To build the effective theory, we must write down all SU(3) x SU(3) invariant interactions involving the i t ' s . L R

This

is easiest in terms of the Z fields, which transform linearly. All nontrivial invariants must involve derivatives.

If there

are no derivatives, the term has a local SU(3)L x SU(3)R symmetry and the Goldstone fields can be rotated away completely.

To put it differently, if the Goldstone boson

fields are constant, they just represent a redefinition of the vacuum, without physical meaning. dependence is important.

Only the spacetime

Thus Goldstone bosons have only

derivative interactions. Still there are an infinite number of invariant terms with derivatives.

But because we are interested only in

low momentum, the terms with fewest derivatives will be the most important. The unique invariant with two derivatives is lf.

tr

^

(II.8)

2

The factor of f /4 is included so that (II.8) contains the normalized kinetic energy term I

3

\ V a -

( I I

'

9 )

In addition, (II.8) describes various derivative interactions which are important in ir-ir scattering at low energy. In our world, the quarks are not exactly raassless. There is a quark mass term in the QCD Lagranglan of the form

232

-tr M 1|>L»|>R + h.c.

(11.10)

where

M =

m u

0

0

0

in, a

0

0

(11.11)

0 m

The corresponding term in the effective theory is 1 2

f

(11.12)

litrME + h.c.

If we stop

where p is a quantity with dimensions of mass.

with (II.8) and (11.12) and ignore terms with more derivatives, the effective Lagrangian describes mesons whose squared masses satisfy the Gell-Mann-Okubo formula, which is very close to what is observed for the it's, K' s and ri.

In addition,

the form of the SU(3)L currents are fixed in the effective +

theory.

Thus the semileptonic weak decays of Tr and K

mesons are completely determined in terms of the parameter f, which turns out to be f = F = 9 3 MeV. IT

(11.13)

To incorporate nonGoldstone matter, such as baryons, into this effective theory, we need to know only how the matter fields transform under the unbroken SU(3) subgroup of SU(3) x SU(3) . L K

One way to see this is to define a matrix

such that (II.14) which transforms as 5

=

(11.15)

Equation (11.15) defines the transformation U which is a

233

function of 0 , U and the IT'S. U IS an SU(3) transformation L R which encodes the SU(3) x SU(3) through its dependence on the L

R

IT'S. Because the IT'S are space-time dependent, U(ir) is also, and this symmetry is local. The SU(3) octet baryons, for example, can be written as a traceless 3x3 matrix field B which transforms as B->UBU+.

(11.16)

We can now build an invariant effective Lagrangian and again keep the terms with the smallest number of derivatives and baryon fields.

Note that an SU(3) invariant baryon mass term, Mg tr BB

(11.17)

is invariant under (11.16). Most of the predictions which can be extracted from the baryon chiral Lagrangian (assuming that all terms with more than one derivative or factor of pM can be ignored) are satisfied by experiment.

But there is one glaring problem.

The nonleptonic decays of the hyperons are not well described. These decays are produced, presumably, by a AS=1 weak Hamiltonian whose most important term is an operator which I will call 0, a sum of four quark and penguin operators incorporating the effect of short distance QCD on the four quark interaction induced by W

exchange.

In the effective

chiral baryon theory, 0 will be represented by some operator 0 built out of the £'s and the baryon fields.

In general, all we

know about 0 is that it transforms like 0 as an (8,1) under SU(3) x SU(3) . Thus we can write L R 0 = Z c a ~ j 3 3 234

(11.18)

where the cr^'s are all the operators in the effective theory which transform as (8,1). condition.

(11.18) is an example of a matching

The constants c_. are, in principle, calculable by

matching the physics in QCD with that in the chiral baryon theory.

Alas, we have no idea how to do such calculations.

All we can do is to order them in terms of number of derivatives.

However, in this case, there are only two

operators involving no derivatives or symmetry breaking and only two baryon fields, ?BB£+

and

£ BB

.

(11.19)

Thus we would guess that the hyperon nonleptonic decays should be described to good approximation in terms of only two parameters, the coefficients (c^'s in (11.18)) of the two operators in (11.19). In fact, the description in terms of (11.18-19) works fine for the s-wave hyperon decays but fails miserably for the p-wave decays.

