A candidate for the confining string of gauge theories is constructed via a
representation of the ultraviolet divergences of quantum field theory by a
closed string dilaton insertion, computed through the soft dilaton theorem. The
resulting (critical) confining string is conformally invariant, singles out
naturally $d=4$ dimensions, and can not be used to represent theories with
Landau poles.Comment: 13 pages, two figures Clarifying remark added. Typos correcte

Alvarez, Enrique

Gomez, Cesar

English

arXiv

A candidate for the confining string of gauge theories is constructed via a representation of the ultraviolet divergences of quantum field theory by a closed string dilaton insertion, computed through the soft dilaton theorem. The resulting (critical) confining string is conformally invariant, singles out naturally $d=4$ dimensions, and can not be used to represent theories with Landau poles.A candidate for the confining string of gauge theories is constructed via a representation of the ultraviolet divergences of quantum field theory by a closed string dilaton insertion, computed through the soft dilaton theorem. The resulting (critical) confining string is conformally invariant, singles out naturally $d=4$ dimensions, and can not be used to represent theories with Landau poles.A candidate for the confining string of gauge theories is constructed via a representation of the ultraviolet divergences of quantum field theory by a closed string dilaton insertion, computed through the soft dilaton theorem. The resulting (critical) confining string is conformally invariant, singles out naturally d =4 dimensions, and can not be used to represent theories with Landau poles

CERN Document Server

IFT-UAM/CSIC-99-27hep-th/9907158The Confining String from the Soft Dilaton TheoremEnrique A´lvarez},| 1 and Ce´sar Go´mez},♠ 2} Theory Division, CERN,1211 Geneva 23, SwitzerlandandInstituto de F´ısica Teo´rica, C-XVI, Universidad Auto´noma de MadridE-28049-Madrid, Spain3| Departamento de F´ısica Teo´rica, C-XI, Universidad Auto´noma de MadridE-28049-Madrid, Spain♠ I.M.A.F.F., C.S.I.C., Calle de Serrano 113E-28006-Madrid, SpainAbstractA candidate for the conning string of gauge theories is constructed via a rep-resentation of the ultraviolet divergences of quantum eld theory by a closedstring dilaton insertion, computed through the soft dilaton theorem. The re-sulting (critical) conning string is conformally invariant, singles out naturallyd = 4 dimensions, and can not be used to represent theories with Landau poles.1E-mail: enrique.alvarez@uam.es,@cern.ch2E-mail: iffgomez@roca.csic.es3Unidad de Investigacio´n Asociada al Centro de F´ısica Miguel Catala´n (C.S.I.C.)1The simplest way to address the problem of a string representation of a non abeliangauge theory [16] consist in looking for a conning string background, determined byvanishing σ -model beta functions, such that the condition of dilaton beta function equalto zero could be reinterpreted as the standard renormalization group equation:µdgdµ= β(g) (1)for the Yang Mills running coupling constant. This program implies a deep geometrizationof quantum eld theory renormalization program, pointing out to gravity as the underlyingdynamics controling the running of coupling constants 2.The next and more dicult step,in the conning string program, ends up stablishing the precise relation between gaugesinglets and the conning closed string spectrum.Back to the early days of string theory [12],[18], [1]the problem of innities of oneloop bosonic string amplitudes, was treated in perfect analogy with the quantum eldtheory renormalization program. An important subproduct of that analysis, controlingthe structure of string renormalizability, was the so called soft dilaton theorem.Let us consider a one loop open bosonic string amplitude for N external gluonsA1(p1, p2, . . . , pN). On the moduli space of the cylinder this amplitude becomes singularwhen the length of the cylinder t−1 ! 1 corresponding to an anulus with the size of theinternal hole going to zero. In string theory the singularity of this amplitude is exactlygiven by the tree level amplitude of emission of a soft dilaton of momentum k = 0 that issubsequently absorbed by the vacuum:Sing (A1(p1, p2, ...pN) = limk!0J (k) A0(p1, p2, ...pN ; k) (2)with J the dilaton-vacuum amplitude, (k) the dilaton propagator and A0(p1, p2, ...pN ; k)the tree level amplitude for N gluons and one dilaton of momentum k.The soft dilaton theorem [1] precisely states that:limk!0A0(p1, p2, ...pN ; k) = pigld−22s [ls∂∂ls− d− 22g∂∂g]A(p1 . . . pN) (3)2Previous work on the relationship between the renormalization group and the holographic principleinclude [19],[2],[3] ,[4] and [6].1with g the open string coupling constant, and α0  l2s , thus implying that the singularityin equation (2) can be absorbed in a double renormalization of α0 and g.As it is clear from equation (2), the singularity of A1(p1, p2, ...pN) comes from the dilatonpropagator (k) at k = 0. We have schematically represented in the enclosed gure thepertinent string diagrams.p1p2p3pNp1p2p3pNJk=0openclosedk = 0JFigure 1: Topology of the sigularities in the open and closed string diagramsUsing the integral representation of (k) and introducing a cuto µ as:µ(k)jk=0 =∫ 11/µdzz(4)(where µ should be interpreted as a cuto on the size of the long tube in Figure 1), therenormalized string tension is given by:1α0R(µ) Jg2α00∫ 11/µdzz+1α00(5)From the soft dilaton theorem (3) and equation (5) we get a running string coupling2constant 3 g(µ) with beta function 4 given by:β(g) = −g3 (d− 2)pi4(6)It is interesting to observe that equation (6) for d > 2 is of asymptotically free type.The modern approach, after Fischler-Susskind ([10]), to the string tension renormaliza-tion (5) stems from identifying log µ with the world sheet Weyl factor (the Liouville eldφ) and to look for a background redenition with a new Ricci tensor determined by:δ( 1α0R(µ))ηµνδlogµ(7)in such a way that the σ-model beta function equations for the string are satised.After the discovery of D-p-branes it is perfectly natural to interpret the one loop openstring amplitude A1(p1, p2, ....pN ) as a world volume amplitude with p1, p2, ..pN momentain p + 1 dimensions. In this case the singularity in the one loop open string amplitude canbe related to a soft dilaton insertion with momentum k in the bulk transversal directions.This D-brane picture naturally yields the interpretation of µ as a transversal coordinate.In this set up the renormalization of α can be interpreted as dening a background metricof the type:ds2 =1α0(µ)d~x2p+1 + dφ2 + d~y226−p−2 (8)(where d~x2d is the flat metric in Rd) for φ = logµ, the world sheet Weyl factor or Liouvilleeld (cf.[16]). (The general idea of using strings with variable tension is due to Polyakov(cf. [17])). The condition of world sheet conformal invariance for the background (8) oughtto be consistent with the dilaton dependence (µ) dictated by soft dilaton theorem (3) i.e3In references [12], [18], the renormalization of the string coupling constant, g, was interpreted as aconsequence of the tachyon singularity in the one-loop open string amplitude, in the t ! 0 limit. In [1],however,the appearance of g in the soft dilaton theorem was related to the transversality of the externaldilaton. We believe that the potential closed string tachyon interpretation of the renormalization of gin parallel to the transversality condition for the dilaton, could benefit from better understanding (cf.[17],[11]).4The first renormalization of the string coupling constant was made in [12] by analytically regularizingthe logarithmically divergent integral. What we have done here is just to regularize it by means of a cutoffin the lower end.3by the renormalized string coupling. As we will see in a moment this is in fact the case,but only if p = 3!Before going into more details let us highlight the main point of this note. From theDirac-Born-Infeld action for a D − 3 brane we get the standard relation between YangMills coupling constant and the closed string coupling constant g2:g2Y M = g2 (9)where in addition g2 is given by the vacuum expectation value of the dilaton eld. Theconning string interpretation of four dimensional pure Yang Mills theory will be based oninterpreting the Yang Mills beta function as a stringy beta function governing the stringone loop renormalization of g as due to soft dilaton insertions. Once we stablish thisidentication we use the soft dilaton theorem to nd the associated running string tensionα0(µ). The conning string background metric will be nally dened by a metric as (8).Asa consistency check of the whole procedure we will show that the σ-model beta functionsvanish for the so dened conning string background.It is important to stress that in this approach we read the conning string geometrydirectly from the perturbative ultraviolet behavior of pure Yang Mills gauge theory. Theidentication of the string beta function, governing the renormalization of the string cou-pling constant, with the pure gauge beta function should be understood as a generalizationto quantum eld theory of expression (2). In fact what we are doing is reinterpreting thestandard ultraviolet singularities of a quantum eld theory in terms of a soft dilaton in-sertion in the corresponding conning string. More precisely what we understand as aconning string is the string interpretation, in terms of a dilaton propagator (in bulk di-rection) and a dilaton tadpole, of the standard ultraviolet singularities of non abelian gaugetheories. In this sense we can think of strings as a physical model of quantum eld theoryrenormalizability. The D-brane description ([15]) of gauge theories will be specially suitedfor our purposes.The DBI action for a D-p-brane:SDBI = Tp∫dp+1ξe−Φ√det[gab + bab + 2piα0Fab] (10)4through an expansion in α0, conveys the relation between Yang Mills bare coupling andthe dilaton vacuum expectation value:1g20= Tpe−Φ0(2piα0)2 (11)The one loop corrections to the Yang Mills coupling are given in the background eldmethod by:(1g20+12CG,1 − CG,0)(F 2 + higher derivatives) (12)where the coecients CG,i represent the contribution of the determinants of the gluon(i = 1) and ghosts (i = 0) quadratic terms:CG,i = ci log(M2/k2) (13)where the scale M can be appropiately chosen to cancel the bare coupling dependence in(12).The simplest possible matching between equations (11) and (12) suggests a renormal-ization group dependence of the dilaton eld such as:Tpe−Φ(k2)(2piα0)2F 2 (14)This dependence will be interpreted as providing a running string coupling constant g(µ2 =k2) that , according to our previous discussion will be identied with the running stringcoupling constant one would obtain in the string renormalization of string one loop innitiessatisfying equation (2). In order to put this identication on more solid grounds we needto match the renormalization group scale in quantum eld theory with the cuto used inequation (4) for the dilaton propagator. In order to do that we will again prot from deD-brane description.To be concrete we will consider the two gluon amplitude corresponding to the processdepicted in gure:A  α01−d/21.2 g2(4pi)d/2∫dt∫ t0dν t−d/21∏n=1(1− e−2nt)2−De2α0p1.p2G(ν)∂2G(ν)e−y2t/α0 (15)5D-BraneGluon Vertex OperatorsyFigure 2: The planar two-gluon annulus amplitudewhere G(ν) is the cylinder Green function, ν is the relative position of the vertex opera-tors, characterized by polarizations and momenta 1, p1 and 2, p2. The D-brane eect isconveyed by the exponential factor depending on y, the distance between branes.Let us now consider the limit:α0 = 0; t−1 = 0; y = 0 (16)withα0t = t;yα0= u (17)In equation (15) d = p + 1 for p the dimension of D-brane world volume space and Drepresent the dimension of the target space-time.In the limit (16), (17) amplitude (15) forp1 and p2 on shell becomes ([9]):A  g2(4pi)d/21.2∫dtτ∫ 10dν^τ 1−d/2e−u2τ¯ [(D − 2)(1− 2ν)2 − 8] (18)where we have used the relationship for the Neumann Green’s functions:[e2τ¯ + (D − 2)]∂2G  (D − 2)(1− 2ν^)2 − 8 (19)From equation (18) we easily get:A  − g2(4pi)d/21.2Γ(1− d/2)(u2)d/2−1[(D − 2)− 4)D − 2)B(2, 2)− 24] (20)6(where B(u, v) is Euler’s Beta function), which by standard dimensional regularizationleads for d = 4 to:A  − g2(4pi)d/21.2u2 log(u2/2)[(D − 2)− 4(D − 2)B(2, 2)− 24] (21)From equation (21) it is clear that variable u as dened in (17) is playing the ro^le ofthe renormalization group variable. On the other hand u as it is plain from denition(17) is nothing else but the near horizon transversal variable introduced in reference [13]possessing the natural meaning of energy in transversal bulk direction which is preciselythe meaning of the cuto variable µ2 introduced in equation (4). Using equation (14) thecorresponding dilaton eld dened by (21) will be given by:e−Φ(u2) = −[(D − 2)− 4(D − 2)B(2, 2)− 24]log(u2/2) (22)Following our strategy what we will do now is to identify the running of the dilaton (22) 5with the string renormalization of the string coupling constant due to soft dilaton tadpole.Next and using the soft dilaton theorem we induce from that a running dependence ofworld volume metric on u. Finally we chek the consistency of the so dened conningstring by solving the σ-model beta functions. Going through these steps the conningstring background metric 6we get is:ds2 = log(u)d~x24 +du2u2+ d~y221 (23)with a dilaton eld of the type:(u) = −log log(u) (24)The σ-model beta equations are:Rµ,ν −rµrν = 0 (25)andr2 + (r)2 = cextra (26)5It is worth remarking that for D = 4 the equation (21) gives directly the well known precise factor forYang Mills (11/3). In D = 26 instead we get zero (the correct value for ns = 22 adjoint scalars). In whatfollows we will stick to D = 4.6With euclidean signature.7It is known in general (cf, for example, [7],[8]) that consistency of the equations imply thatcextra must be constant (equal to2(D−10)3in the (world-sheet) supersymmetric case and to26−D3in the ordinary bosonic string). Curiously enough, the above metric (23) with thedilaton eld exactly as in (24) is a solution of the rst equation (25) for any value of p.The second equation, (26) however can be easily shown to be given by:1− (p + 1)/2(logu)2+1(logu)2(27)conveying the fact that it is a constant for p + 1 = 4 only, in which case it is zero. Thismeans that the string must neccessarily be critical, and we have to add the extra flat(spectator) dimensions, 21 of them in the bosonic case 7. This is one remarkable fact ofthis background. Another is, of course, that the sign of the dilaton in equation (24) isxed, so that it is not possible to construct non-confinig strings to describe ultravioletslave quantum eld theories using this set of ideas.The conning string itself should allow the computation of many gauge invariant ob-servables in gauge theories, such as the Wilson loop. Assuming a static congurationbounded by σ = l/2, and with a temporal span equal to T , it is useful to work in thegauge in which τ = x0 and σ = x1. If at the same time we dene a new coordinate byz  log u such that the metric readsds2 = zd~x24 + dz2, (28)and assuming a symmetric imbedding of the world sheet into the target spacetime of theconning string, z = z(σ), then the induced metric on the worldsheet is given by:ds2 = zdτ 2 + (z + z02)dσ2 (29)It can be argued ([13]) that the semiclassical approximation to the Wilson loop action isgiven by the area of the world sheet computed precisely with the induced metric, that is:S = T∫ l/2−l/2dσ√z(z + z02) (30)7It should be clear that the spectator dimensions change neither the confining string target spacefive-dimensional metric nor the Yang-Mills beta function. It is important to realize that the metric (23)does not correspond to a D−3 brane in a 26-dimensional ambient space. Actually, the only source for anontrivial dilaton is the five-dimensional metric given by the running string tension.8(We are neglecting here the contribution of the dilaton to the action, possibly important).The variational principle (30) enjoys a rst integral, namely:p =−z2pz2 + zz02(31)The formal solution to this dierential equation is:∫ z(σ)df1√f 3 − p2f =∫ σ−l/2dσp(32)The lower limit of the rst integral should be chosen in one of the regions in which z02is allowed to be positive by the rst integral, namely (−p, 0) or (p,1). Choosing byconcreteness it to be 0, the solution can be easily shown to be given by an elliptic integralof the rst kind:σ + l/2 = −(2p)1/2F (cos−1(jz(σ)/pj),p22) (33)which implies that:jz(σ)pj = sn σ + l/2(2p)1/2(34)If we neglect the square of the modulus, m2 = 1/2, the trigonometric approximation canbe applied:z(σ) = psinσ + l/2p2p(35)The relationship between p and l is found by demanding that z(l/2) = 0, yielding p = l22pi2.The action then reduces to:S = T∫ l/2−l/2dσz2p=T l34(36)which indeed corresponds to a conning potential.It would be interesting to check how world sheet supersymmetry (as well as GSOprojections) modify our analysis. In particular, it should be stressed that multiplicativefactors in the beta function of the gauge theory yield additive factors in the formula (24),which do not contribute to the string beta function.A quite surprising aspect of our analysis is the way pure Yang Mills perturbative infor-mation is promoted to string dynamics. This implies a connection between perturbativeaspects of gauge dynamics (such as the beta function) with non perturbative, (stringy)dynamics, such as infrared strong forces. The main ingredient for doing that has been9the soft dilaton theorem that allows to read coupling constant renormalization as variablestring tension and in that sense as conning string background metric. It is perhaps worthremarking that vanishing σ-model beta functions - essential to the quantum consistency ofthe string - have been used in our approach only at the end as a consistency check for themetric and dilaton eld directly derived from the quantum eld theory data and not as anextra governing principle. Moreover four dimensional space time appears in this scheme asthe only consistent solution. All this seems to indicate that some string dynamics is alreadyencoded in the renormalization of asymptotically free four dimensional gauge theories.AcknowledgmentsThis work has been partially supported by the European Union TMR program FMRX-CT96-0012 Integrability, Non-perturbative Effects, and Symmetry in Quantum Field The-ory and by the Spanish grant AEN96-1655. The work of E.A. has also been supported bythe European Union TMR program ERBFMRX-CT96-0090 Beyond the Standard modeland the Spanish grant AEN96-1664. C.G would like to thank the organizers of the AbdusSalam ICTP Extended Workshop in String Theory for hospitality while this work wasbeing written up.References[1] M. Ademollo, A. D’Adda, R. D’Auria, F. Gliozzi, E. Napolitano , S. Sciuto and P. DiVecchia, Soft Dilatons and Scale Renormalization in Dual Theories, Nucl. Phys. B94(1975)221.[2] E.T.Akhmedov, A Remark on the AdS/CFT correspondence and the RenormalizationGroup Flow, hep-th/9906217,Phys.Lett. B442 (1998) 152.[3] E. 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The Confining String from the Soft Dilaton Theorem

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