QCD Sum Rules: Bridging the Gap between Short and Large Distances

0.2 MB·English

by  M. Shifman

FTPI-MINN-10/28,UMN-TH-2921/10
October7,2010
QCD Sum Rules: Bridging the Gap between Short and Large Distances
M.Shifman
WilliamI.FineTheoreticalPhysicsInstitute,UniversityofMinnesota,Minneapolis,MN55455,USA∗
and
JeffersonPhysicalLaboratory,HarvardUniversity,Cambridge,MA02138,USA
1
1 Abstract
0
2 IdiscussaspectsoftheQCDsumrulemethodwhichattractedtheorists’attentioninearnestatarelativelylatestage
n and are not yet fully solved. At first I briefly review such general topics as the structure of the operator product
a expansion in QCD and intrinsic limitations of the quark-hadron duality concept. In the second part I comment on
J holographic constructions — a focus of the current efforts to say something new on QCD at strong coupling. Of

6 particular interest to me is the recent derivation of the vacuum magnetic susceptibility due to Son and Yamamoto.

Remarkably, theirresultisthesameasthatobtainedpreviouslybyVainshteininthefield-theoreticframework. For
]
h reasonswhichIdonotunderstandatthemoment,theVainshteinformula,unexpectedly,isnotsuccessfullnumerically.
p Thisisaslightlymodifiedversionofthetalkdeliveredat15thInternationalQCDConference“QCD10,”June28
- -July3,2010,Montpellier,France. PublishedinNuclearPhysicsB(Proc. Suppl.) 207–208,298–305(2010).
p
e
h
[ 1. Introduction and,finally,

1
v Iwasaskedtoopenthisspecialsessiondevotedtothe • Fast-forward in the past (some remarks on
2 QCDsumrules(sometimesreferredtoastheSVZsum AdS/QCD).
2 rules) with a brief outline of the current status of this
1 Let me first recall that the basic concept of the SVZ
methodinvariousapplicationstohadronicphysicsand,
1 sumrules,itsfoundation,isthefollowingidea:
perhaps,somehistoricalremarks. Instead,Idecidedto
.
1 do something else. I will completely skip the second
The QCD vacuum structure is complicated and not
0
part since quite recently I published a paper [1] where
1 yet fully analytically understood despite significant
thereadercanfindboththedescriptionoftheevolution
1 progress, especially after the advent of supersymmetry
: of the method and relevant anecdotes (see also [2]). I inthisrangeofquestions[3]. Forlimitedpurposesone
v
willnotdwellonvarious(quitefruitful)recentapplica-
i cantrytorepresenttheQCDvacuumbyafewvacuum
X tionswhichwillbe(hopefully)coveredbyotherspeak-
expectation values (VEVs) of local operators intended
r ers. Instead, I will focus on some “afterthoughts” of a to theoretically approximate various correlation func-
a
generalnature,someissueswhichwerenotexplored(or
tionsinanintermediatedomainofdistances—between
notfullyexplored)induetime,whentheSVZsumrules
short and asymptotically large. The gluon and quark
wereinthemaking. Theycameintothelimelightinthe
condensatesaremostimportant,butoneiswelcometo
lasttenyearsorso. Threemaintopicsare
addafewothers. Therearemanyreasonsnottoaddtoo
many,though.Oneofthem,asIwilldiscusslater,isthe
• The structure of the operator product expansion
fact that OPE in QCD presents an asymptotic expan-
(OPE)inQCD;
sion. ThisisincontradistinctionwithOPEinanumber
• Thelimitationsofthequark-hadronduality; of conformal field theories (with no intrinsic scale) in
whichitisbelievedtobebetterconvergentorjustcon-
vergentinthecoordinatespace. (Isit? Agoodquestion
∗Permanentaddress foraseriousreflection...).
PreprintsubmittedtoNuc.Phys.(Proc.Suppl.) January7,2011
thepower-likeapproachtoconformality. Inthiscaseit
ismostnaturalnottointroduceaspecialnormalization
point µ. Instead, it is implicitly assumed to coincide
withtheexternalmomentumqordistance x. OPEwas
thoughtofastheexpansioninsingularitiesat x → 0or
q→∞.
The OPE (fusion rules) in the above form are fitted
for the conformal theories (Polyakov et.al.) where dy-
namicsatallscalesisthesame. Intwodimensionsthe
fusion rules are powerful enough to fully solve some
Figure1: GraphicrepresentationoftheSVZsumrulesinthe
correlationfunction(cid:104)ψ¯γµψ(x),ψ¯γνψ(0)(cid:105). CFTs[5].Infour-dimensionalN =4super-Yang–Mills
theoriesconformalsymmetryleadstomiracles.
Alas... ourworldisfarlessperfect.
Theabovecondensateexpansionintheintermediate
domain of distances can be matched by the sum over • InQCD:
hadronic states, as in Fig. 1. The sum is infinite, but 1) The UV fixed point is at α = 0. Approach to
s
onecanenhancethecontributionfromthelowest-lying the asymptotic limit is very slow — only logarithmic.
statesthroughBorelization. Most operators cannot be defined as absolutely local.
Anomalousdimensionsarealsologarithmic.
2) A dynamical scale of distances Λ−1 is generated
2. OperatorProductExpansion
throughdimensionaltransmutation. Interactionsonthe
oppositesidesofΛ−1aredrasticallydifferent. Nearand
The theoretical basis of any calculation within the
belowΛ−1perturbativecalculationsareinapplicable.
SVZ method is the operator product expansion. It al-
3)Withoutintroductionofanormalizationpointµ,a
lowsonetoconsistentlydefineandbuildthe(truncated)
slidingboundarybetweenthetwodomains—shortvs.
condensate series for any amplitude of interest in the
largedistances—constructionofOPEismeaningless.
Euclidean domain. The physical picture lying behind
As a result, the Wilson–Polyakov original formula
OPE is a consistent separation of short and large dis-
was not implemented as far as subleading power cor-
tance contributions. The latter are then represented by
rections are concerned. An appropriate modification
thevacuumcondensateswhiletheformerareaccounted
for in the coefficient functions. Somewhat symboli- of OPE was needed, fit for a very special QCD envi-
ronment, with its logarithmic approach to the asymp-
cally,theFourier-transformedcorrelationfunctionscan
totic limit. At the same time, the general Wilson’s
berepresentedas
renormalization-group ideas are perfect! We did not
(cid:88)
D(q2)= C (q;µ)(cid:104)O (µ)(cid:105) (1) change them in a conceptual way. Rather, we engi-
n n
neered its implementation most appropriate for QCD.
n
Wilson’sideawasfinallyfullyadaptedtotheQCDenvi-
where D(q2) is, say, a two-point function, and the nor-
ronmentin[6],withquiteanumberofsuccessfulforays
malization point µ is indicated explicitly. The sum in
beginningfrom1978,invariousSVZpapers.
Eq. (1)runsovertheLorentzandgaugeinvariantoper-
ators built from the gluon and quark fields. The oper- Thus, Wilson’s OPE was redesigned, with an im-
atorofthelowest(zero)dimensionistheunitoperator portant technical addition: we understood that power-
I,followedbythegluoncondensateG2 , ofdimension suppressed terms of OPE need not be necessary dis-
µν
four. The four-quark condensate gives an example of carded compared to logarithmically-suppressed terms,
dimension-sixoperators. TheOPEin(1)is,inasense, evenifonecannotsumuptheentirelogarithmicseries.
abook-keepingprocedure. They need not be smaller. I remember that implanting
InthisformWilsondesigneditfortheorieswithaUV this idea in the minds of more formal theorists was a
fixedpointatα (cid:44) 0[4](WilsonconsideredOPEinthe difficulttask. Franklyspeaking,evennowaformalthe-
coordinate space. The same was done by Polyakov in oreticaljustificationforthisprocedureislacking. Butit
unpublishedlecturesonthistopicthatcirculatedabout works!
this time. In this case all x dependence is encoded in • Itwasrealizedthatthedegreeoflocalityofthe
the set of coefficientsC .) Theories with the UV fixed operators O (0) is regulated by an external parameter
n n
point at α (cid:44) 0 are conformal at short distances, with µ. Evenifthefusionoperators Aand Bhavevanishing
2
anomalous dimensions (for instance, conserved vector areortheorderofρ−1. Beinganalyticallycontinuedin
0
currents), the coefficientsCAB(x) and O (0) depend on theMinkowskidomain,theimaginarypartoftheinstan-
n n
µ, in particular, through logarithms of µ, although the toncontributiononthecutoscillatesandisonlypower
overallcombination suppressed.
(cid:88) A very similar situation takes place in an alternative
CAB(x)O (0)
n n construction,theso-calledresonancemodelworkedout
n
in Ref. [10]. Both models are on the market for quite
isµindependent! Inpractice,tomaketheSVZmethod a time, but — alas — there were essentially no recent
workforlow-lyinghadronicstatesoneshouldbeableto advancesinthisdirection. Anyfreshideasonpossible
sail between Scylla and Charybdis of contradictory re- betterormorecompellingmodelsthatwouldliecloser
quirementsonµ. Ifµistoohighortwolow,onelooses tofirstprinciplesaremostwelcome!
the predictive power! The value of µ is to be carefully Now, let me recall that not only the condensate se-
balanced. I won’t go into details here, referring to [7], ries,buttheα seriesperseareasymptotic.Someofthe
s
andassertingthatthenecessarybalancecanbeachieved high-orderdivergencesintheα seriescanbeabsorbed
s
intheso-calledpracticalversionofOPE.Thisisafor- inthecondensatesnormalizedatµ(e.g. infraredrenor-
tuitousspecialfeatureofQCDwhichisnotnecessarily malons). Others still can show up as nonperturbative
shared by simpler theories used to model various as- terms in the expansion coefficientsCAB(0), which, un-
n
pects of QCD. An example of such a simple theoreti- fortunately,continuetoberatherpoorlycontrollabletill
cal laboratory is given by two-dimensionalCP(N −1) present. Ourknowledgeofthesetermsissemi-empiric.
model[8]. UnlikethesituationinCP(N −1),inQCD, Theyareknowntobenumericallysmallandneglectable
if µ is judiciously chosen, the coefficients CnAB(0) are in a variety of channels (but not in all! see [11] where
mostlydeterminedbyperturbationtheorywhilethecon- exceptionalchannelsarediscussedatlength)underaju-
densates(cid:104)On(0)(cid:105)aremostlymostlynonperturbative.We dicialchoiceofthenormalizationpointµ.
arelucky. ConcludingthispartofthetalkIcansummarizethe
• TheOPEexpansionrunsinpowersof1/Q2 and general achievements in understanding OPE in QCD
lnQ2. as follows: It became clear that the overall structure
• The OPE expansion is asymptotic at best. The of this expansion is highly complicated – much more
factthatthecondensateseriesisfactoriallydivergentin complicated than, say, in the (exactly solved part of)
high orders is rather obvious from the analytic struc- N = 2 super-Yang–Mills [3]. Even in the Euclidean
ture of the polarization operator D(Q2). In a nut shell, domain this expansion is asymptotic in various ways
sincethecutinD(Q2)runsallthewaytoinfinityalong andincludescontributionsofdifferentnature. Itisquite
the positive real semi-axis of q2, the 1/Q2 expansion timelytotrytocategorizethosecontributionswhichgo
cannot be convergent. The actual argument is some- beyond the truncated condensate series containing just
whatmoresubtle[9]butthefinalconclusionisperfectly afewterms. Thisisdifficult. Isitdoable?
transparent. Itisintuitivelyclearthatthehigh-ordertail
of the (divergent) power series gives rise to exponen-
3. Quark-hadronduality
tially small corrections ∼ exp(−Qσ) (where σ is some
criticalindex)intheEuclideandomainwhichoscillate Thissectioncouldhavebeenentitled“Parametrizing
andsuppressedbypowersofenergy(orotherappropri- ignorance.” If theoretical calculation is hard enough in
ateparameters)intheMinkowskidomain. the Euclidean domain, the problem is immensely ex-
The numerical value of σ is correlated with the rate acerbated upon transition to the Minkowski kinemat-
of divergence of high orders in the power series. This ics which is necessary for two related reasons: (a) es-
isexplainedingreatdetailinSection8of[10]. Atthe timate of the continuum in the SVZ sum rules (not so
moment very little is known about this rate from first crucialinthemajorityofinstances);(b)predictionsfor
principles, if at all. The best we can do is to rely on highly excitedstates (crucial ina numberof problems,
toy models [10]. The simplest example is provided by suchasrestoration/nonrestorationofchiralsymmetryin
instantons. One has to fix the size of the instanton ρ highexcitations[12],hadronicwidthsofτandsimilar,
by hand, ρ = ρ . Then the fixed-size instanton contri- ReggetrajectoriesatlargeN andsoon).
0
bution in the Euclidean domain is indeed exponential, IfonecouldcalculateD(Q2)intheEuclideandomain
O(exp(−Qρ )). The exponential factor is the price we exactly,onecouldanalyticallycontinuetheresulttothe
0
pay for transmitting the large momentum Q through a Minkowski domain, and then take the imaginary part.
softfieldconfigurationwhosecharacteristicfrequencies Thespectraldensityρ(s) obtainedinthiswaywould
theor
3
present the exact theoretical prediction for the measur- thex2 →∞behavior.
ablehadroniccrosssection. Therewouldbenoneedfor Inbothcasesthedualityviolatingcomponentfollows
duality. thesamepattern:exponentialinEuclideanandoscillat-
Inpractice,ourcalculationofD(Q2)isapproximate, ing in Minkowski. Three distinct regimes were identi-
formanyreasons. First,nobodyisabletocalculatethe fiedandconsideredintheliteraturesofar:
infinite α (Q2) series for the coefficient functions, let
s
•(i)Finite-distancesingularities
alone the infinite condensate series. Both have to be
√
truncatedatsomefiniteorder. Afewlowest-dimension s−κ/2 sin( s); (2)
condensates that can be captured, are known approx-
imately. The best we can do is analytically continue •(ii)Infinite-distancesingularities(N =∞)
c
thetruncatedtheoreticalexpression,termbyterm,from
positive to negative Q2. For each term in the expan- s−η/2 sin(s); (3)
sion the imaginary part at positive q2 (negative Q2) is
well-defined. We assemble them together and declare • (iii) Infinite-distance singularities (Nc large but fi-
the corresponding ρ(s) to be dual to the hadronic nite,s→∞)
theor
crosssectionρ(s) . Inthegivencontext“dual”means (cid:32) (cid:33)
exp 1
equal. exp(−αs)sin(s), α=O (cid:28)1. (4)
N
Let me elucidate this point in more detail. As- c
sume that D(Q2) is calculated through α2 and 1/Q4,
s These regimes are not mutually exclusive – in con-
whilethetermsα3 and1/Q6 (withpossiblelogarithms)
s crete processes one may expect the duality violating
are dropped. Then the theoretical quark-gluon spec-
componenttobeacombinationof(i)and(ii),or(i)and
tral density, obtained as described above, is expected (iii). Fromthetheoreticalstandpointitisquitedifficult
to coincide with ρ(s) , with the uncertainty of order
exp toconsistentlydefinethedualityviolatingcomponentof
O[(α (s))3] and O(1/s3). The uncertainty in the theo-
s the type (3). An operational definition I might suggest
retical prediction of this order of magnitude is natural isasfollows: Startfromthelimit N = ∞andidentify
c
sincetermsofthisorderareneglectedinthetheoretical
the component of the type (2). Follow its evolution as
calculationofD(Q2). Ifthecoincidenceinthiscorridor
N becomeslargebutfinite.