The p-wave decays, in this picture, ought to be

described by pole diagrams in which (11.19) causes a transition from one type of baryon to another and the pion emission occurs elsewhere in the diagram, but the predictions just don't work. This is a long standing puzzle. In a world with exactly massless Goldstcne bosons, we could make the effective chiral description arbitrarily good by simply going to very low energies.

But in our world, the

masses of the pseudoscalar mesons and the mass splittings among the hyperons are nonzero, and therefore the energies and momenta of interest are bounded below by these masses and splittings.

Thus we must be careful about the terms with more 235

derivatives or symmetry breaking parameters, which we have so far ignored.

Consider, for example, a typical term with four

derivatives: „2 k -§— A

3„3 V r .

tr

(11.20)

XSB

I have written the dimensionless coefficient in (11.20) as a ratio of f

2

2 to a dimensional constant A __ times a dimensionXSB 2

less constant k.

The factor of f

is convenient because the

same factor appears in front of (II.8). so that k is of order 1.

Then I choose

The assumption is that I can do this

consistently for all such terms.

That is, if the coefficient

is written as a product of the appropriate number of f's and A ^ ' s times a dimensionless coupling, then all the dimensionless couplings will be of order one.

The idea is that AXSB

represents the scale at which the physics of chiral symmetry breaking becomes important.

At momenta of the order of A

,

all the higher derivative terms are equally important and the effective Lagrangian becomes useless.

Conversely, if AXc n is

much bigger than the momenta of interest, ignoring all but the first few terms in the chiral Lagrangian should be a good approximat ion. If you buy this so far, the question is "How big is A _„?" Unfortunately, it is easy to see that it cannot be XoB arbitrarily large compared to f. renormalization.

The trouble is

All of the couplings in the theory depend on

an arbitrary renormalization scale, p, because all are renormalized by one loop effects.

For example, we change u the

change in the coefficient in (11.20) is about

236

6k «

A2 (4,rf)2

«E . *

(II.21)

2

The (4tt) factor comes from the one loop Feynman integral. It Is just phase space! The point is that if A X

is much

larger than 4irf, then (11.21) is silly. A small change in y causes a large change in k. The largest reasonable value is A x S B « 4«f.

(11.22)

Indeed, if (11.22) is satisfied, then all the dimensionless factors are changed by ~6ji/u by a change in renormalization. This "naive dimensional analysis" is the best we can do. I will simply assume that we are maximally lucky; that A ^ ^ is as large as it can reasonably be, so that (11.22) is a good estimate. If A „„ is really so big, it is quite a bit larger than X!>B the QCD scale parameter A^^.

Perhaps we can build an

effective theory which describes the interactions of quarks below the chiral symmetry breaking scale. be interpreted as constituent quarks.

These could then

They transform as

iJj ->-UtjJ under the nonlinear local SU(3) symmetry which encodes the SU(3)L x SU(3)R. The effective Lagrangian can then be written down as the most general possible Lagrangian which is invariant under chiral SU(3) and conserves C, P, and T. The last requirement follows because the QCD Lagrangian has these discrete symmetries, which are not spontaneously broken by the QCD dynamics.

The first few terms of the effective Lagrangian are

237

= ¡Kwt+yw + g A $ A Ygif» -rn #

-|

+5tMC+) 0 and down is

heavier than up for q < 0.

Because the U(l)

H

coupling is not

asymptotically free, these splittings within doublets are not as large as the splittings between families which come from f. This picture can account for the mass spectrum of the quarks and charged leptons.

However, it is incomplete in

three ways. (A)

The neutrinos are Dirac particles with masses

similar to those of the charged leptons. (B)

There are extra fermions, not associated with

the light families, but which so far have only ordinary SU(2) x U(l) breaking masses. (C)

There is no flavor mixing.

The first two problems can be solved by enlarging the Higgs structure of the model.

Problem (A) can be eliminated

if the RH neutrinos in the (1,2,R) get a large Majorana mass or if they get a Dirac mass that involves not the LH neutrino, but some SU(2)

L

x U(l) singlet fermion.

The first can be

realized if there is a Higgs which transforms as (l,3,N(N+l)/2), 2

and couples to (rp ) . K

(IV.8)

The second can be realized if there

are several Higgs which transform as (2b) and which cquple to neutral singlet leptons.

