A T\bar T-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering
AA T ¯ T -like deformation of the Skyrme modeland the Heisenberg model of nucleon-nucleonscattering Horatiu Nastase a ∗ and Jacob Sonnenschein b, † a Instituto de F´ısica Te´orica, UNESP-Universidade Estadual PaulistaR. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil b School of Physics and Astronomy,The Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University, Ramat Aviv 69978, Israel
Abstract
The Skyrme model, though it admits correctly a wide range of static propertiesof the nucleon, does not seem to reproduce properly the scattering behavior ofnucleons at high energies. In this paper we present a T ¯ T -like deformation ofit, inspired by a 1+1 dimensional model, in which boosted nucleons behave likeshock waves. The scattering of the latter saturates the Froissart bound. Westart by showing that 1+1 dimensional T ¯ T deformations of the free abelian pionaction are in fact generalizations of the old Heisenberg model for nucleon-nucleonscattering, yielding the same saturation of the Froissart bound. We then deformthe strong coupling limit of the bosonized action of multi-flavor QCD in twodimensions using the T ¯ T deformation of the WZW action with a mass term. Wederive the classical soliton solution that corresponds to the nucleon, determineits mass and discuss its transformation into a shock-wave upon boosting. Weuplift this action into a 3+1 dimensional T ¯ T -like deformation of the Skyrmeaction. We compare this deformed action to that of chiral perturbation theory.A possible holographic gravity dual interpretation is explored. ∗ E-mail address: [email protected] † E-mail address: [email protected] a r X i v : . [ h e p - t h ] F e b ontents T ¯ T -like actions as generalizations of the Heisenberg model 7 two ultrarelativistic waves represent-ing high energy nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 On boosting a soliton into a shockwave . . . . . . . . . . . . . . . . . . . . . 133.5 The cross section and saturation of the Froissart bound . . . . . . . . . . . 14 T ¯ T deformed models 15 QCD . . . . . . . . . . . . . . . . . . . . . . 204.4 Boosting the 2d baryon to a shock wave . . . . . . . . . . . . . . . . . . . . 21 T ¯ T -like actions and chiral perturbation theory . . . 295.3 Boosted Skyrme-like soliton vs. shockwave solution . . . . . . . . . . . . . . 36 T ¯ T ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Gravity dual interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 T ¯ T deformation and the Heisenberg model 41B On the Abelian and non-Abelian T ¯ T deformations of Maxwell and Yang-Mills theories 43 Introduction
An important question that has been intensively investigated for about a half a century iswhat happens when nucleons (or, more generally, hadrons) collide at very high energy? It iswell known that the asymptotic total cross section is bounded by the Froissart bound [1, 2] σ tot (˜ s ) ≤ C ln ˜ ss , C ≤ πm , (1.1)where ˜ s is the Mandelstam kinematical variable, s a constant and m is the mass ofthe lightest particle that can be exchanged by the scattering projectiles. But before theFroissart bound, and in fact even QCD were derived, Heisenberg proposed a simple semi-classical model [3] that deals with the regime of high-energy nucleon scattering, leading tothe saturation of the bound. The model, which is defined in terms of a single pion field,described by a massive DBI scalar action in 3+1 dimensions takes the following form: S = (cid:90) d x l − (cid:20) − (cid:113) l [( ∂ µ φ ) + m φ ] (cid:21) . (1.2)This action is then reduced to 1+1 dimensions (time and the direction of propagation).The colliding Lorentz-contracted nuclei and the pion field around them were describedby classical shockwave solutions of this action. Invoking some simple assumptions it wasshown that the total cross section σ tot ( s ) saturates the Froissart bound. In the moderncontext, the Heisenberg model was updated and generalized in [4]. Further work on theHeisenberg model was hindered by the puzzle of how to understand the action (1.2) fromthe point of view of QCD? Possible effective actions for QCD are described in terms of pionsthat are SU (2)-valued. Skyrme-type actions admit static solitons that can be identifiedwith the nucleons, but do not have shockwave solutions. On the other hand SU (2)-valuedgeneralizations