Electromagnetic form factors from the fifth dimension
PPUPT-2262QMUL-PH-08-06
Electromagnetic form factorsfrom the fifth dimension
D. Rodr´ıguez-G´omez a,b, and J. Ward c, a Department of Physics, Princeton UniversityPrinceton, NJ 08544, USA b Center for Research in String Theory, Queen Mary University of LondonMile End Road, London, E1 4NS, UK c Department of Physics and Astronomy, University of VictoriaVictoria, BC, V8P 1A1, Canada
ABSTRACT
We analyse various U (1) EM form factors of mesons at strong coupling in an N = 2 flavoredversion of N = 4 SY M which becomes conformal in the UV. The quark mass breaks theconformal symmetry in the IR and generates a mass gap. In the appropriate limit, thegravity dual is described in terms of probe D AdS × S . By studying the D γπρ and γf ρ transition form factors.At large q we find perfect agreement with the naive parton model counting, which is aconsequence of the conformal nature of both QCD and our model in the UV. By using thesame tools, we can compute the γ ∗ γ ∗ π form factor. However this channel is more subtleand comparisons to the QCD result are more involved. [email protected] [email protected] a r X i v : . [ h e p - t h ] A p r ontents f − ρ transition form factor . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 π − ρ transition form factor . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 q/(cid:126)x dependence of the form factors . . . . . . . . . . . . . . . . . . . . . . 144.4 Field theory expectations for the transition form factors . . . . . . . . . . . 184.5 γ ∗ γ ∗ → π , F πρ and VMD . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 B.1 F µν F µν contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 F µν F αβ (cid:15) µναβ contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Understanding the generic behavior of gauge theories remains as one of the most fun-damental problems in theoretical physics. At weak coupling a perturbative treatment isamenable, however the strong coupling dynamics represents an incredible challenge. It isbelieved that this regime can be understood in terms of a string theory. This correspon-dence has been made more precise for a certain class of gauge theories over the past decade,through the use of gauge/gravity duality [1].It is known that the dynamics of gauge theories differs significantly depending onwhether or not they contain fields in the fundamental representation of the gauge group.One of the most obvious features of having such fields is that there is the possibility offorming bound states. At weak coupling these bound states appear as positronium, i.e. asystem analogous to the hydrogen atom but composed of a quark and an antiquark. Inorder to probe the strong coupling dynamics of flavored gauge theories, it is interesting tostudy these objects at large λ , where λ is the ’t Hooft coupling. A very natural tool toadopt is that of gauge/gravity duality. However including fundamental matter is a difficultproblem. A step forward was taken in [2, 3, 4, 5], were it was suggested to introduce flavor2s a new open string sector coming from an extra stack of branes (so-called ’flavor’ branes)intersecting the color branes. In the limit in which the number of flavor branes is so smallthat they can be considered a small perturbation, we can perform the geometric transitionand replace the color branes by their near horizon geometry - where we should considerthe flavor branes as probes. This sort of quenched approximation has a number of conse-quences, one of which is that the running of quarks in loops is absent. This translates, inparticular, into a vanishing beta function for the gauge coupling. However, in the case ofmassive flavors, the conformal symmetry is broken in the IR, leading to the existence of“mesons”. These bound states were studied for the first time in [6] (for reviews see [7, 8]).It is only very recently that fully backreacted solutions, corresponding to an unquenchedapproach, have been found in [9, 10, 11, 12, 13, 14].In this paper we will be interested in the strong coupling structure of these mesons.Following the approach in [15], we will probe them with photons. As anticipated, inthe approximation we will work on, the beta function for the gauge coupling vanishes.Then, it is to be expected the large momentum transfer regime of the scatterings we wil becomputing, which is insensitive to the IR relevant mass term, to be controlled by conformalinvariance. Related processes have been considered in the literature using a gravity dualfor QCD, such as [16, 17, 18, 19, 20, 21, 22, 23, 24], and also [25, 26] where gravitationalform factors have been computed. Note that in our case, the gravity dual captures thestrong coupling regime of the theory. Thus, as opposed to real QCD, in our case the largemomentum transfer regime will be dominated by a strongly coupled conformal theory.The fact that conformal invariance is recovered in the UV is translated into an appropriatedictionary which allows to use the scaling coming from naive parton counting valid at weakcoupling, along the lines of [27, 28, 15].In order to study the mesons, we will consider the simplest theory admitting a gravitydual and containing a mass gap, which can be engineered as a D D In more adequate terminology, we will be computing electromagnetic transitionform factors. This requires to couple the gauge theory to electromagnetism, howeverfrom the point of view of the SU ( N c ) dynamics, the U (1) EM is just a global symmetry.Technically this allows us to consider the EM current as a U (1) subgroup in the SU ( N f ),corresponding to the gauge field on the flavor brane. This will require us to find theadequate couplings in the meson effective theory, allowing us to compute the desired formfactors. Note that in [15] the vector field probing the mesons was the full SU ( N f ). Afterintroducing the field theory and its gravity dual in section 2, we derive the correspondinginteraction lagrangian allowing us to compute such form factors in section 3. In section4 we compute and analyze these transition form factors. In accordance with the resultsin [15], we are able to match the expectations from QCD at large momentum transfer.This is to be expected since, in that regime, both QCD and our theory are dominatedby conformal invariance. Interestingly we can make use of the interaction lagrangian to The massless limit of this theory was considered in [29], where quark scattering is computed along thelines of [30]. It would be interesting to apply these methods to the massive (non-conformal) case. For example, this approach is similar to that in [31]. γ ∗ γ ∗ π . As opposed to the form factors, this case is more contrivedand we do not have a fully satisfactory field theory picture. On the other hand, this processwill be related to the γ ∗ πρ form factor due to vector meson dominance in much the samespirit as in QCD. In section 5 we examine the full amplitude, in which the analog of thehadronic tensor exhibits a Callan-Gross relation. This is deeply connected with the helicitystructure of our amplitudes. We finish in section 6 with some comments and suggestionsfor future directions. The theory in question consists of N = 4 SYM coupled to N f fundamental hypermultipletsin such a way that the final theory preserves N = 2 supersymmetry. Generically thehypermultiplets will be massive, and we will assume a diagonal mass matrix. It is importantto note that our theory is non-chiral even in the massless limit. In particular this meansthat the flavor symmetry is just SU ( N f ). The field content is SU ( N c ) SU ( N f )Φ I Adj Q i (cid:3) ¯ (cid:3) ˜ Q i ¯ (cid:3) (cid:3) and the superpotential reads W = ˜ Q i ( m q + Φ ) Q i + Φ I Φ J Φ K (cid:15) IJK , (1)The Φ I , I = 1 , , N = 4 SYM sector, whilst the( Q, ˜ Q ) flavor hypermultiplets break the supersymmetry down to N = 2. The mass termadditionally breaks the U (1) R symmetry. Let us set m q to zero for a moment. In thatcase there is an R -symmetry under which R Q = R Φ I = . Assuming we are closeto a conformal fixed point, we can compute the exact beta function of the theory byapproximating γ i ∼ R i −
2. It is then straightforward to see that β g Y M = dd log µ π g Y M = − N f . (2)Thus we see that the theory is not asymptotically free, but rather develops a Landau polein the UV. However we will treat the theory in the large N c limit. The beta function forthe ’t Hooft coupling reads β λ = dd log µ π g Y M N c = dd log µ π λ = − N f N c . (3) This R-charge assignation is the one coming from a-maximization, and is indeed the one adapted tomatch the beta function coming from the gravity description (see for example [32], or [33] for a discussionwith D7 branes). N f /N c ∼ N f /N c ∼
0. Note that even though the gauge coupling has a vanishing beta function,conformal invariance will be broken in the IR by the scale set by m q .It is important to note that in the N f /N c ∼ SU (2) R × SU (2) globalsymmetry, of which the SU (2) R is an R-symmetry (and therefore does not commute withthe supercharges), whilst the other SU (2) is a global symmetry. As noted above in thecase of massless hypermultiplets the R-symmetry is enhanced back to SU (2) R × U (1) R . The theory above can be engineered as a brane web. Consider the D D N c D × × × N f D × × × × × × × Working at small ’t Hooft coupling, upon taking the decoupling limit, the local dynamicson the D N = 4 SYM fields, while the 3-7 strings generate the flavorhypermultiplets. Without loss of generality let us localise the D ,
9) plane.Then the i -th such D (cid:126)z i = ( x i , x i ), which is at a distance L i = (cid:112) ( x i ) + ( x i ) . This distance, in units of 2 πα (cid:48) , defines the mass of the i -th hypermultiplet.However, for simplicity, we will assume that all the masses are equal, corresponding to aconfiguration where all the D (cid:126)z = ( x , x ). In that casewe recover the full SU ( N f ) flavor symmetry with m q = L/ (2 πα (cid:48) ).We can provide a closed string description of the system which captures the strongcoupling regime of the theory by considering the gravity dual of the above system. Inthe N c /N f ∼ D N f probe D N c D AdS × S which has a constant dilaton, translating into a vanishing beta function for the gauge theory’t Hooft coupling in agreement with our discussion above. Note that the 7-7 strings are non dynamical in the gauge theory.
5n order to describe the flavor D7 embeddings, we can write the
AdS × S metric as ds = (cid:126)x + (cid:126)z R dx , + R (cid:126)x + (cid:126)z ( d(cid:126)x + d(cid:126)z ) , (4)where (cid:126)x = ( x , · · · , x ). Working in static gauge, the D x , , (cid:126)x ), whilst sitting at fixed (cid:126)z = L . It is now straightforward to write theinduced metric on them in polar coordinates as ds D = ( r + L ) R dx , + R ( r + L ) ( dr + r d Ω ) , (5)As usual, the radial coordinate on the D r , corresponding to the UV of the field theory, the metricapproaches AdS × S . Additionally, since the ’t Hooft coupling is constant, we see that thetheory approaches a conformal fixed point in the UV. However in the IR, the metric abovedeviates from pure AdS because of the presence of the IR scale L . Since m q = L/ (2 πα (cid:48) )we see that conformal invariance is lost because of the scale m q , which introduces a massgap in accordance with the field theory analysis above.From the supergravity we can also read off the resulting R -symmetry of the field theory.The (cid:126)x coordinates on the D SO (4) ∼ SU (2) × SU (2) symmetry. However the AdS × S background also has a 4-form RR potential which can couple to the D
7. Indeedthe symmetry which interchanges the two SU (2) is broken by the Chern-Simons term onthe D SU (2) becomes the SU (2) R while the other remainsas the global symmetry SU (2). In the case of massless quarks, the D D ,
9) plane, which correspondsto the U (1) R . Since our theory is not conformal in the IR we expect it develops a mass gap, generating ameson spectrum. At weak coupling these mesons are positronium-like systems, however weare interested in their strong coupling description. In order to investigate this we shouldanalyze the 3-7 strings corresponding to the quark fields, but in the dual gravity descrip-tion which captures the strong coupling. As we have argued before, after the geometrictransition the strong coupling gravity dual is in terms of N f probe D D i.e. the mesons we want to study, will correspond to 7-7strings. One can see that these 7-7 strings fall into two distinct sectors: large macroscopicspinning strings corresponding to mesons with large spin; and small strings captured by theflavor D M is of order m M = m q / √ λ , as opposed to the mass of the high spin mesons which is at least of order m M λ .6herefore in the strong coupling regime we see that higher spin mesons are much moremassive than low spin mesons. This hierarchy allows us to concentrate on DBI mesonswhilst forgetting about the more stringy large spin states. Therefore for the mesons ofinterest, the spectrum can be computed by considering fluctuations, up to quadratic order,of the DBI+CS action of the probe flavor branes. Let us consider our D (cid:126)z = ( L,
0) where the scalar fluctuations will be (cid:126)z = ( L + Φ , Φ ). In order to havecanonical mass dimensions we must re-scale the field to Φ i = 2 πl s χ i . Additionally, we haveto take into account the fluctuations of the gauge field on the D
7. After considering thequadratic expansion for the flavor branes action, one can see that the scalar wavefunctioncorresponding to a field of mass m n,l is given by (see [6] for example) χ i = e p f x Φ M ( r ) Y l ; m M = m n,l = 2 m q λ ( n + l + 1)( n + l + 2) , (6)where Y l is the S spherical harmonic which specifies the SU (2) R × SU (2) quantum numbersof the meson ( l , l ). The function Φ M is a radial function with quantum numbers M = { n, l } given by Φ M = Φ n,l = w l (1 − w ) l F ( − − l − n, l + n, l + 2 , w ) , (7)where we have introduced the coordinate w defined through r L = w − w ; w ∈ [0 , . (8)From the asymptotic behavior of this mode one can see that it is dual to a scalaroperator of conformal dimension ∆ = l + 3, which schematically reads( ˜ Q Φ l Q ) θ ¯ θ = ˜ ψ ˜ Q φ l ψ Q + · · · , (9)where ψ Q , ˜ ψ ˜ Q are the fermions in the Q, ˜ Q supermultiplets and φ is the scalar in Φ.From the eigenmodes of the vector field on the D ρ mesons whose wavefunctions are (again see [6] for more detail) ρ µ = (cid:15) µ e px Φ IIM ( r ) Y l ; m M = m n,l = 2 m q λ ( n + l + 1)( n + l + 2) ; (10)where the polarization vector satisfies the gauge condition (cid:15) · p = 0. The Y l is the l-thspherical harmonic specifying the SU (2) R × SU (2) ( l , l ) representation, whilst Φ IIM is aradial function with quantum numbers M = { n, l } given byΦ IIM = Φ
IIn,l = w l (1 − w ) l F (2 + l + n, − − l − n, l + 2 , w ) . (11)This wavefunction corresponds to a spin-1 operator of conformal dimension ∆ = l + 3schematically given by [15] 7 Q † Φ l Q − ˜ Q Φ l ˜ Q † ) θ ¯ θ = q † φ l ∂ µ q − ˜ qφ l ∂ µ ˜ q † + · · · , (12)where q, ˜ q, φ stand for the lowest (scalar) components in the Q, ˜ Q, Φ supermultiplets.Both the scalar and vector meson modes correspond to normalizable fluctuations. How-ever, we can construct the non-normalizable fluctuations starting from the same equationof motion. In particular we will be interested in the vector field non-normalizable modesince, as clear from (12), the l = 0 case reduces to the flavor current. This current is aglobal symmetry, exactly as EM is to QCD. Therefore we will refer to the “photon” asthe non-normalizable mode arising from the vector field on the D SU ( N f ), but we will choose some U (1) subgroup as our electromagnetic cur-rent. Therefore we will neglect the non-abelian dynamics on the D A µ = χ µ e qx A ( r ) Y , χ · q = 0 , (13)where we keep explicit the (trivial) S dependence through Y . However since this sphericalharmonic is a constant we will drop it in our computations. Also note that A = πα (1 + α )sin( πα ) F ( − α, α, , w ) , α = 12 ( − (cid:115) − q λm q ) . (14)By expanding the effective DBI+CS action on the D N c . Therefore we will keep the lowest order terms at which we find the desired interactionvertices as the main contribution to the process in which we are interested. In our particularcase we want to probe the internal structure of mesons with photons. Since our photonactually comes from the non-normalizable mode of the vector field on the brane, the lowestorder interactions will come from terms in the expansion of the DBI+CS which involvetwo (not necessarily identical) mesons plus a gauge field, which we will interpret as theEM current. Clearly at least one of the mesons should be a vector meson in order tocontract the indices of the EM current, so from this point of view, it is clear that we willfind interaction vertices allowing us to compute scalar-vector transition form factors. Wewill confirm this by direct computation. DBI action:
Starting with the DBI lagrangian for the D (cid:15) in such a way that the DBI reads S = − T (cid:90) r (cid:112) ˆ g (cid:112) det(1 + (cid:15) ) ; (cid:15) IJ = g IL ( h ∂ L (cid:126) Φ ∂ J (cid:126) Φ + 2 πα (cid:48) F LJ ) , (15)8ere capital latin indices run over the worldvolume coordinates of the D