Both mechanisms produce RH

neutrino masses which are (in general) of the order of Mj, because most of the RH neutrinos transform nontrivially under H and cannot get mass until it is broken. 263

The Majorana mass

possibility is dangerous (or interesting) because it produces 2

Majorana mass for the LH neutrinos of order m /M^» where m is a corresponding lepton mass. The extra fermions of problem (B) must be present for two reasons.

The heavier families must transform under fairly

large representations of f, such as the 10 of SU(5) in the example we discussed above. extra fermions.

These representations contain

Extra fermions are also required in order

that f be asymptotically flat.

On the other hand, we cannot

rely on the Higgs which gives mass to the light families to give mass to these extra fermions.

They would be too light,

and some would already have been observed. larger mass.

They must get a

The most attractive possibility is that these

extra fermions get SU(2) x U(l) conserving masses of order M^ when H is broken.

Some of these could come from the couplings

of the Higgs, (8), but this cannot give mass to all the extra fermions since it does not couple to iJjL . There must be a Higgs representation which transforms as (l,l,N(N-l)/2).

(IV.9)

2 2 This can couple both to 0|> ) and to (>pn) . To understand Li

R

what is happening, it is convenient to think about the SU(3) x U(l) subgroup of R, where this U(l) is the G G component of the weak U(l) which comes from G.

An ordinary

family of quarks and leptons comes from a piece of R which transforms as (3)

l/6

+

(1)

-l/2

(IV

'10)

under SU(3) x 0(1). Ithe subscript is the U(l) charge]. G G

The

weak U(l) of the LH fields is equal to the U(1)G charge.

The

264

weak U(l) charges of the RH fields are equal to the U(l) charge plus and minus 1/2 [from the RH SU(2)]. produces a normal family plus a RH neutrino.

G

Thus (10) What we want In

order for the couplings of (9) to give mass to all the extra fermions is for R to decompose into three copies of (10) plus a (reducible) representation which is real with respect to SU(3) x U(l) . G

Then the extra fermions break up into pairs

transforming like complex conjugates of one another under SU(3) x 0(1) . G

When (9) develops the most general vev con-

sistent with SU(3) x U(l)„ at M , these pairs get put together G

into Dirac doublets.

X

For example, a pair (3).,., + (3) . in 7/d —7/6

R leads to two Dirac doublets of quarks, each with charges 5/3 and 2/3.

One transforms like a doublet under SU(2) , the Li

other like a doublet under the (badly broken) SU(2) .

All

R

these fermions have mass of order M^.. Now, finally, we can address the question of flavor mixing angles.

The point is that the process of removing the heavy

fermions can introduce flavor mixing angles.

The interesting

sectors of the fermion mass matrix are those which describe the charge 2/3 and -1/3 quarks and the charge - 1 leptons. Consider the quark mass matrices.

Suppose that in addition

to the three light families, there are n Dirac pairs of quarks in R with the conventional charge [ ( 3 ) . ^ + (3)

.

There are then 2n+3 charge 2/3 and -1/3 quarks which can get mixed up by the VeV's of the Higgs, (9).

There are, therefore,

many parameters in the mass matrix and one might think that we could get any masses we want by adjusting the VEV's.

265

That is

probably true, but it is not so easy. Because all the fermions start out degenerate at the unification scale, there is no mixing, or more properly, the mixing at the large scale is undefined.

It turns on and

grows as we come down to low energies. It turns out that if the gauge couplings dominate the renormalization group equations, it is impossible to get mixing which is large enough.

One can show that the mixing angles are smaller than

or of the order of the 3/2 power of the ratio of family masses.

This clearly doesn't work for the Cabbibo angle,

which seems to be of the order of the square root of the family mass ratio. However, when the Yukawa couplings of the Higgs are large enough to have an important effect on the renormalization, then the angles can be large.

It is probably possible

to construct realistic models of this kind. it because they are so ugly.

The ugliness comes from the

mechanism for generating angles. thought of anything better.

We have not done

But for now, we have not

These speculations have been

consigned to the large heap of ideas which seemed very attractive at first but which, at least so far, have not led anywhere. References 1.