of (1.2) that have shockwave solutions were found not to have static solitons[5], which presents another puzzle.Recently [6–8] it was discovered that in 1+1 dimensions there are other interestinggeneralizations of the DBI scalar action, namely T ¯ T deformations of a canonical scalaraction with a potential V . T ¯ T deformations have served as laboratories for investigatingvarious aspects of field theory. See for instance [9–16].A natural question that follows from the generalization of Heisenberg’s action is: cansuch an action be considered instead of the action (1.2) for modeling nucleon scattering, andcan it be related to QCD? After all, in [4] we have already found that various generalizationsof (1.2) work as well, and some have an interpretation from the point of view of AdS/CFT,in possible gravity duals of QCD.With that in mind, we show in this paper that certain T ¯ T deformed actions, indeedadmit shockwave solutions which result in the saturation of the Froissart bound, as doesthe original model of Heisenberg. Moreover, they also have soliton solutions, which uponboosting allow us to interpret the shockwave as a soliton collision.2ur next step is to elevate the 1+1 dimensional abelian deformed theories into a non-Abelian one. For that purpose we make use of the proposal of [7] for a T ¯ T deformation ofthe two dimensional WZW model with a mass term. The deformed WZW action is writtendown for U ∈ U ( N f ) with a level N c where N f is the number of flavors and N c the numberof colors. It was shown in [17] that this describes the strong coupling effective action ofmulti-flavor bosonized QCD in two dimensions. It was further found in [17] that this actionreduces, for the lowest energy configuration, to a sine-Gordon action. In a similar way thedeformed WZW action reduces to a T ¯ T deformation of the sine-Gordon action. In [18] wefound the soliton solutions of that system and we showed that their masses are the sameas of the un-deformed ones. We argue that these may be used to describe the physics oftwo dimensional nucleons.We then uplift this system into a 3+1 dimensional T ¯ T -like deformation of the Skyrmeaction. This action admits both Skyrme-like solitons representing the nucleons, and shock-wave solutions corresponding to their collisions. We argue that the latter can be gottenfrom boosting the former. Upon expanding the deformed action as a power series in thedeformation parameter, we get that the leading order terms are those of the Skyrme modelfollowed by terms with higher power of derivative of the group element field U . These higherorder terms are then compared with the corresponding terms of the chiral perturbationtheory action.We also briefly discuss the description of the model using the holographic gravity/gaugeduality. It turns out that as a result of the deformation the holographic model admits achange of the sign of a brane tension.The paper is organized as follows. In section 2 we review Heisenberg’s model andits generalizations. In section 3 we consider T ¯ T deformed scalar actions, solitons andshockwave solutions, argue that the latter can be understood as boosting the former, andfind that using the mode results in a saturation of the Froissart bound. In section 4 weuse the T ¯ T deformation of the WZW action plus a mass term as a deformation of thestrong coupling limit of multi-flavor QCD in 2 dimensions. We determine the baryon asa soliton solution by transforming the action to that of a deformed sine-Gordon mode.We discuss boosting this solution into a shockwave solution. In section 5 we considerhow to find both Skyrme-like solitons to non-Abelian 3+1 dimensional generalizations ofthe T ¯ T deformed action, and shockwave-like solutions, and how to understand the latterfrom boosting the former. In section 6 we consider the possible interpretations of the T ¯ T deformed actions in QCD, and in AdS/CFT, from the point of view of gravity duals, andin section 7 we conclude and present several open questions. In Appendix A we show thatthe T ¯ T deformation of the scalar actions can’t be used for the Heisenbergmodel (unlike the oxidation to 3+1 dimensions of the 1+1 dimensional actions), and inAppendix B we show that the T ¯ T deformations of Abelian and non-Abelian gauge theories(Maxwell and Yang-Mills) don’t have the needed soliton solutions, so hence cannot be usedas a replacement of the Heisenberg model. 3 Review of Heisenberg model for high-energy nucleon-nucleonscattering and its generalizations
