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Home , Graviton, String (physics), String field theory, String theory

hep-th/9312117 14 Dec 1993 Decem 1 theory In ternet address b er W e clarify the nature of the gra eld The at metric in the app ears as the v State OPENSTRING The University siegelinstiph b ound HIDDEN Institute of state New ysicssun ABSTRA app ears for Y W Siegel ork GRA The viton as a b ound state in op enstring eld ysbedu or Stony as etic CT a 1 VITY al comp osite Br Physics o ok THEOR NY acuum v IN eld in alue of an the Y ITPSB fr e e op eld en

INTRODUCTION

Fundamental strings as opp osed to hadronic strings have b een prop osed as the

solution to two problems unied theory and The use of strings

as a solution to the former problem hinges on its use as a solution to the latter

Since the compactications of string to four are so numerous

and the compactication itself do es not seem to b e predictable it is not clear

that the predictability of for low physics is much greater than

that of renormalizable or lowenergy phenomenological fourdimensional quantum

eld theory of The greater predictability of string theory is exp ected

from the hop efullynotto omuch higherenergy corrections arising from the direct

or indirect eects of gravity including the eects of sup ersymmetry whose breaking

is b est explained through the use of sup ergravity and the sup erHiggs eect In other

words the unication p owers of string theory although originally thought to b e great

b ecause of the few mo dels available in ten dimensions b efore compactication have

b een reduced to a solution to the unication of gravity with sup er grand unied

theories

Therefore string theory is basically a solution only to the problem of quantum

gravity but it is the only known solution Ordinarily that might b e sucient and

p erhaps even desirable but until the exp erimental situation improves it would b e

useful to have an alternative theory of for purp oses of compari

son The only prop osed mechanism free of ghosts and the resultant eectively non

renormalizable ambiguities is the app earance of the as a b ound state in a

renormalizable eld theory One advantage this might have over string theory is the

correct prediction of the dimensionality of space time sup erstring theory has criti

cal while renormalizable eld theory with a nite numb er of elds and

b ounded p otential has critical maximum dimension However the only known

theory in which the graviton has b een demonstrated to app ear as a b oundstate p ole

in the S is op enstring theory Therefore the phenomenon of b oundstate

gravity in op enstring theory warrants a closer study with an eye toward isolating

those asp ects that are essential to this phenomenon but might not require string

theory

In this pap er we p oint out two curious features of this mechanism in op enstring

theory that may b e crucial to the understanding if not the formulation of b ound

state gravity in a more general setting In the following section we discuss the

nature of the value of the op enstring eld Originally this was intro duced

as an analogy to gravity where unlike most other eld theories there is no kinetic

quadratic term in the action until after expanding ab out the at metric However

in classical op enstring eld theory as opp osed to classical closedstring eld theory

there is no gravity We discuss the origin of this phenomenon from the vacuum values

of massive two elds the relation of their Stueckelb erg elds to the Skyrme

mo del and the relation of these massive elds to the massless graviton in the quantum

theory In section we study the generation of the graviton at the quantum level

In op enstring eld theory the graviton app ears in a onelo op diagram rather than

through an innite sum of oneirreducible graphs as in most other known

theories with b ound states such as QED and QCD We p oint out that this implies

that the b oundstate graviton actually app ears in free op enstring eld theory We

discuss the analogy to the theory of a free twodimensional massless spinor which

has a massless scalar as a b ound state The b oundstate graviton in op enstring

eld theory is thus a higherdimensional analog of b osonization Besides this analogy

b etween free theories we also describ e the analogy b etween interacting op enstring

eld theory and the Schwinger mo del which shows how these b ound states manifest

themselves as new p oles in the fundamental elds

It is useful to analyze this phenomenon from the p oint of view of an eective the

ory where a new redundant eld is intro duced for the b ound state and we therefore

discuss the gaugeinvariant eld theory of coupled op en and closed strings in section

In the nal section we give a more general analysis of coupled systems at the free

level which allows us to discuss some examples of the mixing that o ccurs b etween

states of the op en string and the closed string the for the massless

elds of the UN string and the coupling of the graviton of the closed string to the

lowest spin eld of the op en string

THE METRIC AS AN OPENSTRING FIELD

The action for op enstring eld theory has b een rewritten as a single term cubic

in the elds which is apparently indep endent of the atspace metric The at

space metric app ears through the vacuum exp ectation value resp onsible for generating

the kinetic term from the current Q of the BRST op erator Q

m 0 1

X C C hi

Z Z

y y y

1 1 1

S Q

2

3 3

Z Z

2

hi Q I Q d Q Q d Q

L L

0 0

1 m 0 1 0 m n mn 0 1

1 1

Q C X P C B C B C X X P P C B C B

m 0 1 mn m n 1 0

2 2

where I is the identity element for the pro duct the Spinvariant vacuum up to a

m 0 1

BRST transformation and P B B are the momenta conjugate to X C C

m 0 1

Strominger has explained the app earance of the atspace metric in terms of the

coupling of the op en string to a background closed string in a desire to enlarge the

space describ ed by the op enstring eld to include indep endent op en and closed

string comp onent elds or even to describ e only closedstring elds Note that

neither the atspace nor any other metric app ears in the denition of the Hilb ert

