Physics Letters B 729 (2014) 3–8

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Threefold complementary approach to holographic QCD ✩ ∗ Stanley J. Brodsky a,GuyF.deTéramondb, , Hans Günter Dosch c a SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA b Universidad de Costa Rica, San José, Costa Rica c Institut für Theoretische Physik, Philosophenweg 16, D-6900 Heidelberg, Germany article info abstract

Article history: A complementary approach, derived from (a) higher-dimensional anti-de Sitter (AdS) space, (b) - Received 3 August 2013 front quantization and (c) the invariance properties of the full conformal group in one dimension leads Received in revised form 27 November 2013 to a nonperturbative relativistic light-front which incorporates essential spectroscopic Accepted 18 December 2013 and dynamical features of physics. The fundamental conformal symmetry of the classical QCD Available online 27 December 2013 Lagrangian in the limit of massless is encoded in the resulting effective theory. The mass scale Editor: B. Grinstein for confinement emerges from the isomorphism between the conformal group and SO(2, 1).Thisscale appears in the light-front Hamiltonian by mapping to the evolution operator in the formalism of de Alfaro, Fubini and Furlan, which retains the conformal invariance of the action. Remarkably, the specific form of the confinement interaction and the corresponding modification of AdS space are uniquely determined in this procedure. © 2014 The Authors. Published by Elsevier B.V. All rights reserved.

1. Introduction compute physical observables in a strongly coupled in terms of a weakly coupled classical theory. One of the most interesting questions in hadron physics is the Recent analytical insights into the nonperturbative nature of the nature of the interaction which permanently confines quarks and confining interaction follow from the remarkable holographic cor- , the fundamental fields of QCD. In the chiral limit no mass respondence between the equations of motion in AdS space and scale appears in the classical QCD Lagrangian, and its action is con- the light-front (LF) Hamiltonian equations of motion for relativis- formally invariant. Thus a second fundamental question is what tic light hadron bound-states in physical space-time [6].Infact, sets the value of the mass scale of confining dynamics correspond- the mapping of the equations of motion [6] and the matching ing to the parameter ΛQCD that appears in quantum loops. of the electromagnetic [7,8] and gravitational [9] form factors in The AdS/CFT correspondence between gravity on a higher- AdS space [10,11] with the corresponding expressions derived from dimensional anti-de Sitter (AdS) space and conformal field theo- light-front quantization in physical space time is the central fea- ries (CFT) in physical space-time [1] is an explicit realization of ture of the light-front holographic approach to hadronic physics. the , which postulates that a gravitational This approach allows us to establish a precise relation between system may be equivalent to a non-gravitational system in one wave functions in AdS space and the light-front wavefunctions de- fewer dimension [2,3]. The correspondence has led to a semiclas- scribing the internal structure of . sical approximation for strongly-coupled quantum field theories To a first semiclassical approximation, light-front QCD is for- which provides physical insights into its nonperturbative dynam- mally equivalent to the equations of motion on a fixed gravitational ics. In practice, the duality provides an effective gravity descrip- background asymptotic to AdS5. In this semiclassical approach to tion in a (d + 1)-dimensional AdS space-time in terms of a flat AdS gravity, the confinement properties can be encoded in a dila- d-dimensional conformally-invariant quantum field theory defined ton profile ϕ(z) in AdS space which introduces a length scale in on the AdS asymptotic boundary [4,5]. Thus, in principle, one can the AdS action and leads to conformal symmetry breaking, but its form is left largely unspecified. The introduction of a dilaton profile is equivalent to a modification of the AdS metric, even for arbitrary ✩ This is an open-access article distributed under the terms of the Creative Com- spin [12]. mons Attribution License, which permits unrestricted use, distribution, and re- In this Letter we shall show that the form of the confining production in any medium, provided the original author and source are credited. interaction appearing in the light-front bound-state wave equa- Funded by SCOAP3. tions for hadrons follows from an elegant construction of Hamil- * Corresponding author. E-mail addresses: [email protected] (S.J. Brodsky), [email protected] ton operators pioneered by V. de Alfaro, S. Fubini and G. Furlan (G.F. de Téramond), [email protected] (H.G. Dosch). (dAFF) [13], based on the invariance properties of a field theory

0370-2693/$ – see front matter © 2014 The Authors. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2013.12.044 4 S.J. Brodsky et al. / Physics Letters B 729 (2014) 3–8 in one dimension under the general group of conformal transfor- equations to the LF Hamiltonian equation found in Ref. [6].This mations. The construction of dAFF retains conformal invariance of procedure allows a clear distinction between the kinematical the action despite the introduction of a fundamental length scale and dynamical aspects of the LF holographic approach to hadron in the light-front Hamiltonian. Remarkably, the specific form of physics. Accordingly, the non-trivial geometry of pure AdS space the confinement interaction in the light-front equation of motion encodes the kinematics, and the additional deformations of AdS is uniquely determined by this procedure. In fact, by combining encode the dynamics, including confinement [12]. analyses from AdS space, LF quantization, and the dAFF construc- The existence of a weakly coupled classical gravity with negligi- tion we will show that the form of the dilaton in AdS space is ble quantum corrections requires that the corresponding dual field uniquely determined to have the form ϕ(z) = λz2 which leads to theory has a large number of degrees of freedom. In the prototyp- linear Regge trajectories√ [14] and a massless pion [15]. ical AdS/CFT duality [1] this is realized by the large NC limit. In The mass scale λ for hadrons appears from breaking of di- the light-front, this limit is not a natural concept, and the LF map- lation invariance in the light-front Hamiltonian, preserving the ping is carried out for NC = 3, with remarkable phenomenologi- conformal invariance of the action. The result is a relativistic and cal success. Following the original holographic ideas [20,21] it is frame-independent semiclassical wave equation, which predicts a tempting to conjecture that the required large number of degrees massless pion in the massless limit and Regge behavior for of freedom is provided by the strongly correlated multiple-particle hadrons with the same slope in both orbital angular momentum states in the Fock expansion in light-front dynamics. In fact, in the L and the radial quantum number n [15]. The model also pre- light-front approach, the effective potential in the LF Schrödinger dicts hadronic light-front wavefunctions which underlie form fac- equation (2) is the result of integrating out all higher Fock states, tors [16] and other dynamical observables, as well as vector corresponding to an infinite number of degrees of freedom. This is electroproduction [17]. apparent, for example, if one identifies the sum of infra-red sen- sitive “H” diagrams as the source of the effective potential, since 2. Light-front dynamics the horizontal rungs correspond to an infinite number of higher gluonic Fock states [22]. The reduction of higher Fock states to an 1 The central problem for solving QCD using the LF Hamiltonian effective potential is not related to the value of NC . method can be reduced to the derivation of the effective interac- For d = 4 one finds from the dilaton-modified AdS action the LF tion in the LF wave equation which acts on the valence sector of potential [12,24] the theory and has, by definition, the same eigenvalue spectrum 1  1  2 J − 3  as the initial Hamiltonian problem. In order to carry out this pro- U(ζ, J) = ϕ (ζ ) + ϕ (ζ )2 + ϕ (ζ ), (3) gram one must systematically express the higher Fock components 2 4 2ζ as functionals of the lower ones. The method has the advantage provided that the product of the AdS mass m and the AdS curva- that the Fock space is not truncated and the symmetries of the ture radius R are related to the total and orbital angular momen- Lagrangian are preserved [18]. tum, J and L respectively, according to (mR)2 =−(2 − J)2 + L2. In the limit of zero quark masses the longitudinal modes The critical value J = L = 0 corresponds to the lowest possible sta- decouple from the invariant LF Hamiltonian equation HLF |φ= − + ble solution, the ground state of the LF Hamiltonian, in agreement M2|φ with H = P P μ = P P − P 2 . The generators P = LF μ ⊥ with the AdS stability bound (mR)2  −4 [25]. − +  ± = 0 ± 3 (P , P , P⊥), P P P , are constructed canonically from the The choice ϕ(z) = λz2, as demanded by the construction de- QCD Lagrangian by quantizing the system on the light-front at + ± − scribed below, leads through (3) to the LF potential fixed LF time x , x = x0 ± x3 [19]. The LF Hamiltonian P gener- ates the LF time evolution U(ζ, J) = λ2ζ 2 + 2λ( J − 1), (4) −| = ∂ |  P φ i + φ , (1) and Eq. (2) yields for λ>0 a mass spectrum for char- ∂x acterized by the total angular momentum J , the orbital angular + whereas the LF longitudinal P and transverse momentum P⊥ are momentum L and orbital excitation n kinematical generators. We obtain the wave equation [6]   J + L   M2 = 4λ n + . (5) d2 1 − 4L2 n, J,L 2 − − + U(ζ, J) φ = M2φ , (2) 2 2 n, J,L n, J,L dζ 4ζ This result not only implies linear Regge trajectories, but also a a relativistic single-variable LF Schrödinger equation. This equa- massless pion and the relation between the ρ and a1 mass usually tion describes the spectrum of mesons as a function of n,the obtained from the Weinberg sum rules [26] number of nodes in ζ , the total angular momentum J , which rep- 3 3 2 = 2 = 2 = 2 = resents the maximum value of | J |, J = max | J |, and the internal mπ M0,0,0 0, mρ M0,1,0 2λ, orbital angular momentum of the constituents L = max |L3|.The m2 = M2 = 4λ. (6) variable z of AdS space is identified with the LF boost-invariant a1 0,1,1 transverse-impact variable ζ [7], thus giving the holographic vari- 3. Conformal quantum mechanics able a precise definition in LF QCD [6,7]. For a two-parton bound 2 2 state ζ = x(1 − x)b⊥, where x is the longitudinal momentum frac- In the following we shall show that the uniqueness of the dila- tion and b⊥ is the transverse-impact distance between the quark ton profile and the quadratic LF potential λ2ζ 2 follows from the and antiquark. In the exact QCD theory the interaction U in (2) is underlying conformality of the theory according to the algebraic related to the two-particle irreducible qq¯ Green’s function. It repre- construction of dAFF [13]. In fact, the LF Hamiltonian can also be sents the equal LF time qq¯ potential which appears after reducing out the infinite tower of higher Fock states. Recently we have derived wave equations for hadrons with ar- 1 An interesting connection of the AdS/QCD duality using Ehrenfest’s correspon- bitrary spin starting from an effective action in AdS space [12]. dence principle between classical and quantum mechanics in the context of light- An essential element is the mapping of the higher-dimensional front relativistic dynamics is given in Ref. [23]. S.J. Brodsky et al. / Physics Letters B 729 (2014) 3–8 5 derived from a one-dimensional quantum field theory (conformal where J 0i , i = 1, 2, is the boost in the space direction i and J 12 the quantum mechanics) starting from the action [13] rotation in the (1, 2) plane. The rotation operator J 12 is compact    and has thus a discrete spectrum with normalizable eigenfunc- 1 ˙ 2 g S = dt Q − , (7) tions. Since the dimensions of Ht and K are different, the constant 2 Q 2 a = 0 has the dimension of t. In fact, the relation between the generators of the conformal group and the generators of SO(2, 1) where Q˙ = dQ/dt and g is a dimensionless number (the dimen- suggests that the scale a mayplayafundamentalroleinphysical sion of Q is [Q ]=[t1/2]). The equation of motion applications [13]. Thus by superposition of different invariants of g Q¨ − = 0, (8) motion, and thereby introducing a scale, one can hope to come to Q 3 a confining semiclassical theory, based on an underlying conformal symmetry. and the generator of evolution in the variable t, the Hamiltonian   Generally one can construct a new “Hamiltonian” by any su- 1 ˙ 2 g perposition of the three constants of motion, the generators Ht , D Ht = Q + , (9) 2 Q 2 and K , follows immediately from (7). The equation of motion for the field G = uHt + vD + wK, (17) operator Q (t) is given by the usual quantum mechanical evolution which is also a constant of motion. The parameters v and w,like   dQ(t) the constant a in (15) are dimensionful. Using (10) and (13) to- i Ht , Q (t) = , (10) dt gether with the canonical commutation relation for the fields Q (t) with the quantization condition [Q (t), P(t)]=[Q (t), Q˙ (t)]=i. we obtain the evolution generated by the operators D and K IntheSchrödingerpicturewitht-independent operators and   dQ(t) 1 t-dependent state vectors the evolution is given by i D, Q (t) = t − Q (t), (18) dt 2        = d  dQ(t) Ht ψ(t) i ψ(t) . (11) i K , Q (t) = t2 − tQ(t). (19) dt dt The absence of dimensional constants in (7) implies that the Specifically, if one introduces the new variable τ and the rescaled action (7) and the corresponding equation of motion (8) are in- field q(τ ) defined through [13] variant under the general group of conformal transformations in the variable t, dt Q (t) dτ = , q(τ ) = , (20) + + 2 + + 2 1/2  αt + β   Q (t) u vt wt (u vt wt ) t = , Q t = , (12) γ t + δ γ t + δ we recover the commutation relation [q(τ ), q˙(τ )]=i, where q˙ = with αδ − βγ = 1, and there are in addition to the Hamiltonian dq/dτ , and the quantum mechanical evolution for the field opera- tor q(t) Ht two more invariants of motion for this field theory, namely the dilation operator D, and K , corresponding to the special conformal   dq(τ ) transformations in t. In addition to H (9), the operators D and K i G, q(τ ) = . (21) t d can be expressed in the field operators Q (t) applying the Noether τ theorem to the Lagrangian in (7): In the Schrödinger picture the evolution of the τ -dependent state vectors is given by the operator G = − 1 ˙ + ˙ D tHt (Q Q QQ), (13)     4  d  G ψ(τ ) = i ψ(τ ) . (22) 2 1 ˙ ˙ 1 2 dτ K = t Ht − t(Q Q + QQ) + Q , 2 2 If one express the action (7) in terms of the transformed fields where we have taken the symmetrized product of the classical ex- q(τ ) one finds [13] pression Q Q˙ since the operators have to be Hermitean. Using the τ+   commutation relations for the field operators Q (t), one can check 1 g 4u − v2 ˙2 ω 2 that the operators H , D and K do indeed fulfill the algebra of the S = dτ q − − q + Ssurface, (23) t 2 q2 4 generators of the conformal group [13] τ−

[H , D]=iH , [H , K ]=2iD, [K , D]=−iK. (14) = 1 [ + 2 ]τ+ t t t where Ssurface 4 (v 2wt(τ ))q (τ ) τ− . Expressed in terms of the original field Q (t) this action (23), including the surface terms, is The conformal group in one dimension is locally isomorphic to unchanged. However, the scale factor 4uw − v2 in (23) breaks the the group SO(2, 1), the Lorentz group in 2 + 1 dimensions. In fact, scale invariance of the corresponding Hamiltonian by introducing the combinations       1 g 4uw − v2 12 1 1 01 1 1 = ˙2 + + 2 J = aHt + K , J = −aHt + K , Hτ q q , (24) 2 a 2 a 2 q2 4 J 02 = D, (15) since the surface term is not relevant for obtaining the equa- tions of motion. The Hamiltonian (24) is a compact operator for one sees that the generators J 12, J 01 and J 02 satisfy the algebra 4uw − v2 > 0. It is important to notice that the appearance of the of the generators of the group SO 2 1 ( , ) generator of special conformal transformations K in (17) is essen-     tial for confinement. J 12, J 01 = iJ02, J 12, J 02 =−iJ01,   We use the Schrödinger picture from the representation of q J 01, J 02 =−iJ12, (16) and p = q˙: q → y, q˙ →−id/dy, 6 S.J. Brodsky et al. / Physics Letters B 729 (2014) 3–8   ∂ d The Hamiltonian (31) can be expressed as a generalization of (32) i ψ(y, τ ) = Hτ y, −i ψ(y, τ ), (25) ∂τ dy by replacing J 12 − J 01 by J 12 − χ J 01. This generalization yields indeed with the corresponding Hamiltonian   1 2 − 2 J 12 − χ J 01 = a(1 + χ)HLF , (33) 1 d g 4uω v 2 τ Hτ = − + + y , (26) 4 2 dy2 y2 4 with in the Schrödinger representation [13].Forg  −1/4 and 4uw − 1 1 − χ v2 > 0 the operator (26) has a discrete spectrum. λ2 = , (34) 2 + We go now back to the original field operator Q (t) in (7).From a 1 χ (20) one obtains the relations in (31).Forχ = 1werecoverthefreecase(32), whereas for −1 < Q (0) √ v χ < 1 we obtain a confining LF potential. For χ outside this region, q(0) = √ , q˙(0) = u Q˙ (0) − √ Q (0), (27) the Hamiltonian is not bounded from below. u 2 u This consideration based on the isomorphism of the confor- and from (24) follows mal group in one dimension with the group SO(2, 1) makes the   appearance of a dimensionful constant in the Hamiltonian (31) de- ˙ 1 ˙ 2 g 1 ˙ ˙ 1 2 rived from a conformally invariant action less astonishing. In fact, Hτ (Q , Q ) = u Q + − v(Q Q + QQ) + wQ 2 Q 2 4 2 as mentioned below Eqs. (15) and (16), one has to introduce the dimensionful constant a = 0 in order to relate the generators of = uHt + vD + wK, (28) the conformal group with those of the group SO(2, 1). This con- = at t 0. We thus recover the evolution operator (17) which de- stant a sets the scale for the confinement strength λ2 but does not scribes, like (24), the evolution in the variable τ (22) but expressed determine its magnitude, as can be seen from (34). This value de- in terms of the original field Q . pends on a as well as on the relative weight of the two generators With the realization of the operator Q (0) in the state space J 12 and J 01 in the construction of the Hamiltonian (33). with wave functions ψ(x, τ ) and the substitution Q (0) → x and Comparing the Hamiltonian (31) with the potential (4) derived ˙ →− d Q (0) i dx we obtain from the modified AdS geometry (3), we see that we are indeed   2 fixed to a confining term proportional to √ζ , corresponding to a ∂ d = 2 = i ψ(x, τ ) = Hτ x, −i ψ(x, τ ), (29) quadratic dilaton profile ϕ λz ,withλ w/2. We obtain from ∂τ dx (3) the LF potential (4). and from (28) the Hamiltonian     5. Conclusions 2 1 d g i d d 1 2 Hτ = u − + + v x + x + wx . (30) 2 dx2 x2 4 dx dx 2 The main result of this Letter is the unique determination of the confining potential from the underlying conformal symmetry The field q(τ ) was only introduced as an intermediate step in of the classical QCD Lagrangian, incorporated in an effective one- order to recover the evolution equation (21) and the Hamiltonian dimensional light-front effective theory for QCD. In fact, using the (24) from the action (7) with variational methods. This shows that triple complementary connections of AdS space, its LF holographic the essential point is indeed the change from t to τ as the evolu- dual and the relation to the algebra of the conformal group in tion parameter. one dimension we find a dilaton profile zs with the unique power s = 2. This has important phenomenological consequences. Indeed, 4. Connection to light-front holographic QCD for s = 2 the mass of the J = L = n = 0 lowest , which we identify with the pion, is automatically zero in the chiral limit, We are now in a position to apply the group theoretical re- and the separate dependence on J and L leads to a mass ratio sults from the conformal algebra to the front-form ultra-relativistic of the ρ and the a1 mesons which coincides with the result of bound-state wave equation. Comparing the Hamiltonian (30) with the Weinberg sum rules. One predicts linear Regge trajectories [14] the light-front wave equation (2) and identifying the variable x with the same slope in the relative orbital angular momentum L = with the light-front invariant variable ζ , we have to choose u 2, and the radial quantum number n. In the phenomenological appli- = v 0 and relate the dimensionless constant g to the LF orbital cation the constant term 2λ( J − 1) in the potential, which is not = 2 − angular momentum, g L 1/4, in order to reproduce the light- fixed by the group theoretical arguments, is crucial. This term is = 2 front kinematics. Furthermore w 2λ fixes the confining light- fixed by the holographic duality to light-front quantized QCD [12]. 2 2 front potential to a quadratic λ ζ dependence. From (1) and (29) It can be shown that only the power s = 2 in the dilaton term = + +  = we find τ x /P in the LF frame P⊥ 0. The dimension of τ leads to linear trajectories and to a massless pion but without ap- is that of an inverse mass squared, characteristic of the fully rela- parent breaking of chiral symmetry. In addition, since quantum tivistic treatment. fluctuations have no infrared divergences because of color confine- The relation with the algebra of the group SO(2, 1) becomes 2 ment, the gap parameter ΛQCD appearing in the evolution of the particularly compelling when the Hamiltonian Hτ (30) is mapped QCD running coupling will be determined in the present frame- = = = 2 to the light-front. In this case one has u 2, v 0, and w 2λ work by the mass scale λ. and the LF Hamiltonian can be expressed in terms of the genera- The dAFF mechanism for introducing a scale makes use of tors Ht and K the algebraic structure of one-dimensional√ conformal field theory. A new Hamiltonian with a mass scale λ is constructed from the HLF = 2 H + λ2 K . (31) τ t generators of the conformal group and its form is therefore fixed From the relations (15) follows the connection of the free Hamil- uniquely: it is, like the original Hamiltonian with unbroken di- tonian Ht , (9) with the group generators of SO(2, 1) latation symmetry, a constant of motion [13]. The essential point of this procedure is the introduction of a new evolution param- 12 01 J − J = aHt . (32) eter τ . The local isomorphism between the conformal group in S.J. Brodsky et al. / Physics Letters B 729 (2014) 3–8 7 one dimension and the group SO(2, 1) is fundamental for introduc- a prediction which has not been possible within the context of cur- ing the scale for confinement in the light-front Hamiltonian and rent algebra alone. The model described in this Letter provides a fixing the dilaton profile in AdS. The dAFF mechanism is reminis- compelling effective frame-independent effective theory of hadrons cent of spontaneous symmetry breaking, however, this is not the since it predicts essential features of the hadron spectrum, in- case since there are no degenerate vacua [27] and thus a massless cluding the masslessness of the pion, and provides an interesting ++ scalar 0 state is not required. The dAFF mechanism is also dif- alternative approach to the conventional descriptions of nonpertur- ferent from usual explicit breaking by just adding a term to the bative QCD. Lagrangian [28]. The main result of this Letter is that group theoretical argu- A specific value for λ is not determined by QCD alone. The ments based on the underlying conformality of the classical La- scale only becomes fixed when we make a measurement such as grangian of QCD fix the quadratic form of the effective dilaton pro- 2 the pion decay constant or the ρ mass. The scale ΛQCD appear- file and thus the corresponding form of the confinement potential 2 of the light-front QCD Hamiltonian. Though this is particularly im- ing in the QCD running coupling αs(Q ) can be determined from the fundamental QCD confinement scale λ by matching the per- portant for light-front holographic QCD, it is also relevant to other 2 bottom-up holographic QCD models, as for instance the model dis- turbative and nonperturbative regimes. The specific value of ΛQCD 2 cussed in [14]. depends on the choice of the renormalization scheme for αs(Q ), such as the MS scheme. Acknowledgements In their discussion of the evolution operator Hτ dAFF mention a critical point, namely that “the time evolution is quite different We thank Diego Hofman for pointing out the work of de Alfaro, from a stationary one”. 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