In this section we review the model by Heisenberg written in 1952 [3] to describe high-energy nucleon-nucleon scattering, which gives a saturation of the Froissart bound [1, 2],and its modern implementation and generalizations in [4].Heisenberg starts with the observation that when boosting nucleons to ultra-relativisticspeeds, they Lorentz contract, becoming first pancake-shaped, until finally they look likedelta-function shockwaves in the direction of propagation. Moreover, the pion field aroundthem also Lorentz contracts, so finally we can consider that it becomes a delta-functionsourced pion field shockwave.The process of high-energy nucleon scattering then is described by the collision of twosuch shockwaves, and as such can be described by a (classical) shockwave solution of aneffective action, which was chosen to be (1.2), for reasons to be explained shortly. In thenext section, we will see that that in fact the shockwave solution we consider describes thecollision of two such nucleon-sourced pion shockwaves, and not just one pion shockwave, afact that was not clear in previous analyses of the model.The shockwave solution of (1.2) is described as a solution depending only on s = t − x ,where x is the direction of propagation, φ = φ ( s ), with φ ( s <
0) = 0. To constrain theaction, Heisenberg imposes that, while φ ( s ) must be continuous, so φ (0+) = 0, ( ∂ µ φ ) must be finite and nonzero on the nontrivial side ( s >
0) and zero on the trivial side, thejump in the Lorentz-invariant derivative signalling the existence of the (delta function)nucleon-sourced shock. It is found that a canonical scalar with any polynomial potentialdoes not solve it, but the action (1.2) does, as the solution near s = 0 is φ ( s ) (cid:39) l − √ s + ... , (2.1)so that ( ∂ µ φ ) (cid:39) − l − + ... (2.2)On the other hand, in [4] it was found that the condition of jump in derivative, correlatedwith the existence of saturation of the Froissart bound, while it is highly nontrivial, it alsoadmits certain generalizations. The simplest one is to an arbitrary potential V inside thesquare root, L = l − (cid:20) − (cid:113) l [( ∂ µ φ ) + 2 V ( φ )] (cid:21) , (2.3)another being to several scalars with a sigma model, potential and 2 more functions, L = l − ( d +1) (cid:20) h ( φ i ) − f ( φ ) (cid:113) l d +1 [ g ij ( φ k )( ∂ µ φ i )( ∂ µ φ j ) + 2 V ( φ i )] (cid:21) . (2.4)We can also consider an action with an AdS/CFT interpretation, as a D-brane action4oving in a curved background, with S Dd = l − ( d +1) (cid:90) d d +1 xe − ˜ φ (cid:20)(cid:113) − det ( η µν ˜ g ( φ ) + l d +1 ∂ µ φ i ∂ ν φ j g ij ( φ )) − (cid:21) , (2.5)and one can also add some vector mesons (understood as vectors on the D-brane), withLagrangian L = l − (cid:20) − (cid:113) det( η ab + l ∂ a φ∂ b φ + l F ab ) + m φ + M V A a (cid:21) . (2.6)The Hamiltonian (energy density) coming from (1.2) is H = π ˙ φ − L = l − + ( ∇ φ ) + m φ (cid:112) l [( ∂ µ φ ) + m φ ] − l − . (2.7)We interpret the classical shockwave solution φ ( s ), which, as we will explain in the nextsection, describes the collision of two pion schockwaves around the high-energy nucleons,as describing in a general direction in ( x, t ) the pion radiation coming from the collision.Indeed, a classical bosonic field is nothing but the quantum field in the case of many bosonsin each occupied state, so the φ ( s ) solution can describe radiation from a collision. Thenthe Hamiltonian above corresponds to the energy radiated in the collision.Following Heisenberg [3], we can assume that the denominator becomes a non-vanishingconstant near s = 0, thus avoiding the unphysical divergence in the energy density nearthe shock. Fourier transforming in x to momentum k the solution near s = 0 in (2.1), weobtain φ ( k, t ) = l − (cid:90) t dxe ikx (cid:112) t − x (cid:39) l − π | t || k | ( J ( | k || t | ) + i H ( | k || t | )) , (2.8)where J is a Bessel function and H is a Struve function. Expanding at large k , we obtain φ − l − i | t || k | ∝ | t | / | k | − / e − i | k || t | , (2.9)and dropping the non-oscillatory (non-radiative) piece in φ , we get φ ( k ) ∝ k − / .However, in reality (and also as we will see in an example in the next section), theshockwave should have a finite