This talk is based on work done with three students, A. Nelson, A. Manohar and S. Delia Pietra.

See H. Georgi, A.

Nelson and A. Manohar, Phys, Lett. B 126, 169 (1983), and H. Georgi, S. Delia Pietra and A. Nelson, unpublished.

266

V.

A Depressing Speculation In this final lecture, I will talk about an idea which I

have been exploring recently with Murray Gell-Mann.

Our hope

is to understand the light Higgs doublet in a new way which is consistent with, and indeed dependent on the objections of the mathematical physicists to theories with explicit scalar mesons.

It is a speculation because I do not know for sure

whether the kind of field theories to which I will be led exist at all.

It is depressing because, if these ideas are right,

there really is a desert and fundamental physics will be even harder to do in the future than it is now. To motivate what comes later, let m e begin by discussing yet again the various objections to fundamental light scalar mesons.

One objection which is frequently encountered is that

light scalar mesons are unreasonable because their masses are quadratically divergent.

On the other hand, in a mass inde-

pendent renormalization scheme such as dimensional regularization with minimal subtraction, quadratic divergences never appear.

What is going on?

Whom should you believe?

To discuss

this question, I must review the subject of divergences in quantum field theory. There are two kinds of divergences which one encounters in a naive approach to QFT:

Ultraviolet (UV) divergences, asso-

ciated with lack of convergence at high virtual particle momentum; and infrared (IR) divergences, associated with low virtual particle momentum. There is an important difference between these two. Ultraviolet divergences are unphysical. 267

They are a signal that

one has written the theory in an inappropriate way.

The theory

is complaining and trying to get you to do it properly, by expressing everything in terms of measurable quantities. it right requires renormalization, an

To do

extrapolation process in

which physical quantities are expressed as limits.

The need

for renormalization should not bother you any more than the need to take limits in doing calculus. have to be dealt with carefully.

Indeed, the infinities

If, in the calculus, you

defined f'(x) as (f(x)-f(x))/0, you would have similar problems.

The difference between taking a limit in the calculus

and renormalizing a quantum field theory is that you are more familiar with the former. Infrared divergences, on the other hand, have a more direct physical meaning.

They are associated with exchange of

light or massless particles, and the physical fact that such particles have long range effects. Now consider a quadratic divergence (the same arguments will apply to quartic and all higher divergences) in some Feynman diagram of the form d 4 k/k 2

(V.l)

The problem here comes only from large momenta. divergence with no IR divergence. renormalization effect.

This is a UV

It is therefore a pure

In renormalization, it disappears

without a trace, except perhaps for a convention dependent residue associated with definitions of physical quantities. There is no physics in it at allI

That is why there exist

renormalization schemes in which it is simply thrown away. That is what happens in any mass independent scheme. 268

Indeed,

one might argue that it is a yery good property of mass independent renormalization schemes that they eliminate these irrelevant infinities automatically. At any rate, the situation with a logarithmic divergence, of the form 4

4

d k/k is completely different.

(V.2)

This has both UV and IR divergences.

In a typical Feynman integral, the IR divergence is regulated by physical masses or momenta, but it is always there in the limit in which all masses and momenta go to zero.

This must be

so, because the argument in the log of a logarithmic divergence must be dimensionless.

The log of the UV cutoff always comes

along with the log of some physical mass or momentum.

This is

the reason that logarithmic divergences cannot be ignored in the same way as quadratic and higher divergences.

The IR log

which always accompanies a logarithmic UV divergence has real physics in it.

It is this physics which is incorporated in the

scale dependence of the couplings of an effective theory. Now that we have seen that a quadratic divergence is unphysical, what are we to make of the statement that a light scalar is no good because it has a quadratically divergent mass. Out of context, this doesn't mean anything at all.

The

quadratic divergence itself is simply not the problem. However, it is closely related to a real problem for light scalars, which both John Ellis and I have mentioned several times in these lectures:

the fine tuning problem.

Typically,

when one tries to describe physics at a large physical scale such as the unification scale or the Planck mass, the mass 269

squared of the light scalar gets contributions from several different sources.

For example, in the simplest SU(5) model,

the light doublet mass comes from four different terms in the scalar potential: 2 t ,

+ y 2,

+2 Z ,

2 t tr E q>

(V.3)

where