c
doestakeplace,wesaythatthequark-gluonprediction
Nowafewwordsaboutglobal-versus-localdualities
isdualtothehadronicspectraldensity. Iftherearede-
misconceptionsareinorder. (Thistopicisalsorelated
viations going beyond the natural uncertainty, we call
totheissueoforderingofthelimits N → ∞, E fixed,
c
them violations of duality. Needless to say, once our
ortheotherwayaround, E → ∞, N fixed. Thesetwo
calculation of D(Q2) becomes more precise, the defi- c
limits are not interchangeable. As a result, D(Q,N )
c
nition of the “natural uncertainty” in ρ(s) changes
theor mustcontainnonanalytictermsin1/Qand1/N .)
c
accordingly.
Usuallybylocaldualitypeoplemeanpoint-by-point
This is the most clear-cut definition I can suggest.
comparisonofρ(s) andρ(s) ,whileglobalduality
theor exp
Fromtheformalstandpoint,itconnectsthedualityvio-
comparesthespectraldensitiesρ(s)averagedoversome
lation issue with that of analytic continuation from the
ad hoc interval of s, with an ad hoc weight function
EuclideantoMinkowskidomain. Negligiblysmallcor-
w(s),
rections (legitimately) omitted in the Euclidean calcu-
lationsmayanddogetenhancedinMinkowski. Expo- (cid:90) s2 (cid:90) s2
nentiallysmalltermsatpositive Q2 becomeoscillating dsw(s)ρ(s)theor ≈ dsw(s)ρ(s)exp.
andonlypower-suppressed(atbest)atpositiveq2. s1 s1
These oscillating terms defy the hierarchical order- Manyauthorsbelievethatglobaldualitydefinedinthis
ing and can be referred to as “duality violating.” The wayhasamoresolidstatusthanlocalduality.Someau-
physicalpicturebehindthemisasfollows. Theduality thorsgosofarastosaythatwhileglobaldualityiscer-
violations are due to (i) rare atypical events, when the tainly valid at high energies, this is not necessarily the
basic high-momentum quark transition occurs at large caseforlocalduality. Thisbecamearoutinestatement
ratherthanshortdistances;(ii)residualinteractionsoc- intheliterature. Well,routinedoesnotmeancorrect.
curringatlargedistancesbetweenthequarksproduced Infact, bothprocedureshaveexactlythesametheo-
at short distances. In the first case appropriate (Eu- reticalstatus.Thepoint-by-pointcomparison,aswellas
clidean)correlationfunctionsdevelopsingularitiesatfi- the comparison of ρ(s)’s (with an ad hoc weight func-
nite x2,whilethesecondmechanismiscorrelatedwith tion), must be considered as distinct versions of local
4
duality. Thedistinctionbetweenthe“local”quantities, tality and reliability. This is because the duality viola-
suchasR(e+e−)atacertainvalueof sandtheintegrals tionparametrizesourignorance.Ideally,themodelsone
ofthetypeinvolved,say,inR isquantitativeratherthan shouldaimatmusthaveaclear-cutphysicalinterpreta-
τ
qualitative. In both quantities there is a duality violat- tion, and must be tested, in their key features, against
ing component, the only distinction is in the concrete experimentaldata. Thiswillguaranteeacertaindegree
values of the indices of the power fall-off in Eqs. (2) ofconfidencewhenthesemodelsareappliedtotheesti-
and (3), say, 3 vs. 6. Thus, the averaging over s, as matesofthedualityviolationsintheprocessesandkine-
inR ,makesthedualityviolationsomewhatmoresup- maticalconditionswheretheyhadnotbeentested.
