The Sp ectrum and Eective Action of SUSY

Gluo dynamics

G R Farrar G Gabadadze M Schwetz

Department of Physics and Astronomy Rutgers University

Piscataway New Jersey USA

Abstract

We study the lowenergy sp ectrum of SUSY gluo dynamics using

the generating functional for Greens functions of comp osite elds

Two dual formulations of the generating functional approach are given

Masses of the b ound states are calculated and mixing patterns are

discussed Mass splittings of pure gluonic states in the case when

sup ersymmetry is softly broken are consistent with predictions of

conventional Y angMills theory The results can be tested in lattice

simulations of the SUSY YangMills mo del

Sup ersymmetric gluo dynamics the theory of gluons and gluinos is an

extremely useful testing ground for various nonp erturbative phenomena oc

curing in conventional QCD The Witten index of the SU N SUSY gluo

c

dynamics equals to N Thus the ground state of the mo del consists of

c

at least N dierentvacua parametrized by the imaginary phase of a nonzero

c

gluino condensate The dierent vacua are related by discrete Z

N

c

the N vacua is chosen the transformations of gluino elds Once one of

c

Z symmetry group sp ontaneously breaks down to the Z subgroup

N

c

In analogy with QCD one exp ects that in each of those vacua the sp ec

trum of the mo del consists of colorless bound states of gluinos and gluons

Among those are pure gluonic b ound states glueballs gluinogluino mesons

New address Physics Department New York University USA

and gluongluino comp osites These states fall into the lowestspin represen

tations of the N SUSY algebra written in the basis of parity eigenstates

The masses and mixings of these bound states can be given within

the eective Lagrangian approach The eective action for N SYM

was prop osed by Veneziano and Yankielowicz VY The VY action

involves elds for gluinogluino and gluinogluon b ound states However it

do es not include dynamical degrees of freedom which would corresp ond to

pure gluonic comp osites glueballs

At this stage we would like to make a digression and comment on the

physical meaning of the VY eective action This is not an eective action

in the Wilsonian sense In ref the VY action was constructed as a

generating functional of oneparticleirreducible PI Greens functions

That means that the VY action b eing written in terms of comp osite colorless

elds of SYM theory can be used to calculate various Greens functions of

those comp osite variables Performing those calculations however one is not

supp osed to take into account diagrams with comp osite elds propagating

in virtual lo ops Lo op eects are already included in eective vertices and

propagators o ccuring in the action The simplest kind of Greens function

one mightbeinterested in is a twopoint correlator As we mentioned ab ove

the comp osite op erators entering the VY action are the interp olating elds

for the b ound states of N SYM theory Thus atwo p oint correlator or

simply a propagator of those elds can b e used to determine the mass of the

corresp onding bound state Hence the eective action or more exactly the

generating functional of PI diagrams which we deal with in this pap er can

readily be used to deduce masses of comp osite bound states of the theory

In what follows the eective action and eective Lagrangian discussed will

be understo o d in the sense sp ecied ab ove

In this rep ort we describ e briey how one can generalize the VY La

grangian in order to incorp orate the glueball states in to the description

The classical action of N SYM theory is invariantunderchiral scale

and sup erconformal transformations Once quantum eects are taken into

account these symmetries are broken by the chiral scale and sup erconformal

anomalies resp ectively Comp osite op erators that app ear in the expressions

for the anomalies can be gathered into a comp osite chiral sup ermultiplet

TrW W

The eective action of the mo del can be a functional of the sup ereld S

E D

p

g

y F y Ay S TrW W

Q

g

where the VEV is dened for nonzero value of an external sup ersource Q

g stands for the SYM b eta function which is known exactly The

lowest comp onent of the S sup ereld A is bilinear in gluino elds and has the

quantum numb ers of the scalar and pseudoscalar gluinogluino b ound states

The fermionic comp onent in S is related to the gluinogluon comp osite and

the F comp onent of the chiral sup ereld includes op erators corresp onding to

b oth the scalar and pseudoscalar glueballs G and G G resp ectively

Assuming that the eective action of the mo del can b e written in terms of

the single sup ereld S and requiring also that the eective action resp ects