thickness of the order of the Lorentz contracted 1 /m , i.e., √ s m ≡ √ s min = √ − v /m , so at sufficiently large times t , the momenta k are cut offat k m = 1 / √ s m = γm , the relativistic mass of the pion. Then the energy of radiatedpions per unit of spatial momentum, derived from the momentum space Hamiltonian in(2.7) with constant denominator, is dEdk ∝ k φ ( k ) ∼ const .k , (2.10)5or k ≤ k m .Next we relate to the quantum mechanical description of the classical field, by identi-fying k with the momentum k of a pion, and identifying through canonical quantizationthe pion field energy E with the radiated energy E of the pions, the radiated energy perunit of pion energy is d E dk = Bk , m ≤ k ≤ k m , (2.11)integrated to E = B ln k m m = B ln γ. (2.12)This also leads to a relation for the number of radiated pions per unit of radiated energy,since dE = k dn leads to dndk = Bk , m ≤ k ≤ k m , (2.13)integrated to n = Bm (cid:18) − mk m (cid:19) . (2.14)Finally, the average emitted energy per pion is (cid:104) k (cid:105) ≡ E n = m ln( k m /m )1 − m/k m = m ln γ − γ (cid:39) m ln γ , (2.15)which is approximately constant (the dependence on the energy is only logarithmic). Onthe contrary, for a canonical scalar with polynomial potential (and most other canonicalscalar actions), through a similar calculation, we find at leading order in γ , (cid:104) k (cid:105) ∝ γ .To connect to the saturation of the Froissart bound, we need to consider the fully 4-dimensional form of the action (1.2). For a Mandelstam variable ˜ s , the total energy of thecolliding nucleons is √ ˜ s , and we assume that the emitted energy is proportional to it, withconstant of proportionality given by the wave function overlap. Moreover, since at largetransverse distance r (= (cid:112) y + z ), the wave function is small, so satisfies the free massiveKG equation, with solution φ ∼ e − mr , the wave function overlap will be e − mb , where b isthe impact parameter of the colliding nucleons. Then E ∼ √ ˜ se − mb . (2.16)The maximum impact parameter, b max , arises when we emit a single pion, that is, whenthe emitted energy E equals the average per pion emitted energy (cid:104) k (cid:105) , so √ ˜ se − mb max = (cid:104) k (cid:105) ⇒ b max = 1 m ln √ ˜ s (cid:104) k (cid:105) ⇒ σ tot = πm ln √ ˜ s (cid:104) k (cid:105) . (2.17)We see that the saturation of the Froissart bound is equivalent with the (approximate)independence of (cid:104) k (cid:105) of the collision energy √ ˜ s , or in other words, on γ .6 T ¯ T -like actions as generalizations of the Heisenberg model One definition of the T ¯ T deformation of a Lagrangian, proposed by Zamolodchikov [19, 20]is that, for a full quantum theory in 1+1 dimensions, the variation of the Lagrangianwith respect to the deformation parameter equals the determinant of the deformed energy-momentum tensor, i.e, after Wick rotating to Euclidean space, ∂ L ∂λ = − T λzz T λ ¯ z ¯ z − ( T λz ¯ z ) ] . (3.1)Note that here all objects are renormalized and UV finite, and on the right-hand sidewe have an operator regularized by point-splitting. We can solve the equation by expansionin a series in λ (by treating the fields as numbers), if we give a starting point (unperturbedLagrangian L ) and a perturbation parameter λ (then by (3.1), the first order in the series, L is given by the unperturbed det T µν above).Consider as a starting point a real scalar with potential V (still in Euclidean space), L = 12 ( ∂ µ φ ) + V , (3.2)and a perturbation parameter λ .Then, at first one obtained a complicated expression, with an infinite series of com-plicated hypergeometric functions (see eq. 6.34 in [6]), but the series can be summed, toobtain a simple expression [7], giving L ( λ, X ) = − λ − λV − λV + 12 λ (cid:115) (1 − λV ) (1 − λV ) + 2 λ X + 2 V − λV = − (1 − λV ) + (cid:113) λ ( ∂ µ φ ) λ ≡ ˜ V ( φ ) + 12¯ λ (cid:113) λ ( ∂ µ φ ) , (3.3)where we have used the notation X = ( ∂ µ φ ) , as well as defined¯ λ ≡ λ (1 − λV ) . (3.4)Note that when λV →
0, ˜ V → V − / (2 λ ), which cancels the a constant coming fromthe square root, and gives the unperturbed potential. This can then also be written asin [8] (since 4 ∂φ ¯ ∂φ = ( ∂ µ φ ) ), L E = V − λV + − (cid:112) λ∂φ ¯ ∂φ λ . (3.5)This is the Lagrangian in Euclidean space. Going back to the Minkowski signature, weobtain L M = − V − λV + 1 − (cid:113) λ∂ µ φ∂ µ φ λ , (3.6)7ith ