Arbitrary four string scattering at high energy and fixed angle

619 Pages·2008·5.37 MB·English

NuclearPhysicsB729(2005)1–48
Arbitrary four string scattering at high energy
and fixed angle
Nicolas Moeller∗, Peter West
King’sCollegeLondon,DepartmentofMathematics,LondonWC2R2LS,UK
Received17August2005;accepted12September2005
Availableonline11October2005
Abstract
Wecalculate,usingthegrouptheoreticapproachtostringtheory,thetreeandoneloopscatteringoffour
openandclosedarbitrarybosonicstringstates.Inthelimitofhighenergy,butfixedangle,themulti-string
vertexattreeandonelooplevelsthatwefindtakesaverysimpleform.Wepropose,andpresentarguments
for,aformforthehighenergymulti-stringvertexatallloops;inparticular,wegiveapathintegralderivation
ofthisvertex.OurresultsagreewiththoseofGrossandMendefortachyonscatteringamplitudes,butthose
foranyotherstringscatteringaresubstantiallydifferentfromthatdiscussedin[D.J.Gross,High-energy
symmetriesofstringtheory,Phys.Rev.Lett.60(1988)1229].Wealsodevelopsomeofthetechnologyused
inthegrouptheoreticmethodtocomputeloopcorrections.
2005ElsevierB.V.Allrightsreserved.
1. Introduction
Longagointheearlydaysafterthediscoveryofstringtheory,highenergyfixedanglescat-
teringoftheopenstringfourtachyonscatteringwasconsidered[1].Extremisingtheknownone
loopamplitudeitwasfoundthatthenon-planarprocesspossessedasaddlepointandtheampli-
tudewasevaluatedatthatpoint.Thisstudywastakenupinthelateeighties[2–5].Theseauthors
consideredtheclosedstringinthesamelimitandfoundthattheamplitudeforfourtachyonscat-
tering possesses a saddle point at all genus at which the positions of the Koba–Nielsen points
werefixedandwhosecrossratiowasrelatedtotheangleofscattering.Furthermore,aconcise
formula for four tachyon scattering for any genus was derived in the high energy, fixed angle
* Correspondingauthor.
E-mailaddress:[email protected](N.Moeller).
0550-3213/$–seefrontmatter 2005ElsevierB.V.Allrightsreserved.
doi:10.1016/j.nuclphysb.2005.09.036
2 N.Moeller,P.West/NuclearPhysicsB729(2005)1–48
limit. An important role in this analysis was played by the representation of the string world
surface by the Riemann sphere with two branch cuts whose degree was the genus. In Ref. [5]
itwasproposedthatthereexistedrelationsbetweenanystringscatteringamplitudeandthatfor
thesamenumberoftachyonscatteringthatwasindependentofthegenus.Thestudyoftheopen
stringwasalsoextendedinRef.[4].
Someinterestingpapershavealsoconsideredhighenergyscatteringstressingtheroleofthe
decouplingofzeronormphysicalstates[6].
Inatheorythatpossessesasymmetrythatisspontaneouslybroken,thesymmetry,although
present at energy scales below the breaking scale, is obscured; however, above this scale the
amplitudesdisplaythesymmetryinamoretransparentway.Assuch,ithasbeenhopedthatthe
studyofstringtheoryathighenergywillshedsomelightontheunderlyingsymmetries.Inmore
recenttimesithasbeenproposedthatMtheoryshouldpossessaverylargeKac–Moodyalgebra
calledE [7].Thiswasproposedbystudyingthemaximalsupergravitytheoriesthatencodeall
11
perturbativeandnon-perturbativeeffectsofstringtheoryatlowenergy.Themotivationforthis
paperwastostudytheresultsofRefs.[1–5]toseeifthesecanmakecontactwiththisproposed
largesymmetry.
We will study the high energy, fixed angle, scattering of arbitrary physical bosonic string
states at tree and one loop level. The most efficient method to study arbitrary string scattering
istousemulti-stringscatteringvertices.Theseemergedinitiallyfromfactorizingtachyonstring
scatteringamplitudes,andwascarriedoutintheearliestdaysofstringtheory.Bysewingsuch
verticesthepioneersofstringtheorywereabletocalculatemulti-stringamplitudes,solyingthe
foundations for string theory. In particular, they computed the one loop open string scattering
amplitudes [26–32] and developed a projector which was used to eliminate the ghosts for this
amplitude [33]. For two reviews which set out some of this remarkable progress, see Ref. [8].
These results werefound before our present understandingof the systematicBRST techniques
to add ghosts and so the old results often suffered from the presence of spurious ghosts. Their
other problem was that the multi-string vertices they found emerged from factorising tachyon
amplitudesandtheirpropertieswerenotwellunderstood.