all the global continuous symmetries and repro duces the anomalies of the

SYM theory one derives the VenezianoYankielowicz eective sup erp otential

S

W S S ln

VY

e

where N g g stands for the dimensionally transmuted

c

scale of the mo del and e If one uses this sup erp otential along with

the simplest Kahler p otential S S one nds that no glueball op erators

are present in the Lagrangian Indeed all the glueball elds enter through

the F comp onents of the sup ereld S These comp onents have no dynamics

and are thus integrated out from the action We would like to argue below

that one needs to use a bigger representation of sup ersymmetry in order to

accomo date also the glueball degrees of freedom in the eective Lagrangian

In order to determine how glueballs can be included in the action let

us concentrate our attention on the expression for the F eld Using the

equation of motion for the gluino eld and for the D comp onent one gets

g

F iQ G iG G

g

As wehave already mentioned the F eld app ears in the VY action without

a kinetic term The term bilinear in the F eld is prop ortional to F F

In general one is not allowed to use the equation of motion if the VEV of the F eld

is considered

Q Besides that there are terms linear in the F eld in the expression

for the eective action thus the F eld can easily be integrated out by

means of its algebraic equations of motion

In order to reveal subtleties of this pro cedure let us write down the fol

lowing relation

H Q

where H is a eld strength for a threeform p otential C H

C C C C The C eld itself is dened as a color

g

a a a

A singlet comp osite op erator of colored gluon elds A C A

g

a a a a a b c

A A A A f A A A with f being structure constants of

abc abc

the corresp onding SU N gauge group The rightleft derivative in this

c

yy

expression acts as A B AB AB

Using these denitions one nds that the expression bilinear in the F eld

acquires the following form

F F H

The second term in this expression is a kinetic term for the threeform p oten

tial C As b efore the eld can be integrated out however one should

be careful in dealing with the C eld

In ref it was argued that the threeform eld C plays an imp ortant

role in the description of the pseudoscalar glueball The glueball can be

coupled to the QCD meson by means of the C eld In the case

of SYM theory the analog of the meson is the gluinogluino bound state

which acquires mass due to the anomaly in the U current within the VY

R

approach Thus it is natural to attempt to couple the pseudoscalar glueball

to the pseudoscalar gluinogluino b ound state within the VY action using

the threeform p otential C

To elab orate this approach let us rewrite the SUSY transformations for

the comp onents of the S sup ereld in terms of and C instead of F and

yy

The quantity Q can also b e expressed through the ChernSimons current K as Q

K Using this equation one can deduce the relation b etween the ChernSimons current

and the threeform p otential C these two quantities are Ho dge dual to each other



 

K C





F

p p p

i

A i A C

i

p p

C

The set of elds given ab ove forms an irreducible representation of sup ersym

metry algebra All these elds can b e assigned to a sup ermultiplet intro duced

in ref That sup ermultiplet is called a constrained threeform sup er

multiplet The easiest way to present this multiplet is to intro duce the

following real tensor sup ereld U

U B i z i z A A C

p p

z z

B

It is a matter of a straightforward calculation to check that the real sup ereld

U satises the relation

S D U

Thus the real tensor multiplet U dened by the expression includes

all the comp onents of the chiral sup ermultiplet S Consequently using the

relation the VY action can be rewritten in terms of the bigger multiplet

U In addition the multiplet has also a scalar B and fermion z We will

show b elow that this allows one to include glueball op erators in the eective

action

First let us notice some features of SUSY transformations of the com

p onents of the U eld The comp onents which are shared by the tensor

multiplet U and the chiral multiplet S namely A and C transform

among themselves while other elds B and z are connected by SUSY rota

tions to the other four comp onents Furthermore one can dene a gauge

Despite a seeming similarity the tensor multiplet U should not be interpreted as a

usual vector multiplet The vector eld which mightbeintro duced in this approachasa

Ho dge dual of the threeform p otential C would give mass terms with the wrong sign in

 

our approach see section thus the actual physical variable is the threeform p otential

C rather than its dual vector eld the ChernSimons current

 

transformation of the U eld as the following shift U U Y where the

sup ereld Y satises the relation D Y It is imp ortant to notice that by

means of this gauge transformation one can get rid of the B and z elds

in the expression for the U multiplet This is the analog of the WessZumino

gauge for the tensor multiplet U Thus any Lagrangian written in terms of

the S eld only if reexpressed in terms of the U eld is necessarily invariant

under the gauge transformation dened ab ove As a result the B and z

elds can always b e gauged away from that Lagrangian Thus in order to

b e able to retain the B and z elds as dynamical variables one m ust include

terms in the Lagrangian which breaks this gauge invariance The simplest

term of this typ e is the quadratic term U j Once such a term is included

D

in the Lagrangian the gauge symmetry b ecomes explicitly broken and the

B and z comp onents of the sup ereld U survive as dynamical variables

Let us now apply the U eld formalism to the VY action In the case at

hand the chiral symmetry is sp ontaneously broken by the gluino condensate

In terms of the chiral sup ereld this corresp onds to the existence of a nonzero

VEV of the S eld hS i With that in mind the appropriate relation

between the U eld and the chiral multiplet is

D U S hS i

The only result of this mo dication is that the eld A in eq gets replaced

by the quantity A hAi

Now use the relation to write the action in terms of the U eld

In order to break the gauge invariance of the VY action we add a term

prop ortional to U to the VY Lagrangian An appropriate term with zero

Rcharge and correct dimensionalityis

U

j

D

S S

Below we are going to show that once this term is added to the VY action

the following elds b ecome dynamical

The B eld propagates and it represents one massive real scalar degree

of freedom identied later with the scalar glueball

The threeform p otential C which b ecomes massive also propa

gates It represents one physical degree of freedom identied with the

pseudoscalar glueball

The complex eld A being decomp osed into parity eigenstates de

scrib es the massive gluinogluino scalar s and pseudoscalar p mesons

z and describ e the massive gluinogluon fermionic b ound states

Relations b etween masses of these states will b e given in the next section

Based on the arguments given ab owe one can write down the eective

Lagrangian for the lowestspin multiplets of the N SUSY YM theory in

the following form

S U

S S j S log j S j hc L

D D F

S S

where and are arbitrary p ositive constants One obtains the VY La

grangian in the limit In general higher powers of U can also be

added to this Lagrangian Those terms would intro duce new quartic quintic

and other higher interaction terms However the quadratic part of the action

which denes twop oint Greens functions and masses will not be aected

In that resp ect the eective Lagrangian can be considered as the one

describing small p erturbations of elds ab out a vacuum state

Let us determine the SUSY vacuum state dened by the Lagrangian

The p otential for the mo del is a complicated function of the variables present

in the U sup ereld After integration over the auxiliary eld the b osonic

part of the p otential is

C

jj jj cos

V

jj jj

jj jj B

f cos g log

B

jj jj



jj

where we intro duced the notations A and arg

In order to nd the vacuum state one should nd the absolute minimum

of the potential Since we are dealing with a sup ersymmetric mo del the

value of the p otential in that minimum has to b e zero As a result of Lorentz

invariance the VEV of the threeform eld is zero ie hC i The VEV

of Q is also zero due to the CP invariance of the mo del After some algebra

one nds that the only global CP invariant minimum of the potential

i hC i hi The eective Lagrangian is given by hi hB

describ es small p erturbations of elds ab out the vacuum state dened by

these VEVs

Wewould like to make a comment here The singularity in the p otential

at jj B indicates that for large p erturbations the higher dimensional

terms omitted in should b ecome imp ortant The same multiplier

B

app ears in front of kinetic terms for the scalar elds so the physical



jj

potential is always p ositively dened As we mentioned ab ove we are mainly

concerned with the mass sp ectrum of the mo del which can be studied using

small p erturbations ab out the ground state so that the approximation given

in is go o d enough for our goals

The actual physical states describ ed by the action are mixed states Be

low we deduce the masses of these mixed physical states Let us write down

the mass and mixing terms of the Lagrangian separately One nds the

following pairs of b osonic variables b eing mixed with one another

B s system

p

r

s B s Bs

C p system

p

p C p C

In order to nd the physical masses one must diagonalize the corresp ond

ing mass matrices Concentrate for instance on the rst row of these ex

pressions which describ es the mixed state of scalar meson s and scalar

mesonglueball B The former gets mass both from the sup erp otential and

U term while the latter gets mass only from the U term When the mixing

term is switched on the initially heavier state s gets even heavier and

the initially lighter state B b ecomes even lighter than they were originally

Performing the diagonalization one nds that the physical eigenstates are

mixed states with the following mass eigenvalues

s

M

Here the subscript refers to the heavier state s which without mixing

would have b een a pure gluinogluino b ound state the s particle The

subscript refers to the lighter state B B in the absence of mixing

Studying the p otential of the mo del we nd that the physical eigen

states fall into the two dierent multiplets Neither of them contain pure

gluinogluino gluinogluon or gluongluon b ound states Instead the physi

cal excitations are mixed states of these comp osites The heavier set of states

contains

A pseudoscalar meson which without mixing reduces to the l

gluinogluino b ound state the analog of the QCD meson

A scalar meson that without mixing is an l gluinogluino

excitation

A mixed fermionic gluinogluon b ound state

These heavier states become the chiral sup ermultiplet describ ed by the VY

action in the limit that the additional term we have added to the eective

Lagrangian is removed The new states which app ear as a result of our

generalization forms a lighter multiplet

A scalar meson which for small mixing b ecomes a l glueball

A pseudoscalar state which for small mixing is identied as a

l glueball

A mixed fermionic gluinogluon b ound state

We call the readers attention to an interesting feature of the eective

action intro duced here Although the physical states fall into multiplets

P

whose J quantum numbers corresp ond to two chiral sup ermultiplets the

action is written in terms of one real tensor sup ermultiplet U The natural

question arises whether the whole action can be rewritten in terns of two

dierent chiral multiplets The relation between a real tensor and chiral

sup ermultiplets the so called chirallinear duality was established in ref

For SYM theory the chirallinear dualitywas used in refs Applied

to our problem the results of refs and can b e stated as follows One

intro duces into the eective Lagrangian a new chiral sup ereld let us denote

it by

p

y F y y y

One can nd an eective Lagrangian written in terms of two chiral sup er

elds S and which is equivalent to the expression given in In our

case

L S S S S

D D

i h

S

S S hc S log

F F

Comparing this expression to the VY Lagrangian one notices that both the

Kahler potential and the sup erp otential are mo died by new terms The

multiplets S and are indep endent

We would like to relate this expression to the Lagrangian of the the

ory written in terms of the U eld If the U eld is p ostulated as a

fundamental degree of freedom then the S eld is a derivative sup ereld

S D U Using this relation the Lagrangian can b e rewritten as

L S S S S

D D

h i

S

S log S hc U

F D

This expression dep ends on two sup erelds U and S is expressed through

U in accordance with However the dep endence on the chiral sup ereld

is trivial the combination can b e integrated out from the Lagrangian

As a result one derives

U

S S

Substituting this expression backinto the Lagrangian one arrives at the

original expression where the S eld is a derivative eld satisfying the

relation

Let us stress again that the descriptions in terms of the Lagrangian

and are equivalent on the massshell In the Lagrangian the dynam

ical degrees of freedom are assigned to the only sup ereld U while in the

Lagrangian the physical degrees of freedom are found as comp onents of

twochiral sup ermultiplets S and The p eculiarity of the expression is

that the chiral sup ereld enters only through the real combination

That is why it was p ossible to formulate the action in terms only of the real

sup ereld U It is essential from a physical p oint of view since the comp onent

glueball eld must be real

In order to make contact with the results of lattice simulations of SUSY

YM mo del one needs to consider the mo del with soft SUSY breaking

term intro duced via gluino mass

The p otential of the softly broken mo del consists of two parts V dened

in and an additional SUSY breaking term

f

V V m Re

f

where m m

g N

c

One calculates minima of the full scalar p otential V Explicit though

tedious calculations yield the following results The VEV of the eld do es

not get shifted when the soft SUSY breaking terms are intro duced Thus

even in the broken theory hi However the and B elds acquire

nonzero VEVs in the broken case

f f

m and hB i m h i

The shift of the vacuum energy causes the sp ectrum of the mo del to b e also

rearranged Explicit calculations of the masses of all lowestspin states yield

the following results

p

f

p

x M M m

scal ar

x

p

f

p

m M M x

f er mion

x

p

f

p

m x M M

pscal ar

x

where M denote the masses in the theory with unbroken SUSY and

x In these expressions the plus sign refers to the heavier sup er

multiplet and the minus sign to the lighter set of states One can verify that

these values satisfy the mass sum rule to leading order in O m

X

j

j M

j

j

where the summation go es over the spin j of particles in the sup ermultiplet

Let us discuss the mass shifts given in eqs Consider the light

sup ermultiplet In accordance with eqs the masses in the light

multiplet are increased in the broken theory The biggest mass shift is found

in the pseudoscalar channel The smallest shift is observed in the scalar

channel The fermion mass falls in between these two meson states Thus

test state in the sp ectrum of the mo del is the particle which without the ligh

mixing would have been the scalar glueball There is a fermion state ab ove

that scalar Finally the pseudoscalar glueball is heavier than those two

states

Let us now turn to the heavy sup ermultiplet In the broken theory the

masses in that multiplet get pulled down However all states of the heavy

multiplet are still heavier than any state of the lightmultiplet in the domain

of validity of our approximations The ordering of the states in the heavy

sup ermultiplet is just the opp osite as in the light sup ermultiplet the ligh test

state is the pseudoscalar meson the heaviest is the scalar and the fermion

as required falls b etween them The qualitative features of the sp ectrum are

shown in g

It is not surprising that the lowest mass state obtained in is a

scalar particle This is in agreement with the result of ref where it

was shown that the mass of the lightest state which couples to the op erator

G is less than the mass of the lightest state that couples to GG in pure

YangMills theory As a result the lightest glueball turns out to b e the scalar

can apply the metho d of ref to the SYM theory as glueball One

well Due to the p ositivityof the gluino determinant see ref one also

deduces that the lightest state in softly broken SYM sp ectrum should be a

scalar particle The pseudoscalar of that multiplet is therefore heavier

Our result that the multiplet containing glueballs is split in such a way

that the scalar is lighter than the pseudoscalar and vice versa for the mul

tiplet containing gluinogluino b ound states is consistent with exp ectations

from quarkmo del lore In ordinary mesons the l states are heavier

than their l counterparts and the l gluinogluino bound state is a

pseudoscalar while an l gluongluon b ound state is a scalar It is inter M scalar + M+ M fermion +

M H M p-scalar +

SUSY -- unmixed SUSY -- mixed SUSY

M p-scalar - ML

M fermion - M-

M scalar -

Figure Qualitative b ehavior of mass sp ectrum when passing from SYM to

softly broken mo del

esting that in SYM with massless gluinos the l and l bound states

are degenerate but when the gluino masses are turned on one recovers the

exp ected ordering seen in q q states

Summarizing wehaveshown that the generalized VY eectiveactioncan

b e written in two dierentways In one case the fundamental sup ereld up on

which the action is constructed is the real tensor sup ereld U In another

approach all degrees of freedom of the mo del are describ ed by two chiral

cases the sp ectrum consists of two multiplets sup erelds and S In both

which are not degenerate in masses even when SUSY is unbroken The spin

parityquantum numb ers of these multiplets are identical to those of certain

chiral sup ermultiplets

Weintro duced a soft SUSY breaking term in the Lagrangian of the N

SUSY YangMills mo del The spurion metho d was used to calculate the

corresp onding soft SUSY breaking terms in the generalized VY Lagrangian

These soft breaking terms cause a shift of the vacuum energy of the mo del

The physical eigenstates which are degenerate in the SUSY limit are split

when SUSY breaking is intro duced We studied these mass splittings in

detail We have conrmed that the sp ectrum of the broken theory is in

agreement with some lowenergy theorems namely the scalar glueball

turns out to b e lighter than the pseudoscalar one The results of the present

pap er can be directly tested in lattice studies of N sup ersymmetric

YangMills theory

Even when SUSY is unbroken the physical mass eigenstates are not pure

rather the physical gluongluon gluongluino or gluinogluino comp osites

particles are mixtures of them The multiplet which without mixing would

have been the glueball multiplet is lighter As a result those states cannot

be decoupled from the eective Lagrangian This means that comparisons

of lattice results to analytic predictions based on the original VY action are

not justied

The work was partially supp orted by grantNo NSFPHY

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