τ
pressed,butthisisstillsomethingwhichwetotallymiss It is rather discouraging that there was very little
inthepracticalversionofOPE.Inaddition,atthemo- progress,ifatall,inthisdirectioninthelast10years.
ment these indices are model dependent. There is no
waytoreliablydeterminethevalueofdualityviolation,
beitpointbypointasinR(e+e−)orintegrally,asinR , 4. AdS/QCDvs. SVZsumrules
τ
fromtheanalysisofthepracticalOPEperse.
Thegenuineglobaldualityappliesonlytospecialin- Itallstartedin1998whenMaldacena; Gubser, Kle-
tegralswhichcanbedirectlyexpressedthroughtheEu- banovandPolyakov;andWittenargued[15]thatcertain
clideanquantities. Forinstance,iftheintegrationinter- four-dimensional super-Yang–Mills theories at large N
valextendsfromzerotoinfinity,andtheweightfunction could be viewed as holographic images of higher-
isexponential,theintegral dimensionalstringtheory.Inthelimitofalarge’tHooft
(cid:90) ∞ couplingthelatterwasshowntoreducetoanti-de-Sitter
dsexp{−s/M2}ρ(s), (5) supergravity. The framework got the name “Anti-de-
0 Sitter/Conformal Field Theory (AdS/CFT) correspon-
reduces to the Borel transform of D(Q2) in the Eu- dence.”
clideandomain(i.e.atpositiveQ2).Forsuchquantities, Bynow,itisgenerallybelievedthatten-dimensional
dualitycannotbeviolated,bydefinition. 1 string theory in suitable space-time backgrounds can
InthelimitE →∞,N fixed,dynamicsperse“glob- have a dual, holographic description in terms of su-
c
alizes”duality,sincetheresonancewidthsareswitched perconformal gauge field theories in four dimensions.2
onandproduceasmearingofthespectraldensity. This Conceptually, the idea of a string-gauge duality as-
dynamical smearing may increase the indices κ and η cends to ’t Hooft [16], who realized that the pertur-
in (2), and (3), but in no way can eliminate deviations bative expansion of SU(N) gauge field theories in the
fromdualityinthesenseIexplainedabove. Ingeneral, large-N limit (with the ’t Hooft coupling fixed) can be
theindicesκandηaremodel-dependent. Why? reinterpreted as a genus expansion of discretized two-
Bydefinition, onecannotbuildanexhaustivetheory dimensional surfaces built from the field theory Feyn-
ofthedualityviolationsbasedonthefirstprinciples.In- mandiagrams. Thisexpansionresemblesthestringthe-
deed,assumingthereisacertaindynamicalmechanism ory perturbative expansion in the string coupling con-
(which goes beyond perturbation theory and conden- stant. The AdS/CFT correspondence is a quantitative
sates)forwhichsuchatheoryexists,onewouldinclude realizationofthisideaforfour-dimensionalgaugethe-
the corresponding component in the theoretical calcu- ories. In its purest form it identifies the “fundamental
lation. The reference quantity, D(Q2) , will be re- typeIIBsuperstringinaten-dimensionalanti-de-Sitter
theor
defined accordingly. After the analytic continuation to space-timebackgroundAdS ×S5 withthemaximally
5
Minkowski, this will lead, in turn, to a new theoretical supersymmetric N = 4 Yang–Mills theory with gauge
spectraldensitytobeusedasareferenceρ(s) inthe group SU(N) in four dimensions.” The latter theory is
theor
dualityrelation. superconformal.
Thus, by the very nature of the problem, it is bound Shortly after [15] a new ambitious goal was set: to
tobetreatedinmodelsofvariousdegreesoffundamen- get rid of conformality and get as close to actual QCD
as possible. There are two lines of though. Chrono-
logicallythefirstwasthetop-downapproachpioneered
1Inmathematicalliteraturetheyrefertothetransformation(5)as by Witten; Polchinski and Strassler; Klebanov and
theinverseLaplacetransform. Itwasfirstworkedin1930byPost
[13]. Curiously, some questions that are difficult from the general
mathematicalstandpointseemtoberathertrivialfortheoreticalphysi-
cists.IthankMartinBlockandArkadyVainshteinforthisreference. 2Warning: Thisgauge-gravitydualityhasnothingtodowiththe
Seealso[14]. quark-hadrondualitywhichwasdiscussed5minutesago.
5
Strassler;MaldacenaandNun˜ez,andothers. Herepeo- 2005resultsisfullyclear(bothofthemareadmittedly
ple try to obtain honest-to-god solutions of the ten- wrong). Abstractlyspeaking,onecouldhaveimproved
dimensional equations of motion, often in the limit of at least some aspects, for instance, if instead of super-
the large ’t Hooft coupling when on the string side gravityonthestringsidewecouldhavekeptthethefull-
of the theory one deals with supergravity limit. The blownstringtheorystilladheringtotheabovehard-wall
problem is: in many instances these solutions are dual approximation, we would restore asymptotic linearity
to ... sort of QCD ... kind of QCD ..., rather than oftheReggetrajectoriesatlargeangularmomenta J or
QCD as we know it. For instance, Witten’s set-up or excitationnumbersn. Inthislimitveinonecouldthen
theMaldacena–Nun˜ezsolutionguaranteecolorconfine- calculate,inaddition,say,themesondecayrates,aswas
mentbuttheasymptoticallyfreeregimeofQCDisnot done recently by Sonnenschein and collaborators, who
attained. recovered the 1979 Casher–Neuberger–Nussinov [21]
TheKlebanov–Strasslersupergravitysolutionisnear quasiclassicalformula! However,nobodysucceededso
AdS in the ultraviolet limit, a crucial property for the farinobtainingthefullspectrumofcrucialQCDregu-
5
existence of a dual four-dimensional gauge theory. In laritiesfollowingthisroad.
the ultraviolet this theory exhibits logarithmic running This was the reason for the advent of AdS/QCD
of the couplings which goes under the name of the which I have just mentioned a couple minutes ago. I
duality cascade. They start from string theory on a should add that the bottom-up AdS/QCD guess-and-
warped deformed conifold and discover a cascade of trial approach was pioneered by Son and Stephanov
SU(kM)×SU((k−1)M)supersymmetricgaugetheories [22]. If in the AdS/CFT correspondence (string-gauge
ontheotherside. Asthetheoryflowstotheinfrared,k duality) the five-dimensional metric g (x ,z) is “sci-
AB µ
repeatedlychangesbyunity,seethereviewpaper[17]. entifically”obtainedfromtheEinsteinequationsinfive
Intheinfraredthistheoryexhibitsadynamicalgenera- dimensions,inAdS/QCDitispostulatedadhoc. From
tionofthescaleparameterΛ,whichmanifestsitselfin AdS/CFTweget,on“our”sideofduality,agaugethe-
thedeformationoftheconifoldonthestringside. oryinfourdimensionswhichissuperconformal. Toget
Thereisavariantofthetop-downapproachinwhich closertoQCDonemustbreakSUSY,conformalityand
the requirement of the exact solution of the supergrav- buildinthepropertyofasymptoticfreedom.Tothisend
ity equations is “minimally” relaxed. Confinement is peopleabandonthefive-dimensionalEinsteinequations
enforcedthroughacrudecut-offoftheAdSbulkinthe altogetherandtrytoguessappropriatefive-dimensional
infrared,atz ,wherezisthefifthdimension. Thisleads metric. A spectral quantum-mechanical equation with
0
to a “wrong” confinement, however. In particular, the this conjectured metric gives us the hadronic spectral
Reggetrajectoriesdonotcomeoutlinear. Afewyears densities. The guideline in the guessing process is the
ago, preparing for a talk [18], I suddenly realized that “marriage”betweentheholographicrepresentationand
themesonspectrumobtainedinthiswayidenticallyco- OPE-based methods, plus chiral symmetry breaking,
incides with the 30-year-old result [19] of Alexander plusotherknownregularitiesofthehadronicworldone
Migdal, who, sure enough, had no thoughts of super- cansqueeze(withluck)intheconstruction.Thetheorist
gravityinfivedimensions. Hisideawastoapproximate is supposed to make various conjectures en route. The
logarithmsofperturbationtheorybyaninfinitesumof targetistobuildafive-dimensionalmetricencodingas
polesinthe“bestpossibleway.”Thenthisstrategywas much information on real QCD as possible, and then,
abandoned since it contradicts OPE. Now, in essence, with this metric in hands, get new insights and make
theMigdalprogramisresurrectedinanewincarnation newpredictions.
which goes under the name of AdS/QCD. What was It is clear that AdS/QCD and SVZ sum rule method
Migdal’sgoal?Heaskedhimself:“whatisthebestpos- sharesomecommonfeatures.Itisnosurprisethatmany
sibleaccuracytowhichone-looplogQ2canbeapprox- features of the SVZ expansions were recovered. The
imated by an infinite sum of infinitely narrow discrete guessingofthefive-dimensionalmetricissimilartode-
resonances?” and “What are the corresponding values veloping the continuum model in the sum rules. Sum
of the resonance masses and residues?” He answered rule predictions for low-lying states are rather insen-
thisquestionasfollows: “theaccuracyisexponentialat sitive to the continuum model. This is an advantage.
largeQ2andtheresonancesmustbesituatedatthezeros On the other hand, choosing the five-dimensional met-
ofaBesselfunction.”Thisisexactlythepositionofthe ric one completely specifies the relationship between
excited ρ mesons found in the first detailed AdS/QCD thelow-lyingstatesandallhigherexcitations,withab-
work[20]. solute sensitivity to the entire spectrum. Some view
The reason for the coincidence of the 1977 and this as an advantage, others as disadvantage. For in-
6
stance, the original hard cut-off metric [20] gives res- fielddefinedas
onances at the zeros of the Bessel function (remem- 1 (cid:104) (cid:16)
(cid:104)j j5(cid:105) = − w (q2) −q2F˜ +q qσF˜
ber Migdal!). It gives parabolic Regge trajectories, µ ν Fˆ 4π2 T µν µ σν
and Π(Q) ∼ lnQ2 +exp(−Q) in Euclidean (remember
(cid:17) (cid:105)
Migdal!). It was later replaced by a soft cut-off metric − q qσF˜ +w (q2)q qσF˜ , (6)
ν σµ L ν σµ
which gives equidistancy in M2, linear Regge trajecto-
isnotrenormalizedinperturbativeQCD.However,be-
riesandrigid1/Q2 correctionsintheEuclideanexpan-
cause of the chiral symmetry breaking, the above non-
sion,asifallcondensateswereexpressedintermsofthe
renormalizationtheoremisnotextendabletotononper-
lowest-ordercondensate.
turbative effects [25]. Thus, w and w have different
By and large, I cannot say that at present AdS/QCD L T
status: the latter quantity is dynamical. Nevertheless,
gives a better (or more insightful) description of the
thefactthattheα seriesisabsentinw givesonehope
hadronic world, than the good old SVZ condensate- s T
thatonlysomegeneralaspectsofQCDareinvolvedin
based method. Given a rather crude character of the
itscalculation.Underasimpleadditionalassumptionof
hard-wall and similar approximations, perhaps, today
thepiondominance, Vainshteinobtainedthefollowing
one may hope to extract only universal information on
analytic“prediction”forthevacuummagneticsuscepti-
hadronic dynamics, steering clear of all details. How-
bilityχintroducedin[28]:
ever, I do not exclude that further studies of these
N
two approaches, in conjunction, will be beneficial for χ=− c . (7)
bothandwilladdsomesignificantunderstandingtoour 4π2fπ2
knowledgeofthehadronicworld.Iappealtoyoungthe- Here N is the number of colors, and f ≈ 92 MeV is
c π
oristsinvolvedinthisareaofresearchtoinvesteffortin the pion constant. I put “prediction” in the quotation
thissubject. marksbecausetherewasnotheoreticaljustificationfor
the above-mentioned simplest assumption, as was cer-
tainlynotedandemphasizedintheoriginalpaper[25].I
5. Canholographyhelp? guess,thegoaloftheholographyexplorersinthisissue
was to find a justification for this or a similar relation,
As it should be clear, I do not expect for the time perfectthecoefficients,and,ingeneral,elevateitsstatus
beingtheAdS/QCDtoproducedetailedpredictionssu- tothelevelwherethequotationmarkscouldbedropped.
perior (or even close) in their reliability and precision First, itwasrealized[23]thatthegravitydualinthe
to those of the SVZ sum rules. However, I hasten case at hand must be Yang–Mills–Chern–Simons the-
to make a reservation. If one can find such general ory. Addition of the Chern–Simons term turned out to
problemswhosesolutiondoesnotdependonparticular beabsolutelynecessary. Intheso-calledhard-wallver-
choices of the five-dimensional metric, one has much sionofholographythevacuummagneticsusceptibility
better chances to obtain a successful prediction. Such wasfound(numerically)[23]tobecloseto(7),withthe
anattempthasbeenundertakenrecentlyin[23,24]ina coefficientlargerthanN /4π2byabout10%.
c
probleminawayrelatedtothechiralanomaly.3 Aswe Then other versions of holography, such as the so-
willsee,nobreakthroughoccurred,althoughtheresults called soft model of the “bottom-up” AdS/QCD [22]
obtainedin[23,24]givesomefoodforfurtherthought. and the “top-down” Sakai–Sugimoto model [29] (both
It is well known that the longitudinal part of the arepopularinthisrangeofquestions)wereexploredin
fermion triangle graph is unambiguously and exactly [24]. In this rather broad class of Yang–Mills–Chern–
fixed by the chiral anomaly. In essence, this is a topo- Simonsdualtheories,withthechiralsymmetrybroken
logicalratherthandynamicalquantity. Theknowledge by the boundary conditions in the infrared, indepen-
of the above longitudinal part gives us the famous for- dentlyofthechoiceofthegravitymetrics,thefollowing
mulaforπ0 →γγ. Vainshteinaskedhimselfaquestion relationtakesplace[24]:
[25] (see also [26, 27]) whether the transverse part of N N (cid:104) (cid:105)
w (Q2)= c − c Π (Q2)−Π (Q2) , (8)
thetrianglegraphsisalsoconstrained.Hedemonstrated T Q2 f2 A V F
that the transverse part w (Q2) of the current-current π
T foranyQ2.HereΠ arethetwo-pointfunctionsofthe
correlator in an infinitesimally weak electromagnetic A,V
axial-vector (vector) currents in the background (very
soft)electromagneticfield. Thefirsttermin(8)wasob-
3That was probably the original reason behind the belief that tained by Vainshtein. The second term obviously van-
holographywillwork. ishes to any finite order in perturbation theory. This
7
is a nonperturbative correction found through holog- 6. InsteadofConclusions
raphy but independent of the particulars of the five-
dimensional metric. Equation (8) implies a new set of Therichnessofthehadronicworldisenormous.
relationsforvariousresonanceparameters. Itdescribesaverywiderangeofnaturalphenomena,
ConsideringEq. (8)atlargeQ2,usingtheSVZ-type e.g.:
(cid:104) (cid:105)
operator product expansion for Π (Q2)−Π (Q2) , •allofnuclearphysics;
A V
F •ReggebehaviorandReggetrajectories(highlyex-
andfactorizationforthefour-quarkmatrixelement(jus-
tified by the large-N limit) one can derive from (8) citedmesonandbaryonstates);
c
a consistency condition [24] for the vacuum magnetic • strongly coupled quark-gluon plasma; high-T
susceptibility, in an analytic form. Remarkably, this is phenomena; color superconductivity at high density
exactlythesameformula(7)obtainedbyVainshtein. (throughcolor-flavorlocking);neutronstars;
•richnessofthehadronicworld(chiralphenomena,
Anotherexampleofpredictions[24]forphysicalob-
lightandheavyquarkonia,theZweigrule,glueballsand
servablesfollowingfrom(8)arethesumrules
exotics,exclusiveandinclusiveprocesses);
(cid:88) gγViAjgAj = − Nc g , (9) • hadronization of fast moving colored sources, i.e.
m2 −m2 4π2f2 Vi jets (of special interest are, of course, nonperturbative
j Aj Vi π aspectsofthejetphysics);
•interplaybetweenstrongandweakinteractions(in
(cid:88) gγViAjgVi = − Nc g . (10) particular,theso-calledpenguinmechanism);
m2 −m2 4π2f2 Aj
i Aj Vi π andmanyotherissues.
AtshortdistancesQCDisweaklycoupled,allowing
In the first sum rule i is fixed (and arbitrary) while j
high precision perturbative (multi-loop, multi-leg) cal-
runs over all axial-vector resonances. Likewise, in the
culations. However, theadventoftheeraofarbitrarily
secondexpression jisfixedwhileirunsoverallvector
exact analytical computations at all energies and mo-
resonances.Forbroadresonancesonecansubstitutethe
menta, especially in the Minkowski domain, is not ex-
sumsbytheintegralsinthespiritoftheSVZmethod.
pectedintheforeseeablefuture,duetostrongcoupling
Let us ask ourselves how successful numerically is
nature of the large-distance dynamics. Let us ask our-
the Vainshtein formula. The pion constant in (7) is
selves: whatdowewantfromthistheory? Isitreason-
normalized as f ≈ 92MeV. Substituting this number
π
abletoexpecthigh-precisionanalyticpredictionsforall
we arrive at χ ≈ −9.0 GeV−2. At the same time, the
low-energyobservables? Canwe(andshouldwe)com-
magnetic susceptibility had been determined from the
putehadronicmasses,matrixelementsor,say,proton’s
QCDsumruleslongago. Unfortunately,calculationof
magneticmomentuptofivedigits?
the magnetic susceptibility, presented in the main text
UnlikeQED,mostprobablywewillneverbeableto
of [28] is not quite correct since an inappropriate dis-
analytically calculate the above-mentioned and similar
persion relation was used. The correct result is quoted
observables with this precision. But do we really need
in ‘Note added in proof’ in the same paper. The most
this? Tomymind,whatisreallyneededisthecomple-
preciseevaluationofthemagneticsusceptibilitycanbe
tion of the overall qualitative picture of confinement +
foundinSec. 6.3ofthebook[30],
developmentofvariousreliableapproximatetechniques
custom-designedforspecificapplications. Theoriginal
χ =−3.15±0.3 GeV−2 (11)
QCDSR sumrulemethod,extendedbynumerouslaterdevelop-
ments,fitsverywell. InthiscontextQCDsumrulesdo
forthenormalizationpointaround1GeV.Thediscrep-
haveafuture,don’tthey?
ancyisaboutafactorof3,somewhatlargerthancould
havebeenanticipated.
Themainproblemwithholographywhichclearlyre- NoteAddedinDecember
veals itself in confronting (7) and (11) is that hologra-
phy, asweknowitnow, doesnotreproducethosesev- Averyrecentpublication[31]admiredmebyitsel-
eraltermsofOPEwhicharesolidlyestablished. There- egance. Zohar Komargodski implemented, in a bril-
fore, fine details of the hadronic picture obtained from liant way, the old idea [32] that all vector mesons of
holography come out wrong (at least, for the time be- QCD (i.e. ρ and its excitations) are in fact Higgsed
ing),althoughsomeoverallcontoursare,perhaps,cap- gauge bosons of a hidden non-Abelian local symme-
turedright. try (or symmetries) of the hadronic world. This was
8
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tor meson dominance, a phenomenon that is a crucial ory and the V PASCOS Interdisiplinary Symposium, Balti-
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