7, and √ ˆ g is thedeterminant of the internal unit S . Note that any overall factors of the warp factor cancelout because of having D g depends on the warp factor h , which explicitly depends on the fluctuations (cid:126) Φ. Thereforeeven though we will expand in powers of (cid:15) , at each order a further expansion of g is implicit.To lowest order we find that (cid:112) det(1 + (cid:15) ) = 1 + 12 T r ( (cid:15) ) − T r ( (cid:15) ) + 18 ( T r ( (cid:15) )) + . . . (16)Clearly, the linear term will not contribute. From the quadratic terms, to lowest orderin the implicit expansion of g , we will obtain the quadratic action leading to the abovewavefunctions. However we will also get extra terms, which in particular contain theinteraction lagrangian S iDBI = − T (2 πα (cid:48) ) (cid:90) (cid:112) ˆ gr (cid:110) LR ( r + L ) Φ F µν F αβ η µα η νβ + 2 L ( r + L ) R Φ F ri F rj ˆ g ij + L ( r + L ) R Φ F ij F kl ˆ g ik ˆ g jl (cid:111) . (17)Here latin indices run over the S , while greek ones are along Minkowski directions.One can convince oneself that higher orders in the expansion of (16) will contribute tohigher point functions, so the expansion in (16) is indeed enough for our purposes.In the interacting lagrangian we will assume that one of the field strengths correspondsto a non-normalizable gauge field. By inspecting the non-normalizable mode above, it isclear that the only contribution will come from the first term - which in turn requires theother field strength to be that of the massive vector field. Therefore, the vertex on whichwe will focus is S iDBI = − T (2 πα (cid:48) ) (cid:90) (cid:112) ˆ gr (cid:110) LR ( r + L ) χ F µν F αβ η µα η νβ (cid:111) , (18)where we have extracted the 2 πα (cid:48) factor in Φ to write the lagrangian explicitly as acoupling to χ .As advertised, our interaction involves a photon, a vector meson and a scalar meson.This structure is deeply connected with the fact the flavors (and therefore the mesons)come from fluctuations of a D F µν times some tensor The vector field on the D B field ora worldvolume instanton (going to the Higgs phase of the theory [34, 35]), this restrictioncan be avoided. It would certainly be interesting to compare this with our results, whichare essentially probing the Coulomb branch of the theory. CS action:
The
AdS × S background has a non-zero 4-form potential whose electric componentis given by C (4) = ρ R dx ∧ dx ∧ dx ∧ dx . (19)where ρ = r + (cid:126)z .The relevant coupling in the CS of the flavor D πT α (cid:48) (cid:90) C (4) ∧ F ∧ F . (20)which we can integrate by parts to write as a function of the G (5) πT α (cid:48) (cid:90) A ∧ G (5) ∧ F . (21)where we denote by G (5) the 5-form field strength derived from C (4) , and A is the world-volume vector field whose corresponding field strength is F . After some algebra one cansee that G (5) = 4 R r + ( L + Φ ) + Φ ) (cid:18) r ω ∧ dx ∧ dx + r x dr ∧ ω ∧ dx − r x dr ∧ ω ∧ dx (cid:19) (22)Since (20) contains two factors of the gauge field, the lowest order which contributes toour interaction vertex will come from a term in G (5) containing just one factor of the scalarfluctuation. Clearly this can only arise from the last two terms, where the pull-back of G (5) forces us to select the fluctuation through its derivative along the Minkowski directions. Itis not hard to convince oneself that finally the relevant CS contribution is S iCS = 2 T (2 πα (cid:48) ) R L (cid:90) r √ ˆ g ( r + L ) A ∧ d Φ ∧ F . (23)Recalling that d Φ actually stands for the derivative along the Minkowski direction, andextracting the 2 πα (cid:48) dependence from Φ , we can integrate this by parts to get S iCS = T (2 πα (cid:48) ) R L (cid:90) r √ ˆ g ( r + L ) (cid:110) χ F αβ F µν (cid:15) αβµν (cid:111) . (24)10he lagrangian above demands us to interpret χ as a pseudoscalar, since otherwise theeffective meson theory would violate parity. In the UV theory we can set the θ angle tozero, which is dual to taking the RR scalar C (0) to zero. Since in the N f /N c ∼ χ as a pseudoscalar.We can provide an additional motivation for this assignment by assuming that the D (cid:126)L = ( L , L ). It is then straightforward to repeatthe computation above and show that the full interacting lagrangian (DBI+CS) actuallyreads S i = T (2 πα (cid:48) ) R (cid:90) r √ ˆ g ( r + (cid:126)L ) (cid:110) (cid:126)L · (cid:126)χ V F µν F αβ η αµ η βν + 13 (cid:126)L · (cid:126)χ A F αβ F µν (cid:15) αβµν (cid:111) ; (25)where (cid:126)χ V = ( χ , χ ) and (cid:126)χ A = (cid:15) ij χ i = ( χ , − χ ). Thus we see that the sign of theCS term actually depends on the choice of skewness of the (8 ,
9) directions. We canconsider the interacting lagrangian above just at the level of pure field theory, and supposenow that under a parity transformation (cid:126)x → − (cid:126)x we should also consider the combinedtransformation ( L , L ) → ( L , L )( χ , χ ) → ( χ , χ ) (26)Under this transformation it is clear that (cid:126)L · (cid:126)χ V behaves as a scalar, whilst (cid:126)L · (cid:126)χ A picks anextra minus sign compensating the minus sign picked up by F ˜ F . Therefore the transfor-mation (26) allows for conserved parity.We would like to heuristically motivate it yet another way. We could start with theSUGRA background and impose (cid:126)x → − (cid:126)x as a symmetry. Clearly the metric is left in-variant, however the electric part of the 5-form field strength picks up a minus sign. Since G (5) must be self-dual we have to ensure that its magnetic part also picks a minus sign.By inspection of (22) one can achieve this by reversing the skewness of the (8 ,
9) plane.When looking at the linearization of the 5-form one can check that it indeed reduces to(26). If we now choose the particular vacuum L = 0 , L = L we see that effectively it islike considering χ as a scalar and χ as a pseudoscalar.The final interaction lagrangian which we will be using reduces to S i = − T (2 πα (cid:48) ) LR (cid:90) r √ ˆ g ( r + L ) (cid:110) χ ( F A ) µν ( F ρ ) αβ η µα η νβ + 13 χ ( F A ) αβ ( F ρ ) µν (cid:15) αβµν (cid:111) , (27) Even if we took a pure gauge but non-vanishing C (0) , it would not couple to this order; suggestingthat indeed our effective lagrangian is insensitive to the parity-violating sector of the theory. A and ρ to remind the reader that one of the fieldstrengths corresponds to the non-normalizable field dual to the photon, while the othercorresponds to the in/out ρ meson state.It is important to note the difference in the measure with respect to the form factorscomputed in [15]. The additional suppression by ( r + L ) − will be crucial in order toget the expected large q behavior of our form factors. One can heuristically understandthis dependence in much the same spirit as how we motivated the scalar-vector-photonvertex. Because of the DBI+CS structure, as discussed, this is the lowest order term wecould have. Additionally since the scalar appears without derivatives, it could only comefrom the expansion of a term schematically of the form hF . In order to get a singlepower of the scalar it should appear in the combination L Φ in order to have dimensionsof (length) , however on dimensional grounds, each time this combination appears it mustbe suppressed by an extra power of the other dimensionful quantity in the theory, namelythe combination r + L . Therefore an extra suppression on ( r + L ) − with respect to[15] is to be expected. Armed with the interacting lagrangian (27) we can now turn to the actual problem ofcomputing the electromagnetic form factors. As we discussed, we have to understand χ as a scalar and χ as a pseudoscalar. Even though this pseudoscalar is not the pseudo-Goldstone boson of any broken chiral symmetry, we will call it π , since at least its effectivecouplings are identical to those of the neutral pion. In turn, the scalar behaves as a scalarneutral meson such as the f (or σ ).Generically, the situation we will consider is that in figure (1), where the momenta arechosen so that p + q = p (cid:48) . γ ∗ q p p ! f , π M = ( m, l ! ) ρ ! µ , N = ( n, l ) Figure 1: Transition form factors for π/f − ρ mesons Maybe it would be more convenient to call it η (cid:48) , since our theory does not have a chiral non-abeliansymmetry but has an approximate UV chiral U (1) R . It is also possible to argue that there should be apseudoscalar coupling to EM as χF ˜ F . .1 f − ρ transition form factor Using the expressions for the non-normalizable vector field, scalar and vector normalizablemodes, it is straightforward to see that (cid:104) f , M | J µ | ρ, (cid:15), N (cid:105) = 2 T (2 πα (cid:48) ) LR (cid:16) (cid:90) r ( r + L ) A Φ M Φ IIN δ l,l (cid:48) (cid:17)(cid:104) ( p (cid:48) · q ) (cid:15) µ − ( q · (cid:15) ) p (cid:48) µ (cid:105) . (28)Here we have already performed the integral over the S , which is simply (cid:90) (cid:112) ˆ g Y l Y l (cid:48) = δ l,l (cid:48) , (29)since the spherical harmonics are orthonormal eigenfunctions of the laplacian on S .Let us define the radial integral I n,m,l ( q ) = (cid:90) r ( r + L ) A Φ M Φ IIN δ l,l (cid:48) = (cid:90) r ( r + L ) A Φ m,l Φ IIn,l . (30)It is important to notice that it will be a function of the momentum q of the off-shellphoton. Then (cid:104) f , M | J µ | ρ, (cid:15), N (cid:105) = F σρn,m,l (cid:104) ( p (cid:48) · q ) (cid:15) µ − ( q · (cid:15) ) p (cid:48) µ (cid:105) ; (31) F σρn,m,l = 2 T (2 πα (cid:48) ) LR I n,m,l ( q ) . (32)Because of current conservation, q µ (cid:104) f , M | J µ | ρ, (cid:15), N (cid:105) = 0. Let us concentrate on the tensorstructure of the form factor. If we go to the rest frame of the vector meson the form factorreduces to (cid:104) f | J µ | ρ (cid:105) ∼ m ρ (cid:104) − q (cid:15) µ − ( q · (cid:15) ) δ µ (cid:105) = m ρ (cid:104) − q (cid:15) δ µ − q (cid:15) i δ µi + q (cid:15) δ µ (cid:105) = − m ρ q (cid:15) i δ µi . (33)We might now choose to align (cid:126)q with the z direction. Current conservation requires thenthat q µ (cid:104) f | J µ | ρ (cid:105) = 0, which in turn implies that (cid:126)(cid:15) = ( (cid:15) x , (cid:15) y , (cid:15) givesa non-zero contribution to the transverse part of the off-shell current, reflecting the factthat the transition is between a spin 0 and a spin 1 state, and thus should involve the spin1 part of the current.For later purposes, let us note that, in the Breit frame where (cid:126)p + (cid:126)p (cid:48) = 0, we will findthat all the momenta are of order q . In that case, we see that roughly speaking, thelarge q behaviour of the matrix element will be given by I n,m,l ( q ) q . In that frame (cid:126)p (cid:48) = − (cid:126)p = (cid:126)q , and we may choose (cid:126)q = (0 , , q ). A boost along the z direction connectsthis frame with the rest frame of the vector meson. Since in the rest frame (cid:15) = (0 ,(cid:126)(cid:15) ⊥ , q ∼
0, then p = ( | (cid:126)q | , − (cid:126)q ), p (cid:48) = ( | (cid:126)q | , (cid:126)q ), q = (0 , (cid:126)q ). .2 π − ρ transition form factor It is also straightforward to evaluate the CS interaction term. It reads (cid:104) π , M | J µ | ρ, (cid:15), N (cid:105) = F πρn,m,l (cid:15) µναβ (cid:15) ν q α p (cid:48) β ; (34) F πρn,m,l = 43 T (2 πα (cid:48) ) LR I n,m,l ( q ) ; (35)where I n,m ( q ) is the same integral (30) as above. One can check that also current conser-vation is satisfied and q µ (cid:104) π , M | J µ | ρ, (cid:15), N (cid:105) = 0.Again we can go to the rest frame of the vector meson where (cid:104) π | J µ | ρ (cid:105) ∼ m ρ (cid:15) µνα (cid:15) ν q α ∼ m ρ (cid:15) µij (cid:15) i q j (36)and we again see that only the vector part of the polarization of the vector meson isinvolved - and therefore only the spin 1 part of the current is involved in the interaction.Note that again, in the Breit frame, the magnitude of the whole matrix element forlarge q will be of the order I n,m,l ( q ) q . We also point to the results obtained in [16, 17,18, 19, 20, 20] for the form factor, albeit in a slightly different set-up. q/(cid:126)x dependence of the form factors By inspection of the two form factors, we have that F πρn,m,l = F σρn,m,l . Since in addition F σρn,m,l ∼ I n,m,l , we will loosely identify I n,m,l with the form factors of interest. The factthat both form factors are proportional one to the other should be due to SUSY, sinceboth the two scalars are in the same N = 2 massive supermultiplet.In order to go further, we should study I n,m,l . Following the method suggested in [15],we can do the j -th integration of A with respect to w - which we will call a j . Recall that A is a function of the photon momentum, so for integration we must recall that a j = a j ( w, q ).Denoting by F the rest of the hypergeometric functions under the integral in (30), we caniteratively integrate this by parts to obtain I n,m,l = (cid:90) A F = (cid:90) ∂ ω a F = ( a F ) | w =1 w =0 − (cid:90) a ∂ ω F == ( a F ) | w =1 w =0 − ( a ∂ ω F ) | w =1 w =0 + (cid:90) a ∂ ω F = · · · . (37)It can be easily checked that a j | w =0 = 0. The crucial observation then is that only a finitenumber of derivatives of F are non-vanishing when evaluated at w = 1. We find that thelast non-zero derivative is the j max = 2 l + n + m + 3. Analogously one can check that thefirst non-zero derivative is the j min = l + 2. This way we see that indeed I n,m,l = j max (cid:88) j min ( − j a j +1 ( q ) ∂ jw F ; (38)14here both a j +1 and ∂ jw F are evaluated at w = 1. By iteration one can, in principle,determine the value of the integral by obtaining all the higher order coefficients.We could take a seemingly different approach and make direct use of VMD (see appendixA). It is possible then to rewrite (30) as I n,m,l = m q λ k max (cid:88) k =0 f k, R n,l,m,l,k, q + m k, ; R n,l,m,l,k, = (cid:90) r ( r + L ) Φ m,l Φ IIn,l Φ IIk, , (39)where R n,l,m,l,k, is proportional to the coupling constant between hadrons with the speci-fied quantum numbers, and f k, is the decay constant of the vector meson with quantumnumbers ( n, l = 0). A priori k in (39) should take values in the range k ∈ [0 , ∞ ). Howeverit can be checked that R n,l,m,l,k, = 0 for k > k max , with k max = 2 l + n + m + 2, whichtruncates the sum and ensures its finiteness. Additionally we should point out that theminimal k depends on the particular choice of ( n, m ). For example, for n = m the sumextends all the way down to k = 0.Expression (39) should coincide with (38). However, it has a compelling interpretationsince it explicitly allows us to see the dependence on the decay and coupling constants.Indeed expression (39) is the manifestation of VMD in gauge/gravity duality. It allowsus to regard the photon-hadron interaction as a photon going into a vector meson whichis indeed the one which interacts with the hadrons, in accordance with the vector mesondominance principle. The pictorial representation is in figure (2). ! ρ int ρ int ρ N ( f , π ) M γ ∗ ρ N ( f , π ) M γ ∗ = f R Figure 2: Vector meson dominance.It should be pointed that even though VMD naturally falls out in the gravity construc-tion, universality does not generically hold. See [36] for further details on this.The necessary agreement between (39) and (38) requires some, a priori, non-obviousrelation between masses, decay constants and coupling constants. We will make explicituse of some of these properties below.It is expected that form factors in position space carry information about the chargedistribution of the hadrons. This interpretation is most straightforward for diagonal formfactors ( i.e. form factors in which the in meson is identical to the out meson). Howeverin the case of off-diagonal form factors (transition form factors) we could think of it as thedistribution of charge at the interaction point.There are, however, subtleties in how one should extract this kind of information inposition space from a form factor computed in momentum space. For example, in a non-15elativistic system the 3-dimensional Fourier transform of the form factor with respectto (cid:126)q would give the spatial charge distribution. However since our mesons have a veryhigh binding energy, one should expect our system to be highly relativistic. It has beenargued in [15] that, in order to have the right probabilistic interpretation (as well as aconnection to generalized PDF, see for example [37, 38, 39]), it is natural to switch tothe large momentum frame, and interpret the photon as probing the transverse structureof the hadron. As suggested in [15], we might then consider aligning the initial hadronmomentum along the z direction, and boosting the system to large momentum along it.Then choosing q = (0 , (cid:126)q ⊥ ,
0) we can perform a 2-dimensional Fourier transform
F T (cid:16) f ( q ) (cid:17) = 12 π (cid:90) d (cid:126)q ⊥ e i(cid:126)q ⊥ (cid:126)x ⊥ f ( q ) ; F T (cid:16) q + m k, (cid:17) = K ( m k, r ) , (40)where K ( x ) is the corresponding Bessel function. The function obtained by means of this2d Fourier transform should be interpreted as a charge density in the transverse spaceparametrized by the transverse radius r . Restricting to the case at hand we have, from(39) I n,m,l = m q λ k max (cid:88) k f k, R n,l,m,l,k, K ( m k, r ) . (41)In the case of diagonal form factors, one possible definition of the size of the hadron is (cid:104) r (cid:105) = 4 ∂∂q F diag ( q ) | q =0 , (42)where the factor of 4 accounts for the fact that this is a transverse (2-dimensional) chargedistribution. We will use this definition and interpret it as a measure of the size of theregion where the interaction takes place. Then using (39) we have, for the σρ transitionform factor (cid:104) r σρ (cid:105) = 8 T (2 πα (cid:48) ) LR m q λ − k max (cid:88) k f k, R n,l,m,k, m k, . (43)Unfortunately we have been unable to explicitly perform the above sum.Finally let us note that, from the relation between F σρ and F πρ , we have (cid:104) r σρ (cid:105) = (cid:104) r πρ (cid:105) ,which follows trivially from the definitions of the two form factors. Large q behavior: Let us concentrate on the large q behavior of (30). The j -th integration of A , a j ,behaves like a j → j j ! (cid:16) m q λq (cid:17) j +1 . (44)16t is clear that the large q behavior of (30) will be controlled by the first non-zero derivativeof F at w = 1 (at w = 0 a j vanishes) since it will be the least suppressed term. Asanticipated, the first non-zero derivative is the l + 2. Therefore we see that at large q , theintegral (30) runs like I n,m → C n,m,l (cid:16) m q λq (cid:17) l = C n,m,l (cid:16) m q λq (cid:17) ∆ , (45)where C n,m,l is a numerical coefficient depending on n, m, l . Note that the q dependence iscompletely independent of n, m , and just relies on the conformal dimension of the operatorsinvolved. This is connected to the fact that the theory flows to a UV conformal point andso we see that conformal invariance alone governs the structure of the form factors.We could just as well use the alternate expression (39). Expanding (39) for large q wehave I n,m,l → I n,m,l = m q λ (cid:88) j ( − j ( q ) j +1 (cid:16) (cid:88) k f k, R n,l,m,l,k, ( m k, ) j (cid:17) . (46)We can re-write this as I n,m,l → I n,m,l = (cid:88) j ( − j j j ! (cid:16) m q λ q (cid:17) j +1 (cid:16) j j ! (cid:88) k f k, R n,l,m,l,k, ( √ λ m k, m q ) j (cid:17) , (47)so we conclude that the coefficients in (38) can be written as ∂ jw F | w =1 = 14 j j ! k max (cid:88) k f k, R n,l,m,l,k, (cid:32) √ λ m k, m q (cid:33) j . (48)This allows us to write the leading term for large q fixing the C n,m,l above I n,m,l → ( − l +2 (cid:16) k max (cid:88) k (cid:16) λ m k, m q (cid:17) ( l +2) f k, R n,l,m,l,k, (cid:17)(cid:16) m q λq (cid:17) ∆ . (49)Since from (38) we know that the first non-zero term is that with 1 / ( q ) l +3 , we concludethat (cid:88) k f k, R n,l,m,l,k, ( m k, ) j = 0 ∀ j < l + 2 . (50)This is an important constraint on the algebraic structure. For later purposes, let usconsider the function ( r (cid:54) = 0) h j ( r ) = (cid:88) k f k, R n,l,m,l,k, ( m k, ) j log( m k, r , (51)17aking its r derivative we have dh j ( r ) dr ∼ r (cid:88) k f k, R n,l,m,l,k, ( m k, ) j ; (52)so that by using (50) we see that when j < l + 2, h j is actually a constant.Let us analyze the form factor in position space. As argued above in order to havea sensible physical interpretation we must boost the system to the infinite momentumframe and assume q = (0 , (cid:126)q ⊥ , r . The large q region corresponds to small r . Expanding (41) andre-writing it in a suitable form, we see that for r ∼ I n,m,l → m q √ λ (cid:88) k (cid:88) j f k, R n,l,m,l,k, j j ! (cid:8) ψ ( j ) − log (cid:0) m k, r (cid:1)(cid:9)(cid:0) m k, r (cid:1) j , (53)where ψ ( j ) are numerical coefficients depending on j . The term dominating the sum abovewill be the one with the lowest exponent for r . Using (50) and the fact that h j is constantfor any j < l + 2, we conclude that, up to a constant (which on physical grounds must bezero), the small r dependence in transverse space is F σρn,m,l ∼ F πρn,m,l ∼ I n,m,l ∼ r l +2) log r ∼ ( r ) ∆ − log r , (54)where we have re-written the r power in terms of the conformal dimension of the operatorsinvolved. Note that the scale in position space at large q is set by 1 /m l +2 , . General behavior going towards the IR:
Let us now analyse the IR behaviour, i.e. the small q region. In position space thiscorresponds to the asymptotically large r region. From the asymptotic behavior of theBessel function we see that I n,m,l → √ πm q √ λ (cid:88) k f k, R n,l,m,l,k, e − m k, r √ m k, r ∼ √ πm q √ λ f ˆ k, R n,l,m,l, ˆ k, e − m ˆ k, r √ m ˆ k, r . (55)Where ˆ k is the lowest k for which R n,l,m,l,k, does not vanish. As we pointed out, thisminimal ˆ k depends on the particular choice of ( n, m ), which in turn sets the scale 1 /m ˆ k, of the measured charge distribution in position space for small q . As we have discussed, the UV of our theory is described by a conformal point. Thereforewe expect the large q behavior of our form factors to be controlled purely by conformalinvariance, in much the same way as in QCD - where asymptotic freedom is responsible forthe vanishing beta function at large q . However in that case the theory is weakly coupled18nd one can make use of perturbative tools to study the behavior of diverse processes atlarge q [40, 41] (see [42] for an exhaustive review).Rather than looking directly to form factors, it is useful to consider the full matrixelement, i.e. taking into account the scaling of the tensor structure. In the Breit frame,where all the momenta are of order q , we can identify the q -dependence of our matrixelement (recall equation (36)) as (cid:104) π , f | J µ | ρ (cid:105) ∼ q ) ∆ − . (56)On the other hand, for a conformal field theory at weak coupling, the expected scaling forthe transition form factor between a hadron h of helicity s and a hadron h of helicity s is [42] (cid:104) h , s | J µ | h , s (cid:105) ∼ q n − | s − s | ; (57)where n is the number of partons. Additionally, (57) requires us to impose the selectionrule that current helicity is given by λ = s + s . We can can heuristically understandthis formula in a free parton model. Assume that h , h are composed of n partons, eachcarrying a fraction of the total momentum q of the hadron. The off-shell photon wouldstrike one of them which, in the Breit frame, forces the struck parton to recoil. Since weare looking into elastic processes, for the hadron not to break we require the struck partonto emit a gluon to force the other partons to recoil. After power counting in this naiveparton model it is easy to see that one recovers (57).From (57) it is also clear that form factors in which helicity change is involved aresuppressed by additional powers of q [43, 44]. It is easy to understand this in the naiveparton model. The reason is that the vector boson vertex does not change helicity unlessthe partons are massive. In that case helicity flipping processes are suppressed by an extrapower of m/q .The discussion above is not limited to weak coupling, since in the end the is tied toconformal invariance ( i.e . naive dimension counting as if the beta function was zero).Indeed, it can be extended to strong coupling by replacing n by the twist of the lowesttwist operator capable of creating both hadrons [27, 28]. In the case at hand we have aspin 0 hadron (conformal dimension ∆) whose twist is τ S =0 = ∆, and a spin 1 hadron(conformal dimension ∆) whose twist is τ S =1 = ∆ −
1. Even though τ S =1 is smaller, thelowest twist operator capable of creating both hadrons has τ = ∆. Thus we can extend(57) to strong coupling by replacing n → ∆.It remains to discuss the helicities of the in and out hadrons. We saw that in both the π − ρ and f − ρ cases the transition was purely transverse. This means that s = 0 forthe f /π while s = ± ρ . According to the selection rule this implies that thepart of the current involved in the transition is the λ = ± For example, our matrix elements are schematically I ( q ) (cid:15) µναβ q α q β ∼ I ( q ) q .
19e found (we will see additional consequences of this when we study the full amplitude).Therefore on general grounds, we expect the matrix element to scale like (cid:104) h , s | J µ | h , s (cid:105) ∼ q − = 1( q ) ∆ − , (58)which is indeed the scaling we obtained.Note that in order to get this precise scaling the extra suppression by ( r + L ) − in (27)is crucial. At this point it is instructive to compare with the form factors computed in [15].For simplicity let us consider the spin 0 case in that paper. The corresponding integralleading to the form factor was very similar to (30), but without the extra suppression by( r + L ) − . This has the non-trivial effect of making the matrix element scale with anextra power of 1 /q − (technically it is due to the fact that the first non-zero derivative ofthe equivalent F would appear one order beyond). On the other hand these form factorsare between spin 0 states, and thus we expect that the extra suppression due to helicityflip in (58) to be absent. This justifies the extra power of 1 /q − . We can now re-analysethe appearance of this form factor in view of these results. The scalars of the theory arereal implying that there will be no minimal coupling to the vector field on the brane. Thusthe only possible trilinear combination is the one we obtained which, due to dimensionalreasons, requires the extra suppression with ( r + L ) − . Now we re-discover that the dualstatement is that the theory recovers conformal invariance in the UV, which dictates thescaling of the form factors. γ ∗ γ ∗ → π , F πρ and VMD The interacting lagrangian (27) allows us to study the process γ ∗ γ ∗ → π by consideringthe two vector fields to be non-normalizable modes. To be more precise, we will considerthe process shown in figure 3. γ ∗ γ ∗ π p p ! q Figure 3: γ ∗ γ ∗ → π process to be considered.By using the CS interaction (27), it is straightforward to see that this amplitude isgiven by i M γ ∗ γ ∗ → π = 43 T (2 πα (cid:48) ) LR ˆ I M ( q , p ) (cid:16) (cid:90) (cid:112) ˆ g Y l (cid:17)(cid:2) (cid:15) µναβ (cid:15) µ ( q ) (cid:15) ν ( p ) q α p β (cid:3) , (59) With a little bit of more work one can argue the same is true for the other form factors. I M ( q , p ) is given byˆ I M ( q , p ) = (cid:90) r ( r + L ) A ( q ) A ( p )Φ M . (60)and Φ M is the radial wavefunction of the π with quantum numbers M = ( n, l ).We can go to the rest frame of the final hadron, where we have (cid:126)q = − (cid:126)p , q + p = m π .It is straightforward to check that the tensor structure here reduces to m π (cid:15) ijk (cid:15) i ( q ) (cid:15) j ( p ) q k .Forgetting for a while about the angular integral, let us define the following form factor F γ ∗ πn = 43 T (2 πα (cid:48) ) LR ˆ I M ( q , p ) . (61)Using the VMD decomposition (see appendix A) we haveˆ I M ( q , p ) = m q λ (cid:88) n (cid:48) f n (cid:48) , p + m n (cid:48) , (cid:90) r ( r + L ) Φ n,l Φ IIn (cid:48) , A ( q ) ; (62)where f n, is the decay constant of the vector meson with quantum numbers n, l = 0. Wecan interpret this as in figure 4. = ! Figure 4: γ ∗ γ ∗ → π after using VMD.Each term in the sum is then the transition form factor for ρπ , with the caveat that theintermediate vector meson has l = 0, which follows trivially from VMD since the photonis the non-normalizable mode with l = 0 of the vector field - and as such can only mixwith vector mesons of l = 0. This suggests that we interpret the integral over the angularcoordinates as (cid:90) (cid:112) ˆ g Y l Y ; (63)where the Y spherical harmonic would correspond to the intermediate vector meson (whichhas l = 0). Then the normalisation condition requires (cid:90) (cid:112) ˆ g Y l Y = δ l, ; (64)so the final state will only contain the l = 0 π meson. Therefore we can define the fullform factor as F γ ∗ πn = T (2 πα (cid:48) ) LR ˆ I n, ( q , p ), where we make explicit the fact that onlythe l = 0 mode contributes. Explicitly 21 I M ( q , p ) = m q λ (cid:88) n (cid:48) f n (cid:48) , p + m n (cid:48) , (cid:90) r ( r + L ) Φ n, Φ IIn (cid:48) , A ( q ) = m q λ (cid:88) n (cid:48) f n (cid:48) , I n,n (cid:48) , ( q ) p + m n (cid:48) , . (65)We can re-write our form factor as F γ ∗ πn = f , m q λ − F ρπ , , p + m , + (cid:90) ∞ ds ρ h s + p . (66)where we have separated out the contribution of the ρ meson with quantum numbers (0 , i.e. the lowest one in the KK-tower. This is formula is analogous to the one obtained inQCD arising from VMD (see for example [45]).The spectral density reads in this case ρ h = 8 cT (2 πl s ) R Lm q λ − (cid:88) m (cid:48) (cid:54) =0 f m (cid:48) , δ ( s − m m (cid:48) , ) (cid:90) r ( r + L ) f n, Φ IIm (cid:48) , A ( q ) . (67)We can use once again the decomposition formula, and write ρ h = 8 cT (2 πl s ) R Lm q λ − (cid:88) m (cid:48) (cid:54) =0 (cid:88) m (cid:48)(cid:48) f m (cid:48) , f m (cid:48)(cid:48) , q + m m (cid:48)(cid:48) , R n, ,m (cid:48) , ,m (cid:48)(cid:48) , δ ( s − m m (cid:48) , ) . (68)The expression (68) for the spectral density follows from vector meson dominance. Theinteraction γ ∗ γ ∗ π (or γ ∗ γ ∗ σ ) can be seen as γ ∗ → ρ and ρρπ . = ! i,j i j R n, ,i, ,j, f i, f j, Figure 5: γ ∗ γ ∗ → π in the light of VMD.Then after separating out the lowest mass state, we have that the spectral density isjust the sum over higher mass states.It is interesting to look at the large momentum behavior of the above form factor. Notethat, in fact, we could switch the scalar with the pseudoscalar here, and the calculationproceeds in exactly the same fashion albeit with an additional factor of 2 /
3. From (65) wesee that for large q ˆ I n,n (cid:48) , → m q λ − (cid:88) n (cid:48) f n (cid:48) , p + m n (cid:48) , (cid:16) k max (cid:88) k (cid:16) λ m k, m q (cid:17) f k, R n, ,n (cid:48) , ,k, (cid:17)(cid:16) m q λq (cid:17) . (69)From (69) we can extract the relevant behavior when one of the virtualities is large andthe other is small to obtain F γ ∗ π ∼ / ( q ) . We can cross-check this result by using the22terative integration method of section 4. We would like to note that it is straightforwardto reapeat a similar computation with the scalar meson getting the same result up to anumerical factor.It is instructive to consider the corresponding process with a vector meson as the finalstate, i.e. the process γ ∗ γ ∗ ρ . This can be obtained from (5.43) in [15] if we assume thephoton is valued in SU ( N ). The crucial difference would be that the analog of (65) nowreads ˜ˆ I M ( q , p ) = m q λ (cid:88) n (cid:48) f n (cid:48) , p + m n (cid:48) , (cid:90) r ( r + L ) Φ IIn, Φ IIn (cid:48) , A ( q ) . (70)Restricting this to the case of one (almost) on-shell photon and the other with large (virtual)momentum ( p ∼ q (cid:29) /q . Therefore when both photons andvector meson are polarised in the transverse direction, we find that F γ ∗ ρ ∼ / ( q ) .Let us return to the pseudoscalar (or scalar) case ( γ ∗ γ ∗ π or γ ∗ γ ∗ σ ) assuming one pho-ton with large virtuality and the other almost on-shell. We found that the form factorscales like 1 /q . This scaling is different to that obtained in QCD, where the γ ∗ γ ∗ π formfactor goes like 1 /q . In principle one would expect these two results to match due toconformal invariance. Nevertheless, the γ ∗ γ ∗ π is more subtle than the form factors dis-cussed above. In pQCD (perturbative QCD) it is dominated by the one-quark propagator(see for example [42]), which in turn arises from the fact that the π meson is a 2-quarkbound state. By comparison with the case of the form factors, we see that in order tocompare weak coupling results with strong coupling results one needs, at least to replace τ ↔ n . As opposed to the form factors, in the case at hand the fact that the selection rulesets l = 0 obscures the identification of τ . However we can perform the integrals abovebefore taking l = 0. One can see that, considering (60), in order to have a well-behavedintegral we have to restrict ouselves to even values of l . Under that assumption, one cancheck that ˆ I ∼ /q l +6 , which upon taking l = 0 coincides with the result obtained usingVMD. Since l has to be even, we can re-write it as l = 2 l (cid:48) in such a way that the integralgoes like 1 / ( q ) l (cid:48) +3 . Defining a new twist operator τ = l (cid:48) + 3, the actual behavior of theintegral in which we are interested is 1 / ( q ) τ min , where τ min is the minimal twist, i.e. theone corresponding to l = l (cid:48) = 0. One could also consider the vector meson case for generic(again even) l , which goes like 1 /q l +4 . In terms of l (cid:48) this reads 1 / ( q ) l (cid:48) +3 − . Define now τ = l (cid:48) + 3 −
1, where the − / ( q ) τ min . Note how in this case the different suppression factor( r + L ) in the integrand, is crucial to obtain the extra factor of q which allows to inter-pret the exponent as a spin 1 hadron. Thus we see that the integral actually scales like1 / ( q ) τ min for both the vector and scalar cases. The “twist” is defined in terms of half of the l corresponding to the actual meson state (the factor of 3 is related to the dimensionalityof the “basic” l = 0 state, and it seems reasonable that it should be kept). This suggests The γ ∗ γ ∗ π channel is more sensitive to details of the theory than the form factors. It might be thatour theory, being non-perturbatively trivial, simply has a different structure than QCD. τ min ↔ n min /
2, where n min is the minimal number of valence partonsin a QCD hadron ( i.e. /q , which is precisely the QCD result. However we must warn the readerthat we do not have any compelling explanation for this identification.This process has been recently considered [46] in the context of the hard wall model of[47], obtaining that the large q behavior of the form factor matches that of QCD. Howeverin that case the model is designed to capture the same symmetries as low energy QCD, soit is expected a good agreement. It is interesting to compute the complete amplitude for the processes above. The physicalprocess which we are actually looking at is really either eπ → eρ or ef → eρ (or itscrossed channel). Suppose we are interested in the unpolarized cross-section in figure (9). e f , π e ρ k, s k ! , s ! Figure 6: Physical process.The matrix element comes from i M = − e q ¯ u s (cid:48) ( k (cid:48) ) γ µ u s ( k ) (cid:104) h | J µ | ρ (cid:105) . (71)where h stands for the initial hadron (either f or π ). Actually, the matrix elements (cid:104) h | J µ | ρ (cid:105) are nothing buth the ones we already computed.Squaring, summing over polarizations and averaging over spins, this takes the usualform |M| = e q L µν W µν , W µν = (cid:88) pol. |(cid:104) h | J µ | ρ (cid:105)| . (72)We can now compute W µν for each of the two cases in our theory. Interestingly in both casesthe tensor structure leads, after summing over polarizations, to a Callan-Gross relationbetween the “structure functions” (see appendix B). This should come as no surprise. We24xplicitly saw how the transverse character of our transitions was responsible for the large q behavior of the form factors. The fact that we recover Callan-Gross scaling here isanother consequence of having a transverse transition. Summarizing our results; f meson as in state: F n,m,l = q x ( F σρn,m,l ) ; F n,m,l = q x ( F σρn,m,l ) . (73) π meson as in state: F n,m,l = q x (cid:0) x m ρ q (cid:1) ( F πρn,m,l ) ; F n,m,l = q x ( F πρn,m,l ) ; (74)where we indicate the quantum numbers of the (pseudo)scalar n and vector meson m ,which share the same l . Note that the x above is the Bjorken x , which in our quasielasticcase is fixed eventhough we will keep it as open.The results above are quite reminiscent of DIS (Deep Inelastic Scattering) structurefunctions. Summing over possible final states we can construct an inelastic scatteringamplitude. However we have to remember that our computation does not allow for theproduction of high spin states. Therefore if we want to interpret our results in terms ofDIS we have to restrict ourselves to a regime in which the production of such states ishighly suppressed. These high spin states are much more massive than the low spin stateswe have been considering in this paper. Therefore if we consider the DIS experiment inwhich we have a final state with N particles (in our theory to leading order in λ − N = 1)whose 4-momenta add to W , whilst the initial hadron h and off-shell photon have momentarespectively p and q - we have the trivial relation W = p + q . Squaring this we find W − m h = q (1 − x ) . (75)Since we don’t want to allow for final state masses much larger than the initial statemass, we should only consider DIS in the region where we take q → −∞ and x → W − m h remains finite and small. Thus we see that we could only accessthe x ∼ i.e. quasi-elastic) regime of DIS. In principle it should be possible to connectthe threshold regime of the DIS with the form factors. At weak coupling this was studiedin [48, 49], however at strong coupling one must be more careful since the calculationproceeds slightly differently [28]. Generically one would expect F DIS ∼ F F orm F actor G ( q (1 − x − )) . (76)However a detailed discussion of these issues is beyond the scope of the current work. We thank G. Gabadadze for pointing this out to us. Conclusions
In this paper we have been concerned with the structure of quark-antiquark bound states(mesons) at strong coupling. In order to study them we have probed these mesons with anexternal electromagnetic field. Together with the results [15, 27], the picture that emergesis that the large q behaviour of the matrix elements is dominated entirely by conformalinvariance.This provides some justification as to why the scaling at weak coupling, based on a naiveparton model, extrapolates to the strong coupling regime upon the replacement n ↔ τ . Inthe particular cases we have studied the helicity dependence of the matrix elements appearsin a very explicit manner. The U (1) form factors we computed are non-vanishing in thelarge N limit, only for different in and out hadrons ( i.e. they are transition form factors).As we pointed out earlier, this is determined by the precise structure of the SUGRAlagrangian. It also follows from the index structure and reality of the worldvolume fieldson the brane, that the transitional form factors involve fields of different spin. Therefore thehelicity dependence of the matrix elements appears in an explicit manner. The necessarypowers of q , required to account for the helicity change in the amplitude, have a preciseSUGRA origin in that they arise from terms that are suppressed by additional powers ofthe warp factor.We can imagine a way in which a different structure could arise. Refs. [34] and [35]studied the Higgs phase of the theory, in which the quarks have a non-trivial VEV andtherefore the theory has a different vacuum structure. In the gravity side this is achievedby means of a worldvolume instanton on the flavor branes (in order to go to the Higgsbranch of the N = 2 theory one needs at least two flavors - see for example [7]). In thepresence of the instanton, we could imagine new interaction operators emerging such as ∂ i Φ F irinst F Arµ ∂ µ Φ . This term would capture a form factor for Φ. Therefore studying themeson structure in the Higgs branch, and comparing it with what it was obtained in theCoulomb branch could be very interesting.By using the same tools, we computed the γ ∗ γ ∗ π form factor. However, in that casethe comparison with the QCD result is more obscure. A priori it looks like our scaling iscompletely different (1 /q as oposed to 1 /q ) from that in QCD. Alerted by the experiencewith the form factors, where the strong/weak coupling matching in the light of conformalinvariance demands τ ↔ n , we provided a first attempt of such a map by computing theform factor for generic l . Eventhough the matching requires some unjustifyied identifica-tion, we feel that it should be possible to have a deeper understanding of the particularscaling we obtained in the light of conformal invariance. We leave that issue open for futurework. A consequence of gauge/gravity duality is that it satisfies VMD due to the (rather)generic properties of Sturm-Lioville operators. Since both the gauge field and the vectormeson come from the same PDE, as described in appendix B, VMD follows directly. Thisallows us to relate F γ ∗ π with F ρπ in very much of the same spirit as in QCD - where VMDalso holdsThere are a number of things which could be studied further. One is to understand theconnection with inclusive processes (in particular DIS). It should be possible to compute26IS amplitudes directly in this model. One could consider computing the current-currentcorrelator using a given hadron and non-normalizable mode wavefunctions, by employingthe usual AdS/CFT methods. Understanding the Bjorken x behavior of the DIS andcomputing the actual behavior of G in (76) to compare with [28] would be very interestingand would surely shed more light on the structure of the mesons.Another extension of our work could be to study the structure of the hadrons at hightemperature. It has been suggested that these mesons could play an important role in thecontext of the QGP (see for instance [50] or [51] and references therein) In the Minkowskiphase of [52] mesons still exist. It would be interesting to study the structure of the hadronsin that phase by probing them with photons. We expect the large q behavior should notdiffer too much from the zero temperature result, since at q (cid:28) T one would expect torecover conformal invariance. However the IR behaviour will differ substantially. It wouldbe interesting to check whether the relations between F ρπ and F σρ continue to be valid,and if any new form factors appear. A naive analysis in the light of our computationsseems to suggest that there will be no modifications along these lines.Finally we note that another way of getting vertices allowing for the same meson toappear as in and out state, could be obtained by considering a pure gauge B -field alongthe Minkowski directions - in much the same spirit as in [53]. There are a number of thingswhich could be studied this way, for example one could study the emission of photonsby mesons in an external magnetic field. Interestingly in this case, even at zero massthe theory develops a condensate which breaks the U (1) R symmetry. It would be veryinteresting to check if one can reproduce more accurately the QCD γ ∗ γ ∗ π behavior in thisinstance. Acknowledgments
We are grateful to G. Gabadadze, A. Garcia-Garcia, C. Herzog, I. Klebanov, P. Kovtun,M. Papucci, A. Ramallo, A. Ritz, A.Scardicchio and H. Verlinde for useful discussions andcomments.D. R-G. acknowledges financial support from the European Commission through MarieCurie OIF grant contract no. MOIF-CT-2006-38381.
A A brief review of VMD in gauge/gravity
The hypothesis of vector meson dominance (VMD) [54] regards the photon-hadron inter-action at low energies in terms of an intermediate vector meson (see [55] for a review).In our D D U (1) gauge field on the flavour brane with l = 0.Restricting to the sector with l = 0, both normalizable ( i.e. vector mesons with l = 0)and non-normalizable ( i.e. EM current) modes come from solving the same PDE. Thisequation was first written down in [6]. Writing Ψ = e iqx ψ , where Ψ stands for either27ormalizable or non-normalizable mode, it can be recast in terms of w as follows ∂ w (cid:0) w ∂ w ψ (cid:1) − w − w R q L ∂ µ ψ = 0 , (77)For generic q we would obtain the non-normalizable mode, whilst the normalizable modeappears when q = m n, . Let us now define L ψ = ∂ w (cid:0) w ∂ w ψ (cid:1) ; λ q = R q L ; ρ = w − w ; ˜ L ψ = L ψ − λ q ρψ ; (78)in such a way that the equation for the non-normalizable modes is just ˜ L A = j upon taking j = 0. As usual we now write j ( w ) = (cid:90) dw (cid:48) δ ( w − w (cid:48) ) j ( w (cid:48) ) , A = (cid:90) dw (cid:48) G ( w, w (cid:48) ) j ( w (cid:48) ) ; (79)where G is the Green’s function.Clearly ˜ L satisfies Green’s theorem, so (cid:90) dw (cid:16) ϕ ˜ L χ − χ ˜ L ϕ (cid:17) = 4 (cid:16) ϕ∂ w χ − χ∂ w ϕ ) | w =1 ; (80)where we have already eliminated the vanishing contribution from w = 0. We can use(80) with G and A , recalling that the non-normalizable mode satisfies Neumann boundaryconditions at w = 1 - and taking j → A ( w, q ) = N lim w (cid:48) → ∂ w (cid:48) G ( w, w (cid:48) ) ; (81)where N is a constant. Thus we see that the non-normalizable mode is determined interms of the Green’s function of ˜ L . In order to find an expression for G we might considerthe equation L Φ IIn, + λ n ρ Φ IIn, = 0 ; (82)where λ n = λ q | q = m n, . This is nothing but the equation satisfied by the normalizablemodes Φ IIn, . We keep the subscript 0 to remind the reader that these normalizable modescorrespond to vector fields with l = 0. They satisfy the following completeness and or-thonormality conditions (cid:88) n ρ ( w )Φ IIn, ( w )Φ IIm, = δ ( w − w (cid:48) ) , (cid:90) dw ρ ( w )Φ IIn, ( w )Φ IIm, ( w ) = δ nm . (83)Then if we consider G = − (cid:88) n Φ IIn, ( w )Φ IIn, ( w (cid:48) ) λ n + λ q , (84)28t is straightforward to check that this solves the Green’s function equation for ˜ L . Thereforethe non-normalizable mode is given by A ( w, q ) = m q λ (cid:88) n f n, Φ IIn, ( w ) q + m n, . (85)and the decay constant of the ( n,
0) vector meson is given by f n, = N lim w (cid:48) → ∂ w (cid:48) Φ IIn, ( w (cid:48) ) . (86) B On the Callan-Gross relation
We now turn to the appearance of the Callan-Gross relation result in a deeper way. In thecontext of VMD it was argued that this type of relation should naturally appear [56]. Alsodue to the structure of effective lagrangians we are considering, this could be thought of asthe low energy coupling of the Higgs to photons (see [57, 58]). From that perspective onecould argue that the effective vertex hF involves, in particular, h going into γγ througha top quark loop. Since the top quark is a spin 1 / B.1 F µν F µν contribution Consider (cid:88) (cid:15) | ( q µ ξ ν − q ν ξ µ )( p µ (cid:15) ν − p ν (cid:15) µ ) | , (87)where ξ is the polarization vector of an external photon of momentum q and (cid:15) the polar-ization of a massive vector particle of momentum p and mass M . In addition ξ · q = 0, (cid:15) · p = 0 and − q = Q <
0. This can be expanded as4 (cid:88) (cid:16) ( q · p ) | ξ · (cid:15) | − ( q · p )( ξ ∗ · (cid:15) ∗ )( q · (cid:15) )( p · ξ ) − ( q · (cid:15) ∗ )( p · ξ ∗ )( q · p )( ξ · (cid:15) ) +( q · (cid:15) ∗ )( p · ξ ∗ )( q · (cid:15) )( p · ξ ) (cid:17) = (88) ξ ∗ µ ξ ν (cid:110) (cid:88) (cid:16) ( p · q ) ( (cid:15) µ ) ∗ (cid:15) ν − ( q · p ) p µ q α ( (cid:15) α ) ∗ (cid:15) ν − ( q · p ) p ν q α ( (cid:15) µ ) ∗ (cid:15) α + p µ p ν q α q β ( (cid:15) α ) ∗ (cid:15) β (cid:17)(cid:111) The sum runs over (cid:15) polarizations, so we have to use (cid:88) (cid:15) ( (cid:15) µ ) ∗ (cid:15) ν = ( − η µν + p µ p ν M ) . (89)After a little bit of algebra, one can show that the terms with M cancel out, and one isleft with 29 ξ ∗ µ ξ ν (cid:16) − ( p · q ) η µν − q p µ p ν + ( q · p ) ( p µ q ν + p ν q µ ) (cid:17) . (90)Consider now ( p · q ) (cid:0) − η µν + q µ q ν q (cid:1) − q ( p µ + q µ x )( p ν + q ν x ) , (91)where x = − q p · q ) . After expanding this − ( p · q ) η µν − q p µ p ν + ( p · q ) ( p µ q ν + p ν q µ ) , (92)so finally we have4 ξ ∗ µ ξ ν (cid:16) ( p · q ) (cid:0) − η µν + q µ q ν q (cid:1) − q ( p µ + q µ x )( p ν + q ν x ) (cid:17) . (93)This we can re-write as4 ξ ∗ µ ξ ν (cid:16) p · q ) q q (cid:0) − η µν + q µ q ν q (cid:1) + 2 xQ q x ( p µ + q µ x )( p ν + q ν x ) (cid:17) . (94)where, as already illustrated, − q = Q . Then (cid:88) (cid:15) | ( q µ ξ ν − q ν ξ µ )( p µ (cid:15) ν − p ν (cid:15) µ | = 4 ξ µ ξ ν ˆ W µν , (95)with ˆ W µν = q x (cid:0) − η µν + q µ q ν q (cid:1) + 2 xQ q x ( p µ + q µ x )( p ν + q ν x ) . (96)Therefore F = q x , F = q x ; (97)so clearly F = 2 xF . B.2 F µν F αβ (cid:15) µναβ contribution Consider now (cid:88) (cid:15) | ( q µ ξ ν − q ν ξ µ )( p α (cid:15) β − p β (cid:15) α ) (cid:15) αβµν | (98)where again ξ is the polarization vector of an external photon of momentum q , and (cid:15) is thepolarization of a massive vector particle of momentum p and mass M . In addition ξ · q = 0, (cid:15) · p = 0 and − q = Q > q µ ξ ν − q ν ξ µ )( p α (cid:15) β − p β (cid:15) α ) (cid:15) αβµν = 4 q µ p α ξ ν (cid:15) β (cid:15) µναβ , (99)we have, using the expression for the sum over polarizations16 q µ q ˆ µ p α p ˆ α ξ ν ξ ∗ ˆ ν (cid:15) µναβ (cid:15) ˆ µ ˆ ν ˆ α ˆ β ( − η β ˆ β + p β p ˆ β M ) . (100)Therefore we get − ξ µ ξ ∗ ν (cid:16) ( q ρ p α (cid:15) µραβ )( q ˆ ρ p ˆ α (cid:15) ν ˆ ρ ˆ α ˆ β ) η β ˆ β (cid:17) (101)Making use of the properties of the (cid:15) -tensor( q ρ p α (cid:15) µραβ )( q ˆ ρ p ˆ α (cid:15) ν ˆ ρ ˆ α ˆ β ) η β ˆ β = p q ( − η µν + q µ q ν q ) + ( p · q ) η µν + q p µ p ν − ( p · ) ( p µ q ν + p ν q µ )(102)Adding and subtracting ( p · q ) q µ q ν q we can re-write the expression above as( q ρ p α (cid:15) µραβ )( q ˆ ρ p ˆ α (cid:15) ν ˆ ρ ˆ α ˆ β ) η β ˆ β = ( p q − ( p · q ) )( − η µν + q µ q ν q ) + q ( p µ + q µ x )( p ν + q ν x ) . 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