Inthelateeightiesthestudyofmulti-stringverticeswastakenupagainintheso-calledgroup
theoreticapproach[9–22].Theessenceofthisapproachwasthatthemulti-stringverticeswere
the simplest objects in string theory. It was found that multi-string vertices satisfied some very
simplepropertiescalledoverlaprelations,andthesewerethenformulatedinsuchawaythatthey
wereusedtodefinewhatamulti-stringvertexwas.Thisallowedforasystematicconstruction,
andelaborationoftheproperties,ofmulti-stringvertices.Forexample,itallowedtheincorpora-
tionofghostoscillators,firstgiveninanearlierpaper[21],andasystematicmethodtochange
thedependenceofthevertexontheKoba–Nielsenpointsandmoduli.Theamplitudewasfound
bysaturatingthemulti-stringvertexwiththedesiredphysicalstateswhichweretobescattered
and integrating, with an appropriate measure, over the Koba–Nielsen points and moduli of the
Riemannsurfacesweptoutbythestring.Aswehavementioned,thevertexisdeterminedbycer-
tainverysimpleoverlaprelationswhilethemeasureisdeterminedbydemandingthatzeronorm
physical states decouple. Using this technique, the scattering amplitudes for the open bosonic
string[10,12–14],includingghosts[15,16,21],andsuperstring[10,19,20]werecalculated.The
treelevelscatteringfortheclosedstringwasfound,butthisstringwasnotassystematicallyin-
vestigatedastheopenstring.Werectifythisomissioninthispaper.Thegrouptheoreticmethod
was developed in the rather long series of papers [9–22]. Although the early papers contained
theessentialsteps,theyoftenlackedsomeoftheinsightsthatweregainedlateron,andsothey
arenotalwayssoeasytoread. AreviewwasgiveninRef.[18],butwegiveanaccountofthe
N.Moeller,P.West/NuclearPhysicsB729(2005)1–48 3
group theoretic method in Section 2 of this paper in such a way that the rest of the paper can
be understood in a self-contained manner. Some aspects of the group theoretic approach, such
astheintegratedoverlapidentities,wereconsideredintheapproachofRef.[23]whichorigined
[25]inthepropertiesofthesumoverworld-sheetsapproachtostringtheory.
In Section 3 we use the group theoretic method to calculate the high energy, fixed angle,
behaviour of open and closed bosonic strings scattering at tree level, while in Section 4 we
calculatefouropenandclosedbosonicstringscatteringatoneloop.Themulti-stringverticeswe
findinthehighenergylimitdohaveanextremelysimpleform, thatfor oneloopclosedstring
scatteringisgivenby
(cid:2) (cid:4) (cid:5)
(cid:1)N (cid:3)(cid:3)∞
1 1
V = (cid:2)0|exp − αµ(i) α(j)+α¯µ(i) α(j)
c (k) 2n 2n(ς −ς )n 0µ 2n 2n(ς¯ −ς¯ )n 0µ
k=1 i,j n=1 j i j i
(cid:8)
(cid:3) (cid:6)(cid:7) (cid:7)
+ αµ(i)ln (cid:7)(ς −ς )(cid:7)α(j) , (1.1)
0 j i 0µ
i,j
where ς are the positions of the Koba–Nielsen points, which are fixed to be at the beginning
j
andendofthetwobranchcutsontheRiemannspherethatrepresentsthestringworld-sheet.We
conjecture,withsupportingevidence,theresultsforarbitrarygenusinSection7.
In Sections 5 and 6 we further develop some of the aspects of the group theoretic method
associatedwiththecalculationofloopstringscattering.
Ourresultsforfourtachyonscatteringinthehighenergy,fixedangle,limitagreewiththose
ofRefs.[1–4]andsoweagreewiththemajorityoftheworkinthesepapers.However,wefind
thatourresultsforphysicalstatesotherthantachyonsdiffersubstantiallyfromthoseofRef.[5].
Indeed,wefindthatsomeoftheamplitudeswhicharenon-zeroinRef.[5]vanishintheextreme
highenergylimit.Asaresult,wedonotfindtheallgenussymmetryrelationsfoundinRef.[5].
AttheendofsectionfourwepinpointwherewebelievetheproblemsariseinRef.[5].