R

y

space inner pro duct measure h j i The string

1 2 1 2

eld theory formulation is one based on rst and thus uses the Zinn

JustinBatalinVilkovisky ZJBV formalism This means that nonvanishing inner

pro ducts exist only b etween elds and their ZJBV antields eg a covariant vector

n

A has a contravariant antield A so the corresp onding inner pro duct involves

m

n

only Kronecker deltas

m

Here we give a dierent interpretation of Stromingers result Writing the op en

string eld as

Z

2

d Q I

0

1 m 0 1 0 m n mn 0 1

1 1

Q C X P C B C B C g X X g P P C B C B

m 0 1 mn m n 1 0

2 2

hg X i

mn mn

we interpret the eld g as an open string eld In fact if we compare with the

mn

usual oscillator expansion of it is clear that this eld minus its vacuum value

is a combination of the usual op enstring elds Ie the linearized metric is an

innite sum with appropriate co ecients of the massive spin elds that are mass

eigenstates in the free theory Equivalently it corresp onds to eliminating the vacuum

exp ectation values of all spin except one by simple eld redenitions For

any g including traces of higherspin elds indep endent of g that has hg i

mn mn mn

k replaceg with a new eldg g k g so hg i Thus if we start with

mn mn mn

mn mn

many metrics ie spin elds with nonvanishing vacuum values we can always

redene them so only one remains Similar remarks apply to gauge parameters such

as those for general co ordinate transformations where vacuum value refers then to

the invariances of the vacuum ie the global part of the lo cal transformation

The consistency with Stromingers interpretation in terms of closed string elds

follows from the fact that in the eld theory of b oth op en and closed strings the

closedop en twop oint vertex that follows from quantum corrections implies a

direct nonderivative twop oint coupling b etween these op enstring elds and the cor

resp onding closedstring elds like the nondiagonal massterm typ e of coupling that

can o ccur b etween two spin elds as here for the op enstring eld g and its

mn

closedstring analog We will discuss this further in later sections However here we

are considering a eld theory of op en strings only This is actually more consistent

with resp ect to an expansion inh since the relation of the closedstring selfcoupling

to the op enstring one ish dep endent Closedstring states still app ear in the theory

but only as b ound states so they are not describ ed by fundamental elds but rather

by comp osite elds just as eg the in QED In this interpretation

the op enstring eld g intro duced ab ove is the only available in the

mn

theory the only spin eld which couples universally to the energy ten

sor of the op en string When onelo op corrections to its are calculated

one nds a new massless spin p ole that didnt app ear in the classical treelevel

theory ie the graviton

At the linearized level it is already clear that the op enstring eld theory has

gauge invariances for massless spin elds that resemble those of massless spin

In string eld theory expansion in the co ordinates gives Stueckelb erg elds

The interacting case requires only a generalization of the Higgs mechanism to gravity

This can b e derived in the same way as for massive spin by p erforming a gauge

transformation on a gauge noninvariant action consisting of the gaugeinvariant

massless interacting action plus a mass term The gauge parameter then b ecomes

a new eld In the spin case the appropriate gauge parameter is the transformed

co ordinate This construction for massive spin in string theory was mentioned in

The gaugeinvariant action is the EinsteinHilb ert action and the mass term is

the FierzPauli term

2 2 a 2

1

L M h h

FP a

ab

4

which is the same as the mass term obtained from by dimensional

reduction The generalization of the mass term to the interacting case is somewhat

ambiguous In an expansion of the elds ab out their vacuum values the constant

term can always b e canceled the linear term must vanish no cosmological term

and the quadratic term must t the FierzPauli term but higher terms are

arbitrary In the mo del we write to study this sector of the string we cho ose these

terms to have as few derivatives as p ossible a typ e of lowenergy limit Eectively

this means just dening

ab mn a b ab

h g A A

m n

p

in L and throwing a g in the measure

FP

The unique result in D dimensions is then

Z

p

2 D

g R M L S d x

M

D (D 1)

mn a b

D 1

g A A L

m n ab M

4 2

mp nq a b c d

1

g g A A A A

ac bd

[m n] [p q ]

8

a a

hg i hA i x

mn mn

This is exactly the Skyrme mo del for the translation It has not only

a

general co ordinate invariance under which A transforms as scalars but a separate

a a b a a

internal global Poincare on A A A This is similar

b

to a vierb ein formalism with curved and at indices but here the tangentspace

invariance is only global Unlike the analog for massive YangMills the quartic term

is necessary here and is required to b e the Skyrme term to repro duce the FierzPauli

term Unfortunately this action has explicit atspace metric factors and it

ab

do es not seem p ossible to eliminate them They cant b e replaced with other tensor

elds since after the ab ove eld redenitions g is the only tensor with nonvanishing

mn

vacuum value Similar remarks apply to eliminating the vacuum values of the other

vectors A A In fact all at indices can b e removed from everything except A by

a m a

using A as a vierb ein eg A A A The string action consists of a sum

m

of such terms for dierent M from the eld redenitions g g and A A

at dierent mass levels The sum for the mass term is less convergent than that for

the R term and the sum of the co ecients might regularize to zero

3

This suggests that the interpretation of the action as background indep endent

is singular However we can still interpret g or p erhaps all the spin elds to

mn

avoid singular eld redenitions as a metric tensor since it transforms covariantly

under general co ordinate transformations and has a nontrivial vacuum exp ectation

value Furthermore b ecause of the Stueckelb erg elds supplied by string eld theory

the action is general co ordinate invariant The interpretation is then that A

metric tensors and general co ordinate invariance exist in the classical op enstring

eld theory but do not describ e the graviton b ecause of the Stueckelb erg elds

However quantum contributions to the eective action cause this metric to describ e

a massless graviton in addition to massive spin with the dep endence on the ab ove

tangentspace at metric disapp earing at low energy near the graviton p ole Thus the

metric tensor and general co ordinate invariance are classical features of the theory

and it is only the massless p ole that is the quantum feature

Another interesting feature of the vacuum value of the op enstring eld is that it

can b e written as a pure ab elian gauge transformation In terms of the ghostnumb er

op erator J the BRST op erator Q and left half of the BRST op erator Q can b e

L

expressed as

0 1

J C B C B J Q Q

0 1

Z Z

2

J Q Q J Q Q J J J J

L L L

0 0

hi QJ I

L

This is also true in ordinary gravity In terms of the ab elianized gauge transformation

1

g we have hg i x

mn mn mn

(m n) (m n)

2

The origin of the b oundstate massless graviton from massive spin also explains

the origin of the b oundstate The Stueckelb erg formalism for massive spin

can b e generalized from the ab ove to include a scalar representing massive spin in

terms of massless spins and all three of these elds app ear automatically

in op enstring eld theory In the massless limit of massive spin the vector

decouples but the scalar remains It is the dilaton

THE GRAVITON IN FREE OPENSTRING THEORY

The most unusual feature of closedstring b oundstate generation in op enstring

theory is that it o ccurs at one lo op in constrast to more familiar b oundstate mech

anisms such as the hydrogen atom where the b ound states when represented by

p erturbation theory are generated essentially by an innite sum of ladder graphs or

the random lattice formulation of the string or the QCD formulation of the

hadronic string where the string is generated as an innite sum of all the leading

graphs in the N expansion The reason is simple The closed string is actually a

b oundstate of the free theory This follows from the fact that any amputated one

lo op propagator correction in an interacting theory is equivalent to the propagator

3

of a comp osite eld in the free theory Consider any p oint coupling for sim

plicity we hide all derivatives indices etc The corresp onding lo op propagator

2

correction is then in co ordinate space hjJ xJ y ji where J S is the

current coupling to In evaluating this lo op correction we use the free propa

2

gator x y hjxy ji hjJ xJ y ji x y This is clearly the

0 0

same as evaluating the propagator for the comp osite eld J in the free theory of the

eld

This bizarre mechanism is actually well known in twodimensional eld theory

In the Schwinger mo del D massless QED the propagator gets a scalar

p ole from the lo op it eats the scalar to b ecome massive The app earance of

this scalar is now well understo o d It is b osonization The vector current of the

electron is equivalent to the gradient of a scalar eld even in the free theory of the

electron The photon couples to this current This onelo op correction to the classical

ab

QED lagrangian F A

a b

2

1

L F i A

0

2

can b e expressed by intro ducing the b oundstate eld through the additional terms

L L hL

0 1

1

L F

1

2

resulting from the b osonization of We have intro duced as a redundant eld to

Intro ducing a separate eld for a b ound state is always redundant but not incor

rect although coupling constants need to b e mo died b ecause of double counting

1

1

F Integrating out classically tree graphs pro duces a term F in the eective

2

action the same as integrating out at one lo op

We therefore have three twodimensional eld theories that describ e this b ound

state phenomenon In the theory of a free massless L i a com

p osite eld describing the scalar b ound state is dened by the usual b osonization

b

formula The consistency of such a denition follows from just current

a a b

conservation in any dimension but the fact that the propagator of the comp osite

eld has p oles follows from kinematics Classical massless particles in D travel

at the sp eed of to either the left or right picks out the comp onents of and

for electron and p ositron b oth traveling to the left or b oth to the right as follows

from their equations of motion Two massless particles starting at the

same p oint and traveling in the same direction are never separated and therefore

act as a b ound state In massless QED L L the photon couples to this

0

comp osite eld so at one lo op the b ound state shows up in the propagator of the

fundamental eld A In the eective eld theory describ ed by L L hL is

a 0 1

a fundamental eld rather than a comp osite one but the physics is identical to that

describ ed by just L If is eliminated by its equation of motion we get back the

0

formulation in terms of just L but including the onelo op contribution that shows

0

the presence of the b ound state Although the free fermion theory is sucient to

describ e the b ound state the interacting theory of QED automatically p oints out the

existence of the b ound state by showing it in the onelo op propagator of a fundamen

tal eld and the eective eld theory helps to elucidate the mechanism by which this

happ ens

A similar mechanism o ccurs for the string An op enstring current is equivalent

to a closed string even in the free theory of the op en string This current can b e

3

found from the of the interacting theory As for the Schwinger mo del

where the b ound state can b e describ ed as a comp osite quadratic eld in the theory

of a free fermion b osonization the closedstring b ound state can b e describ ed as a

comp osite quadratic eld in the theory of a free op en string and our discussion of

the interacting op enstring eld theory and in the following section the interacting

eld theory of op en and closed strings is for p edagogical purp oses and b ecause those

theories are interesting in their own right

The Schwingermo delstring analogy is then

string eld theory Schwinger mo del

photon A op enstring metric g

a mn

electron rest of op enstring eld

scalar b ound state closedstring b ound state

current current

a

b y

b osonization G

a a b

where is the op enclosed p oint vertex op erator in our notation it takes

an op enstring eld to a closedstring eld and G takes care of the zeromo des

asso ciated with global translation invariance on the world sheet

Z

G b N b d B N fQ b g

1 1 1  1

Unlike the Schwinger mo del contains no derivatives so we can

lo cally invert the expression for the current in terms of the closedstring eld as

y

b b b b G

0 0 0 0

is essentially the pro duct of two functionals that pro jects out op enstring states

that can couple to closed strings The ghost insertions b are gauge dep endent as

0

exp ected from the fact that has a gauge parameter indep endent of s gauge



parameter They are exactly those that app ear in the iteration of the lowestorder

y

propagator corrections expressed by the op enclosed coupling G

1

b b c

0 0



2 2

2 2

p M p M



y y 1

G G G c G

 

Thus is just a eld redenition of at least for the propagating comp onents

The analogy also relates to the interpretation of the previous section For the

Schwinger mo del in terms of elds A and L hL it is that has the

0 1

physical b osonic p olarization which shows up in the A propagator only b ecause of

its coupling to On the other hand for the formulation in terms of just the elds A

and L the electromagnetic eld which classically has no physical p olarizations

0

in D has gained a physical p olarization at one lo op For the string in terms of

and it is that describ es the graviton but in terms of just the op enstring eld

the metric tensors g in describ es only massive spin classically but develops

mn

a massless graviton p ole as well as a dilaton p ole at one lo op

OPENCLOSED

Unlike the Schwinger mo del where the eld that generates the b ound state

has no self the selfinteractions of the op en string generate closed

string b oundstate self interactions at higher lo ops so our analysis in terms of an

eective action of b oth op en and closed strings requires an analysis b eyond one lo op

In string theory the order in h of a graph is related to the Euler numb er In eld

Sh

theory language the contributions to the eective action app earing as e are of

the form for arbitrary nonnegative m and n

m+1 n n+1

h h

plus terms of this form but higher order inh As in ordinary eld theory h and the

2

g app ear in Sh only in the combination h g There might also

b e a tadp ole term h but not in the sup ersymmetric theory Again the normal

h counting with h s app earing only in lo ops is consistent only in the formulation

where only app ears as a fundamental eld The eld theory of closed strings was

develop ed in Field theory of op en and closed strings has b een discussed in

but in a form where the existence of gauge invariance was unclear

For our purp oses it will b e sucient to consider adding to the classical op enstring

lagrangian

y y

1 1

L Q

0

2

3

2

the eective closedstring terms L L h L h L

0 1 2

y y n+2 y

1 1

L GQ G terms V

1 

2 2

y 3 2 n+1

1

GV G L term terms

 2

2

We could replace with the eld G satisfying the constraints b N

1

1

y

which would simply absorb all Gs except for the replacement GQ



2

1

y 1

c Q This eld is more useful for conformal eld theory b ecause its Hilb ert



2

space more closely corresp onds to the direct pro duct of two op enstring Hilb ert spaces

Its use is analogous to the use of chiral sup erelds in sup ersymmetry The term

1

y 1 1

c Q could then b e expressed without the c as integrated over the chiral



2

subspace whereas all other terms including all quantum corrections would b e inte

grated over the full space However we prefer to work with unconstrained elds

Like and I is a functional times ghost factors that follow from including

y

the ghost coupling to the worldsheet If we write the G term as a

y

double integral with G evaluated on closedstring co ordinates and evaluated on

0 1

op enstring co ordinates again using C and C as the ghost co ordinates is a

0

functional of the closed minus the op en co ordinates times a factor C where

is the interaction p oint where the worldsheet curvature is This follows from consid

ering ghost zeromo des For physical elds and G are indep endent of them the

op enstring co ordinate integration integrates over one while the closed integrates over

0

two This leaves three for two for the functional one for C The counting

is dierent for b osonized ghost co ordinates In this form intro duces no spacetime

derivatives in contrast to the Schwinger mo del The exact vertex that app ears

y y

in the onelo op diagram as the eective action contribution G G dif



fers from by a conformal transformation that includes derivatives which is just an

invertible eld redenition

essentially pro jects out the op enstring states that couple directly to the closed

string states ie the singlets like the op enstring metric g in

mn

analogy to the way that in the Schwinger mo del pro jects out the chiral part of

a

Furthermore whereas in the Schwinger mo del integrating out the b oundstate

eld gives an identical result to a fermion lo op with two external lines integrating

out the closedstring eld gives at b est only the part of the nonplanar onelo op graph

containing the physical closedstring p oles This is probably related to the fact that

the b osonization expression for in terms of is invertible while the expression of

in terms of might not b e invertible to express the op enstring eld in terms of

1 1

y y

the closed string Therefore we include terms V and GV G representing



2 2

the lo cal parts of onelo op propagator corrections corresp onding to contracting the

y

lengths of the in terms of prop erworldsheet time while the G

term incorp orates the physical p ole parts of those corrections

For studying the eective action we can concentrate on the quadratic terms

dropping theh s

1 2 y y

L G Q Q GQ GQ Q G

2

G

y

V G Q

Q Q Q Q

0 1 2

V G Q

 

1 3

resp ectively in terms Q G V V have ghost numb ers

0 

2

2

of the ghost numb er op erator that is antihermitian with resp ect to the Hilb ertspace

metric ie the integration measure is dened to have vanishing ghost numb er

Unlike the A coupling of the Schwinger mo del the op enclosed coupling is not

gauge invariant under the indep endent gauge transformations of the



R

uncoupled op en and closed string eld theories The string action S L is actually

the ZJBV op erator which includes not only the classical gaugeinvariant action but

also the BRST gauge transformations In the quadratic part of the action this is

simply the statement that Q is b oth the kinetic op erator and the generator of gauge

0

transformations Through their dep endence on c the string elds contain b oth the

0

usual elds and their ZJBV antields In the expansion c contains

+ +

only elds while contains only antields The usual ZJBV antibracket of elds

y

and antields then follows from the string eld antibracket in terms

0

of a functional of all the co ordinates including c The closedstring eld b ehaves

y 1 1 1

similarly except c has the factor c b ecause the c indep endent part of

y

drops out of the action Equivalently G where G is analogous to the

factor app earing in the denition of functional dierentiation of chiral sup erelds to

maintain the constraints on the eld Gauge invariance is then the usual statement

y

S S The term G Q thus contributes a crossterm not only to the action

1

but also to the gauge transformations We consider an eective action calculated by

background eld gauge metho ds so gauge xing do es not break the gauge invariance

of the eective action

2

We now examine the gauge invariance condition Q p erturbatively This

14 14

can b e considered an h expansion if we redene h h followed

2 2

byh h Then Q Q h Q h Q Of the resulting relations

0 1 2

2 2 2

Q fQ Q g Q fQ Q g fQ Q g Q

0 1 0 2 1 2

0 1 2

2

is the rst two expressions are known to vanish The middle identity states that Q

1

BRST trivial In the op enstring sector this follows from the fact that it is BRST