2. Reviewofthegrouptheoreticapproach
Here we review the group theoretic approach to string theory, which was developed in
Refs. [9–20]. The followingresults will be validfor bothopen andclosed bosonicstrings, and
with suitable extensions the superstrings. As N strings scatter they sweep out a world-sheet
whichisatwo-dimensionalsurfaceofgenusgwithsomemarkedpoints,orpunctures,wherethe
outgoingstrings leavethesurface. We refer to these points as Koba–Nielsenpoints.The string
theory possesses certain conformal operators, denoted generically by the symbol R(z), which
depend on the world-surface, and act on the string Hilbert space H. By definition a conformal
operator of weight d behaves under a conformal transformation ξ →ξ(cid:5)(ξ), where ξ is a local
coordinateontheworld-surface,as
(cid:4) (cid:5)
dξ d
R(ξ)→R(cid:5)(ξ(cid:5))=UR(ξ)U−1=R(ξ) . (2.1)
dξ(cid:5)
HereU istheoperatoronthestringHilbertspacethatcorrespondstotheconformaltransforma-
tionz→z(cid:5)(z).TheoperatorU canbewrittenlocallyintheform
(cid:2) (cid:8)
(cid:3)∞
U =exp a L . (2.2)
n n
n=−∞
4 N.Moeller,P.West/NuclearPhysicsB729(2005)1–48
For the open string we only consider operators of this type, but for closed strings conformal
¯
operators have left and right weights (d,d), but we will mainly only require operators with a
¯
weightwhichiseither(d,0)asaboveor(0,d)inwhichcasewewillhavetheconformaltrans-
formationonthecoordinatez¯ beimplementedbyL¯ .Forexample,fortheopenbosonicstring,
n
the conformal operator is given by ∂Qµ whichhas conformal weightone. Eq. (2.1) also holds
fortheembeddingfielditself Qµ withweightzero.Theenergy–momentumtensor T(z) trans-
formswithweighttwo,butalsohasananomaloustermwhichinvolvestheSchwarzianderivative
oftheconformaltransformation.Inparticular,itobeysT(cid:5)(z(cid:5))=T(z)(dz)2+ c (Sz)(z(cid:5))where
dz(cid:5) 12
(Sf)(z)= f(cid:5)(cid:5)(cid:5)(z) − 3(f(cid:5)(cid:5)(z))2 and c is the central charge associated with T. We also have the
f(cid:5)(z) 2 f(cid:5)(z)
ghostsbandcofconformalweights2and−1,respectively.
Inthegrouptheoreticapproach,thescatteringamplitude,atgenusg,ofN arbitraryon-shell
physicalstates|χ (cid:6)isgivenintermsofavertexV,bytheexpression
i
(cid:11)
(cid:9) (cid:10)
A |χ (cid:6),|χ (cid:6),...,|χ (cid:6) = Π dv Π(cid:5)dz f (z ,v )V|χ (cid:6) |χ (cid:6) ···|χ (cid:6) .
1 2 N r r i i m i r 1 (1) 2 (2) N (N)
M
g,zk
(2.3)
HereM isthemodulispaceofRiemannsurfacesofgenusg withN markedpointsz ,i.e.,
g,zk i
theKoba–Nielsenpoints.Theintegrationrunsoverboththez andoverthemoduliv .Theways
i r
of representing the two-dimensional surface will be discussed later. The function f (z ,v ) is
m i r
the measure, and V is what we call the vertex. It is a multi-linear map from N copies of the
Hilbertspaceofthestringintothecomplexnumbers:
V :H ⊗···⊗H →C. (2.4)
1 N
It depends on the Koba–Nielsen variables and on the moduli. The same vertex provides the
scatteringforanyphysicalstatesonceitissaturatedwiththecorrespondingphysicalstatesasin
Eq.(2.3).Thusthevertexdoesnotdependonwhatphysicalstatesarescattering,andtherefore
onceweknowthevertexandthemeasure,allscatteringamplitudesatgenusgaretreatedonthe
samefooting.Wewillnowdefinewhatismeantbythevertexandshowhowitandthemeasure
aredetermined.
ThestringvertexV isdefinedbydemandingthatitsatisfytheoverlapequations
(cid:4) (cid:5)
dξ d
VR(i)(ξ )=VR(j)(ξ ) j . (2.5)
i j
dξ
i
Hereξ andξ denotecoordinatepatchesonthestringworld-surfacethatarevalidintheneigh-
i j
bourhoodoftheKoba–Nielsenpointsz andz ,respectively.Thecoordinateξ associatedwith
i j i
the Koba–Nielsen point z is required to vanish at this point, i.e., ξ (z ) = 0, and similarly
i i i
ξ (z )=0. Furthermore, the two coordinates are related in their overlap region by an analytic
j j
transformation which we denote by ξ =τ (ξ ). In Eq. (2.5) it is important to realise that the
j ji i
conformal operator R(j) of weight d acts on the jth leg, or string Hilbert space of the vertex.
Thisequationstatesthatwhentheactionoftwoconformaloperatorsonthevertexaredefinedin
anoverlapregiontheneachoperatorgivesthesameresultwhenreferredtothesamecoordinate
system.Eq.(2.5)holdsfortheconformaloperatorsusedtoconstructthestringunderconsidera-
tion.Fortheclosedbosonicstringtheseincludexµ and∂xµ andtheenergy–momentumtensor
T providedoneincludesitsanomalousterm.Ofcourse,someoverlapidentitiescanbederived
asaconsequenceofsomeothers.Theobviousexamplebeingthat∂xµ followsfromtheidentity
for xµ bydifferentiationoftheoverlapequation.AtfirstsightitmayappearthatEq.(2.5)just
N.Moeller,P.West/NuclearPhysicsB729(2005)1–48 5
relatestheconformalpropertiesofanoperatorR ofweightd,however,theoperatorsR onthe
left- and right-hand side of the equation act on different one-string Hilbert spaces in a highly
non-trivialmanner.
Oneisfreetochoosewhatevercoordinatesystemonelikesprovideditobeystheabovecri-
terion, and one finds correspondingly different string vertices. In fact the vertex only depends
onthetransitionfunctions τ betweenthecoordinatepatchesratherthanthelocalcoordinates
ji
themselvesasonlytheformerquantitiesactuallyenterintotheoverlapequations.Toseethiswe
writeξ inEq.(2.5)asτ (ξ )andwenotethatξ playstheroleofadummyvariable.Itisoften
j ji i i
usefultoreferthecoordinatesξ tosomeunderlyingcoordinatez,onechoiceofwhichwewill
j
discussshortly,andthenwemaywrite
(cid:3)∞
ξ (z)= ai(z−z )n≡(V )−1(z), ai (cid:9)=0. (2.6)
i n i i 1
n=1
Intermsofthecoordinatez,theKoba–Nielsenpointsaredenotedbyz=z ,i=1,...,N,and
i
it has the above form as ξ (z )=0. We note that the transition functions are given by τ =
i i ji
(V )−1V . Thechoiceof anylocalcoordinates ξ , subjecttotheaboveconditions,corresponds
j i i
to taking any values of the constants a . However, as we will see different choices of local
n
coordinatesleadtothesamephysicalscatteringamplitudes.
TheexpressionVR(i)(ξ )canbeexpressedasaLaurentseriesinξ aroundξ =0,butwhen
i i i
ξ approaches another Koba–Nielsen point, say z , this series diverges in general. To obtain a
i j
convergent expression as one approaches z one uses the overlap identity of Eq. (2.5) in the
j
overlap region where both VR(i)(ξ ) and VR(j)(ξ ) are convergent, and then approaches the
i j
pointz usingthelatterexpression.Assuch,thedomainofvalidityofthecoordinateξ isdefined
j i
in this sense of convergent series rather than in the sense of coordinate patches as used in the
theoryofdifferentiablemanifolds.
Forscatteringatgenuszerotheworld-sheetisjusttheRiemannspherewiththemarkedKoba–
Nielsen points and we can introduce a coordinate z which covers the whole sphere with the
exception of the point at infinity where we use the coordinate ζ =1/z. If none of the Koba–
Nielsen points are at infinity, one can use the local coordinates ξ =z−z , but there are of
i i
courseaninfinitenumberofotherchoices.ShouldithappenthatonehasaKoba–Nielsenpoint
atinfinitythenonecanusethelocalcoordinateζ forthatstring.
For surfaces of genus greater than zero the string world-surface is a Riemann surface with
marked points and handles. We may use either the Schottky or Fuchsian representation of the
surface.IntheFuchsianrepresentationoftheRiemannsurface,everyRiemannsurfaceofgenus
g (cid:1)2 is conformally equivalent to the quotient of the upper-half plane by a fixed point free
Fuchsian group (a discontinuous subgroup of PSL(2,R)). While a genus one surface can be
mappedtoaparallelogramonthecomplexplanewhoseopposingsidesareidentified.Hence,for
allgenussurfacesweneedonlyonecoordinatez,withcertainidentifications
z∼P z, n=1,...,2g, (2.7)
n
whereP aregeneratorsoftheFuchsiangroupifg(cid:1)2,orinthecaseofgenusonethetwotrans-
n
lationsthatidentifytheopposingsidesoftheparallelogram.Whenconsideringtheopenstring
wewilldoubletheopenstringworld-surfacetoworkwithamoreconventionalRiemannsurface.
Wenowdemandthatthevertex,bydefinition,satisfythefurtheroverlapconditions,namely
(cid:4) (cid:5)
(cid:9) (cid:10) d(Piξ ) d
VR(i)(ξ )=VR(i) Piξ n i , n=1,...,2g. (2.8)
i n i dξ
i
6 N.Moeller,P.West/NuclearPhysicsB729(2005)1–48
EssentiallythisequationjuststatesthatVR isdefinedontheactualsurface.UsingEq.(2.5)to
changeEq.(2.8)fromlegi tolegj wefindthat
Pj =τ Pi(τ )−1 or V Pi(V )−1=V Pj(V )−1≡P . (2.9)
n ji n ji i n i j n j n
In carrying out this step we have expressed ξ as ξ =τ ξ , the fact that the argument Piξ of
i i ij j n i
R(i)changesusingtheoverlaptoR(j)withargumentτ (Piξ )andthat(τ )−1=τ .Inthelast
ji n i ji ij
partofEq.(2.9)wehaveusedtheexpressionforthetransitionfunctionsintermsoftheV and
i
asaresult
Pi =(V )−1P V . (2.10)
n i n i
We recognise P as the identification expressed in terms of the z coordinate system as in
n
Eq. (2.7). The story for the Schottkyrepresentationof the surface is similar, but one has fewer
overlap conditions as one has fewer identifications. The reader is referred to Ref. [13] for an
account.
Hence, the vertex V is just defined to satisfy Eq. (2.5) and, in the case of loop corrections,
Eq. (2.8). What is far from obvious, but true, is that these two equations determine the string
vertex V uptoafunction(cid:12)of theKoba–Nielsencoordinates zi andmoduli vr. Onemaychoose
this function by taking V |0(cid:6) =1, but any other choice of the function is acceptable and
i (i)
willbetakenintoaccountwhenonecomputesthemeasure.
TheamplitudeofEq.(2.3)alsoinvolvesthemeasuref whichisdeterminedfromtherequire-
mentthattheamplitudeshouldvanishifanyofthe|χ (cid:6)isazeronormphysicalstate.Thismeans
i
that
(cid:9) (cid:10)
A |χ (cid:6),|χ (cid:6),...,|χ (cid:6) =0 (2.11)
1 2 N
if any of the |χ (cid:6) are physical zero norm states. This requirement is similar to the decoupling
i
ofsuchstatesinquantumelectrodynamicsandsowecanthinkofEq.(2.11)asakindofWard
identity.Infact,thisfixesthemeasuref uptoaconstant.Theconstantisdeterminedasinthe
m
old days of sewing, or the sum over world-surfaces approach, by requiring that the scattering
amplitudeisunitary.
To summarize the group theoretic approach, the string vertex V is by definition an object
which obeys the overlap Eqs.(2.5) and (2.8) corresponding to the conformal operators used to
definethestringtheory,thestringscatteringamplitudeisgivenbyEq.(2.3)andthemeasureis
determinedbydemandingthatzeronormphysicalstatesdecouple.Thisisthecompletesystem
requiredtofindthescatteringofarbitrarystringstates.
Mostofthevertexcanbefoundmoreeasilyfromtheso-calledintegratedoverlapidentities
whichwenowderivefromtheunintegratedoverlapidentitiesabove.Letusconsidertheexpres-
sion
(cid:4)(cid:13) (cid:5)
V dξ ϕR(i)(ξ ) , (2.12)
i i
0
where ϕ is a meromorphic tensor of weight 1−d, that is under a conformal transformation it
transformsas
(cid:4) (cid:5)
dξ 1−d
ϕ(ξ)→ϕ(cid:5)(ξ(cid:5))=ϕ(ξ) . (2.13)
dξ(cid:5)
N.Moeller,P.West/NuclearPhysicsB729(2005)1–48 7
As a result, VϕR(i)(ξ ) is a tensor of weight one, or simply, a one form. Let us now con-
i
sider deforming the contour around z such that it encloses any other non-analytic behaviour
i
of VϕR(i)(ξ ). We take an arbitrary ϕ of weight 1−d provided it is analytic except for poles
i
aroundtheotherKoba–Nielsenpoints.ThuswefindasetofcontourintegralsaroundtheKoba–
Nielsen points. In carrying out this contour deformation one approaches the neighbourhood of
otherKoba–Nielsenpointsand,asexplainedabove,onemustusetheoverlapEq.(2.5)tomain-
tainconvergence.Thustheoverlaprelationsaretheessentialingredientinthederivation.Forthe
case of genus zero the contour is pulled off to infinity where it vanishes, provided this is not a
Koba–Nielsenpoint,whileathighergenuswefindacontouraroundtheboundaryoftheregion
which represents the Riemann surface. Since ϕ is defined on the surface, that is it is suitably
periodicmeaning
(cid:4) (cid:5)
(cid:9) (cid:10) d(Piξ ) 1−d
ϕ(ξ )=ϕ Piξ n i , (2.14)
i n i dξ
i
thiscontourvanishesusingEq.(2.8).Thuswefindtheintegratedoverlapequation
(cid:4)(cid:13) (cid:4) (cid:5) (cid:5)
(cid:3) dξ d−1
V dξ R(j)(ξ )ϕ j =0, (2.15)
j j
dξ
i
j
0
wherewehavekeptexplicitthetransformationofϕunderachangeofcoordinatesbywritingthe
derivativeterminfrontofϕ;thisiscons(cid:14)istentw(cid:15)iththepreviousliteratureonthegrouptheoretic
approach. We could also have written V( dξ R(j)(ξ )ϕ(ξ ))=0, but then we have to
j 0 j j j
rememberthatϕ isatensorthattransformsasin(2.13).
ThevertexbydefinitionsatisfiestheoverlapEqs.(2.5)and(2.8),andtodiscoveritsproperties
it is often simpler to find how these equations behave under the desired transformation than to
look into the details of the vertex itself. In particular we now consider the effect of conformal
transformationsonthevertex.Letusconsideraconformaltransformationξ →N(ξ)anddenote
bythesamesymbolN theoperatoroftherealisationofthesameconformaltransformationon
thestringFockspace.ThereadercandistinguishfromthecontextwhichN isrequired.Wecan
carryoutdifferentconformaltransformationsonthedifferentstringHilbertspaces,andasbefore
thesuperscriptindicateswhichHilbertspacetheoperatorisactingon.Forinstance,N(i)actson
theithHilbertspace.Wecandefineanewvertexby
V(zˆ ,vˆ ,ξˆ )=V(z ,v ,ξ )Π N(k). (2.16)
i r i i r i k k
TheactionoftheconformaltransformationmaychangetheKoba–Nielsenpointsz ,themoduli
i
v andthelocalcoordinatesξ usedtodefinethevertex.Wedenotethenewquantitieswithhats
r i
and show the explicit dependence of the vertex on them. The subscript on N(k) indicates the
k
possibilitytocarryoutdifferentconformaltransformationsondifferentstringHilbertspacesor
stringlegs.IngeneraltheoperatorN hastheform
(cid:2) (cid:8)
(cid:3)∞
N(k)=exp ckL(k) , (2.17)
k n n
n=−∞
whereck arearbitraryconstants.Undertheaboveconformaltransformationthevertexmayalso
n
scale by a function of v and z which we have not explicitly indicated. As mentioned above,
r j
thebehaviourofthevertexunderaconformaltransformationcanbeunderstoodbystudyingits
effectontheoverlapequations.UsingthepropertiesofconformaloperatorsofEq.(2.1)wefind
8 N.Moeller,P.West/NuclearPhysicsB729(2005)1–48
thattheoverlapofEq.(2.5)becomes
(cid:4) (cid:5)
(cid:9) (cid:10) (cid:9) (cid:10) d(N−1(ξ )) d
V(zˆ ,vˆ ,ξˆ )R(i) N−1(ξ ) =V(zˆ ,vˆ ,ξˆ )R(j) N−1(ξ ) j j . (2.18)
i r i i i i r i j j d(N−1(ξ ))
i i
Writingξ =τ (ξ )andmakingthechangeinthedummyvariable(N(i))−1ξ toξˆ wefindthat
j ji i i i i
theoverlaprelationbecomes
(cid:4) (cid:5)
(cid:9) (cid:10) d(N−1τ N (ξˆ )) d
V(zˆ ,vˆ ,ξˆ )R(i)(ξˆ )=V(zˆ ,vˆ ,ξˆ )R(j) N−1τ N (ξˆ ) j ji i i . (2.19)
i r i i i r i j ji i i ˆ
dξ
i
Interpreting the argument of the operator R on the right-hand side of this equation as the new
local coordinate, ξˆ =τˆ (ξˆ ), we find that the new local coordinates and Koba–Nielsenpoints
j ji i
ofthevertexaftertheconformaltransformationaregivenbytheequation
τˆ =(N )−1τ N (2.20)
ji j ji i
foralli,j =1,...,N.TheKoba–Nielsenpointsareencodedastheyarethepointswhereξˆ =0
j
for each j. Carrying out a similar computation on the overlap Eq. (2.8), we conclude that the
moduliofthevertex,whichareencodedintheidentificationsPi,aftertheconformaltransfor-
n
mationaregivenbytheequation
Pˆi =(N )−1PiN , i=1,...,N. (2.21)
n i n i
Althoughonecanworkentirelyintermsoftheoverlapfunctions,itisadvantageoustoworkin
termsofthecoordinatezdiscussedabove.Tothisendweuseτ =(V )−1V andthenEq.(2.20)
ji j i
becomes
V N (Vˆ )−1=V N (Vˆ )−1≡N or N =(V )−1NVˆ . (2.22)
j j j i i i i i i
We recognise N as a conformal transformation acting on the z coordinate system. Using
Eqs.(2.9)and(2.22),wefindthatEq.(2.21)canbewrittenas
Pˆ =(N)−1P N. (2.23)
n n
Letusconsideraconformaltransformationwhichleavesthemoduliv inert,butchangesthe
r
positionsoftheKoba–Nielsenpointsandthelocalcoordinatesystemsξ →ξˆ =(Vˆ )−1zaround
i i i
eachKoba–Nielsenpoint.FromEq.(2.23)suchatransformationwillhaveP =Pˆ andsowe
n n
maytakeN =I and,fromEq.(2.22),haveN =(V )−1Vˆ .Actingonthevertexwehave
i i i
(cid:1)N
V(v ,zˆ ,ξˆ )=V(v ,z ,ξ ) (V )−1Vˆ . (2.24)
r j j r j j i i
i=1
ˆ
The new Koba–Nielsen points are given by the points where ξ vanish for each j. This is
j
ξˆ (zˆ )=0, or equivalently (Vˆ )−1(zˆ )=0. Transformations N of this kind are all analytic
j j j j i
transformations. Those for which N (0)=0 do not change the positions of the Koba–Nielsen
i
pointsandjustdefinenewlocalcoordinatesaroundeachKoba–Nielsenpoint.Theseareofthe
formof(2.17)withthesumforn(cid:1)0.ThosethatalsoshifttheKoba–Nielsenpointsalsoinclude
L−1 inthesum.SincephysicalstatesareannihilatedbyLn−δn,0,n(cid:1)0theformertransforma-
tionsdonotaffectphysicalamplitudesexceptforascalingduetotheL term.However,theL
0 0
N.Moeller,P.West/NuclearPhysicsB729(2005)1–48 9
termmayscalethevertexbyafactor,butthisiscompensatedforwhenthemeasureiscalculated.
ThesameconclusionappliestothetransformationsthatshifttheKoba–Nielsenpoints.
Nowweconsideraninfinitesimalconformaltransformationwhichdoesnotchangez ,orthe
j
localcoordinates,butchangesthemoduliv →vˆ =v +(cid:11)h (v).Inthiscase,V =Vˆ ,andwe
r r r r j j
areleftwithN.ThenEq.(2.23)appliedtothecoordinatezbecomes
P (z)+(cid:11)f(z)dPn(z) =Pˆ (z)+(cid:11)f(P z), (2.25)
n n n
dz
where
N(z)=z+(cid:11)f(z). (2.26)
Aninfinitesimalconfo(cid:15)rmaltransformationz→z+(cid:11)f(z)isrealisedonthestringHilbertspace
bytheoperatorI+(cid:11) T(z)f(z).Consequently,implementingtheconformaltransformationof
0
Eq.(2.26)onthevertexwefindthatEq.(2.16)isgivenby
(cid:2) (cid:8)
(cid:13)
(cid:3)g (cid:3)N
∂V
h (v) =V dξ T(j)(ξ )fj(ξ )+c , (2.27)
r j j j f
∂v
r=1 r j=1ξj=0
where c is a function of the moduli and Koba–Nielsen points and ξ +(cid:11)fi(ξ )=N (ξ )=
f i i i i
(V−1NV )(ξ ).Theconstantc isrequiredtotakeaccountofthepresenceofamultiplierwhich
i i i f
has not been explicitly written in Eq. (2.16) and the fact that the vertex is only determined up
to a function of the moduli and Koba–Nielsen points by the overlap relations. The value of c
f
is easily determined by taking a suitable state, such as the vacuum state, on either side of this
equation.Werecall,however,thattofindtheactualscatteringamplitude,wemultiplythevertex
bythemeasuref andintegrateoveralltheKoba–Nielsenvariablesandmoduli.Thefinalresult
m
isindependentofwhatchoiceofmultiplicativefactorinfrontofV onetakes.
It is instructive to consider conformal transformations that do not change the vertex, that is
theychangeneitherv ,z ,northelocalcoordinatesystem.Whenwehaveloopsthismeansthat
r j
P =Pˆ andsowerequire
n n
P−1NP =N. (2.28)
n n
IftherearenoloopsthentherearenoidentificationsP andsoN isunrestricted.Theconformal
n
transformationsoneachlegaregivenbyEq.(2.22)withV =Vˆ andso
i i
N =(V )−1NV . (2.29)
i i i
For infinitesimal transformations, we take Eq. (2.25), with P =Pˆ , which means that f(z)
n n
transforms as a vector field (i.e., a tensor of rank −1) and is defined on the Riemann surface
itself.Thuswefindanequationoftheform
(cid:16) (cid:13) (cid:4) (cid:5)(cid:17)
(cid:3)N
dξ c dξ
V dξ ϕ Tj(ξ ) j + (Sξ )(ξ ) i =0. (2.30)
j j j i
dξ 12 dξ
j=1ξj=0 i j
ThisisjustanexampleoftheoverlapEq.(2.5)foundbytakingR(z)=L(z)whichhasconformal
weight2andwehaverecognisedtheconstanttobetheSchwarzianderivative.Thesetofsuch
transformations which preserve a g loop vertex with N external lines form a group which is
discussedinRef.[17].
10 N.Moeller,P.West/NuclearPhysicsB729(2005)1–48
Underthetransformations
V →SV , P →SP S−1 (2.31)
j j n n
the vertex is inert as it only depends on Pi of Eq. (2.10) and the transition functions τ =
n ji
(V )−1(V )andsincetheseareinertundertheabovetransformationstheresultfollows.Indeed,
j i
the transformation properties of the vertex can be summarized by writing the dependence of
the vertex V on v and z , ξ through P and V , i.e., V(P ,V ) and the SU(1,1) invariance
r j j n j n j
becomes V(SP S−1,SV )=V(P ,V ). This invariance is a feature of the redundancy of the
n j n j
representationintermsofP andV thatwehaveused.Theconformaltransformationproperty
n j
ofthevertexembodiedinEqs.(2.22)and(2.23)maybewrittenas
(cid:1)N (cid:9) (cid:10)
V(P ,V ) N(i)=V N−1P N,Vˆ . (2.32)
n j i n j
i=1
WehaveagainnotexplicitlywrittenthepossiblescalingofV byafunctionofv andz .
r j
Asstatedabove,themeasureisdeterminedbyensuringthatzero-normphysicalstatesdecou-
ple.Fortheopenbosonicstringthesestatesareoftheform
(cid:4) (cid:5)
3
L−1|Ω(cid:6), L−2+ 2L2−1 |Ω(cid:5)(cid:6) (2.33)
where
L |Ω(cid:6)=L |Ω(cid:5)(cid:6)=0, n>0, L |Ω(cid:6)=(L +1)|Ω(cid:5)(cid:6)=0 (2.34)
n n 0 0
together with higher level zero norm states. The states for other strings have a similar form;
they consist of operators which generate the symmetry algebra of the string in question acting
onastatewhichishighestweightwithrespecttothealgebra.Saturatingthevertexwithoneof
thesestateswemayinterpretitasaninfinitesimalconformaltransformationappliedtothevertex
times |Ω(cid:5)(cid:6) or |Ω(cid:6). Such transformations will either preserve the vertex, or change its Koba–
Nielsencoordinatesormoduli.Changesoflocalcoordinatesystemsareirrelevantasthevertex
issaturatedwithphysicalstates.Consequently,wefind,generically,that
(cid:11) (cid:4) (cid:5)
(cid:1) (cid:1) (cid:3) (cid:3)
(cid:5) ∂V ∂V
dz dv f h + g −cV |χ (cid:6) ···|Ω(cid:6) ···|χ (cid:6) =0,
k r m r j 1 (1) (i) N (N)
∂v ∂z
k r r r j j
(2.35)
whereallthestates|χ(cid:6) arephysicalandsoobey(L −δ )|χ(cid:6)=0,n(cid:1)0and
(i) n n,0
v →v +h (v ,z ), z →z +g (z ,v ) (2.36)
r r r r j j j j j r
arethechangesthattheinfinitesimalconformaltransformationinducesonthevertex.Wefind,
byintegratingbyparts,that
(cid:3) (cid:3)
∂ ∂
0= (h f )+ (g f )+cf (2.37)
r m i m m
∂v ∂z
r r i i
in other words a first-order differential equation for the measure, f . In fact, one has as many
m
differentialequationsasonehasKoba–Nielsenvariablesandmoduliandsothemeasureisde-
terminedbythem.Inthiscalculationwehaveneglectedboundarytermswhichingeneralarenot
zeroandcorrespondtoaB.R.S.Tanomaly.Indeed,allopenstrings,exceptforthegaugegroup
SO(213),havesuchananomaly[12,25].