2

invariant but not in the op erator BRST cohomology Otherwise Q ji would b e a

1

Poincare invariant state in the cohomology but there are none of that ghost num

b er Since on shell and translation invariant means massless the only onshell zero

momentum scalars in the cohomology are those of YangMills theory the YangMills

ghost at zero momentum ji corresp onding to the identity op erator and its anti

eld CC C ji corresp onding to the op erator CC C with ghost numb er whereas

y y

G has ghost numb er Thus G fQ V g for some V In the closed

y

string sector in geometrical terms describ es a closed string that breaks into

an op en string and instantaneously reconnects into a closed string In other words

the cylindrical worldsheet describing the spacetime propagation of the closed string

0 y

has an innitesimal hole Because of the ghost insertions C in and b eing

multiplied at the p oint of that hole resulting in the squaring of an anticommuting

cnumb er this diagram vanishes We can thus set V The same result holds if



we use U U instead of since it diers only by a unitary transformation U



1

2 y y

The remaining identities then b ecome V V which U U





we exp ect to follow also from squaring of fermionic insertions There are additional

V and V terms in nonorientable string theories such as the SO sup erstring



At the free level this calculation should b e identical in an appropriate represen

tation of the ghost co ordinates to the calculation of closure of the Lorentz algebra in

the lightcone theory At the quadratic level the BRST op erator derived by covari

antizing the nontrivial lightcone Lorentz generators is generally identical

to that obtained by the usual BRST metho ds in this case with and not as

the op enclosed vertex It has b een shown that the Lorentz algebra closes for

op enclosed b osonic lightcone string eld theory although there the cancella

y

tion of the G term was with an anomalous term coming from a nonplanar

3

lo op diagram generated by the squaring of the term However the generation

of the closedstring b ound state from the nonplanar graph in the lightcone string

eld theory if it o ccurs at all is dierent than in b oth the usual covariant one and

the covariantized lightcone one Still it should b e p ossible to obtain the analog of

our result for the light cone by adding a corresp onding lo cal term to the classical

lagrangian and subtracting the same term from the onelo op corrections In any

case we will assume the V term is sucient to solve the problem as implied by

our op enstring cohomology argument which do es not require consideration of lo op

diagrams to show the gauge invariance of the free theory In the following section

this assumption will b e supp orted by examples from nite subsets of the op en and

closedstring elds There we will give a general analysis of quadratic lagrangians ex

hibiting nonderivative couplings b etween two dierent sectors of which op enclosed

string theory should b e a sp ecial case

GENERAL ANALYSIS OF FIELD MIXING

Rather than delving into the technical details of this quadratic coupling b etween

elds of the op en and closed strings we analyze this phenomenon in a more general

setting We rst consider the most general p ossible relativistic free lagrangian up

to eld redenitions for any set of elds This result is known from rstquantized

BRST metho ds

y cm 2 2 ca cc

1 1

Q Q Q iM M M c Q cp iM p bM L

0 0 a

2 2

ij

where M are the Ddimensional generators of OSpDj an extension of the

Lorentz group to include ghosts The index c lab els one of the two ghost direc

tions and m lab els the single reduced dimension used in the generation of mass by

0

p M c and b are c and b for the string We have

m 0

explicitly separated out the mass dep endence writing the general BRST op erator

Q in terms of the massless BRST op erator Q Irreducible representations of the

0

ij

OSpDj subgroup generated by M for i j m are representations of the

massless version of this extended consisting of a single gauge eld and

its ghosts and antields through dep endence on c Such representations can eat

each other to b ecome massive representations of OSpDj

2

We next consider the most general way to break up Q Q h Q h Q

0 1 2

2

such that Q p erturbatively in h Since Q has the most general form Q can

0

ij ij

dier only in how the op erators M and M are represented However M for

i j m the OSpDj subgroup must b e represented in the same way to

preserve the Lorentz subgroup SOD thus their Q terms are the same In

0

the lightcone string theory from which the covariant result can b e derived this

is the statement that the generators of SOD are the naive free rstquantized

op erators and get no quantum corrections while the remaining Lorentz generators

include terms from the op enclosed coupling as well as other interaction terms Since

cm

each representation of OSpDj and in particular M requires an appropriate

set of elds its representation in Q can dier from that in Q only by mixing b etween

0

OSpDj representations of the same typ e However it seems that the only way

to break up an individual OSpDj representation that is consistent with satisfying

2

Q p erturbatively is the separation of its elds into those with o dd and even

numb ers of m indices We therefore restrict the massless representations of Q ie

the subset or our original elds with zero eigenvalue of the op erator M to b e only of

the form of an even or o dd part of a representation of OSpDj so they can

mix with the corresp onding parts of the massive representations This holds for the

massless sectors of the string the vector and are already the

even parts of the corresp onding massive representations the graviton and the dilaton

together corresp ond to the even part of massive spin Furthermore for the string

the separation into o dd and even numb ers of m indices corresp onds to separation

cm

into o dd and even numb ers of oscillators In Q MM is cubic in oscillators while

0

2 cm

M is quadratic in Q the MM cindep endent term in the op erator is o dd

1

2

in oscillators while the M term is even as follows from the representation of the

overlap integrals used in evaluating functionals as Gaussians

cm

The most general form of a representation Q of the iM M term in any of the

pieces Q Q Q is then

0 1 2

y y cm y

Q A A iM A A

cm

where we have broken up the original M in Q into the piece A that takes the even

y

part of an OSpDj representation into the o dd part and the piece A that takes

o dd into even is a matrix that mixes the dierent copies of an even part and

maps them to the dierent copies of the corresp onding o dd part and similarly for

y

mapping o dd to even Thus Q Q and Q are blo ck diagonal with resp ect to

0 1 2

OSpDj The elds for any one blo ck consist of all the copies of any particular

OSpDj representation but of dierent masses plus all the massless elds cor

resp onding to the even or o dd part of that representation Restricting our attention

to such a set of elds for a particular representation of OSpDj we can divide the

elds into even and o dd as where e lab els the copies of the even part

e d

and lab els the comp onents of an even part and similarly for the o dd stu d but

d has a dierent range than e and dierent than Then the index notation for

the ab ove equation is

y y

0 0 0 0 0 0

Q Q A

ed ee ed

0 0 0 0 0 0

Q Q A

d d d e de

2

Solving Q p erturbatively then completely determines the Qs in terms of the s

and requires that Q have no Q term

3

Q Q Q c Q Q c Q c

0 0 0 0 1 1 1 2 2

y y

Q A A

k k k

y y y y y y

0 0 0 0 0 1 2 1 1 1 1

(0 1) (0 1)

Lastly we break up our elds into two sets and which will b e the analog

of the op en and closedstring elds This division of elds is chosen to preserve even

and o dd pieces For any representation of OSpDj any linear combination of the

copies of the even part including those of dierent mass may go into and any

linear combination of the o dd the indep endent combinations go into In partic

ular we may break up an OSpDj representation so some of its OSpDj

representations are in and some are in ie the gauge eld and its even Stueck

elb erg elds are in one place the rest of its Stueckelb erg elds in the other Finally

we cho ose an that is diagonal with resp ect to this breakup and an that

0 1

is odiagonal That makes Q and Q diagonal Q odiagonal We exp ect this

0 2 1

description to b e equivalent to that just given for the string as we now demonstrate

with two examples

For our rst example we consider the generation of mass by dimensional reduction

and cho ose the ab ove p erturbation expansion as simply an expansion in the mass

ie we dene our separation of elds into those with o dd and even numb ers of m

indices so that and M

0 1

cm 2

Q Q Q iM M Q M c

0 0 1 2

The lagrangian is then

y 2 y cm y 2

1 1

Q M c i MM Q M c L

0 0

2 2

For the sp ecial case of the antisymmeric tensor A where dimensional reduction

ab

gives also the vector A this agrees with results for the massless level of the UN

ma

b osonic string where the antisymmetric tensor from the closed string mixes with

a

the singlet part of the vector from the op en string through a term B A

ma

involving A s NakanishiLautrup NL eld B

ab a

2

We next eliminate by its equation of motion inverting Q M c in the usual

0

1

cc 2 2 cm 2

M gauge to get the propagator bp M and using the identity M

2

to nd the equivalent lagrangian in terms of just

y

1

L Q

2

b b

cc 2 cc 4 2 2

1 1

M Q M c M O M Q Q Q Q M c M

0 0 1 0

2 2

2 2 2

p M p

1

y 2

Q Q is the result that would b e obtained for Q if the term M c were ignored

0 1

2

as in string theory where the analogous V term do es not o ccur Since fQ Q g

 0 1

Q represents a correction to the lagrangian consistent with global BRST invariance

1

2

to that order in M However the complete Q is necessary for total BRST invariance

1

y 2

M c term is necessary in L for gauge invariance Although in string and the

2

theory the analogous term is unnecessary the corresp onding term for is This

0

mo del is unlike the full string in that Q for the string includes c c terms

1

which contribute to the propagator as dened by L L L L in the b

0 1 2 0

gauge c terms in Q contribute terms to the gaugeinvariant action containing just

physical elds while other terms in Q contribute terms containing NL auxiliary elds

However in the calculation of for the massless sector of the UN b osonic string

cm

the p ortion of actually used was exactly the M used here and the only part

cc 2

of Q obtained by explicit calculation was exactly the bM p obtained here while

1

y

the c term analogous to V was inferred to also follow from G G by BRST



invariance of Q

1

We now discuss an example that exhibits direct coupling b etween physical elds

of the same spin but dierent masses as opp osed to the previous example which

illustrated coupling b etween particles of the same mass but coupled a physical eld

to an NL eld In the more interesting cases one particle is massless so well assume

that restriction here for convenience In the sp ecial case of spin this is analogous

to the coupling b etween the massive elds of the op en string and the graviton and

dilaton of the closed string on which we fo cused in section As for the case of spin

vector dominance such actions are obtained simply by writing the

kinetic terms for two particles of the same spin but dierent masses coupling b oth

gauge covariantly to two sectors with dierent couplings for the dierent

sectors and making eld redenitions that diagonalize the couplings to the two

dierent matter sectors which makes the gauge elds mass terms odiagonal For

example for the spin case we start with the usual free lagrangians for a massless

photon A and massive rhomeson couple A to strongly interacting matter and

A to weakly interacting matter and then redene A and A A as

2 2

our new elds so the mass term A has cross terms In the spin case

the gravitons coupling is nonab elian so we have to b e more careful with interactions

but well again restrict ourselves here to just the quadratic part of the lagrangian

We b egin by dividing up the elds of the massive states into with even numb ers

of m indices and with o dd In the case of spin is a vector while is a scalar

for spin is a tensor plus a scalar while is a vector We next write the lagrangian

y 2 y 2 y y y

1 1 1 1

Q m Q m Q c c m A Q L

0 0 0

2 2 2 2

2 y

Q m c mA

0

C B C B

2

mA Q m c Q

A A

0

Q

0

where m is the value of the mass for the massive states To mix the massless with

in a way that preserves the matrix structure we must cho ose it to b e the same

representation of OSpDj Thus for spin must include the dilaton as well

as the graviton Of course it is still p ossible to pro duce such couplings b etween

massive and massless spin without a dilaton but the dilaton is necessary for

2

Q to b e satisifed orderbyorder in the p erturbation expansion Q Q Q Q

0 1 2

We then undiagonalize by

y

1

p p

Q Q Q Q Q L

0 1 2

2

1 1

2 y

p

Q m c mA

0

2

2

B C

1 1

2

p

Q m c mA Q

A

0 0

2

2

Q

0

1

2

m c

2

C B B C

1 1

2

p

m c Q Q mA

A A

1 2

2

2

1 1 1

2 2 y

p

m c m c mA

2 2

2

1

p

We could have obtained a mass other than m by using a dierent undiagonalization

2

and a corresp ondingly mo died splitting of Q into Q Q and Q Q gives the

0 1 2 0

1

p

m and in BRST form standard lagrangian for the decoupled elds with masses

2

cm

Now Q contains not only the MM term of the previous example but also

1

a c term that gives the desired direct coupling b etween physical elds Examination

of the explicit of shows that these are exactly the terms it contains that couple

the rst massive level of the op en string to the elds of the closed string describing

cm

the graviton and dilaton To the gaugeinvariant action the MM term contributes

a

B A in terms of the NL eld B of the graviton and linearized Stueckelb erg eld

a a

A of massive spin and B in terms of the NL eld B of A and the dilaton

a a

ab

while the c term contributes h h in terms of the linearized graviton h and

ab ab

linearized massive spin h and in terms of the other Stueckelb erg eld of

ab

2 2 2

massive spin Q gives further diagonal mass terms A h and We can also

2

a

ab

recognize this as a sp ecial case of the general mixing describ ed ab ove which gives a

1 1

p p

m more direct derivation of this result given by m

0 1

2 2

ACKNOWLEDGMENT

I thank Martin Rocek Fred Goldhab er and for discus

sions This work was supp orted in part by the National Science Foundation Grant

No PHY

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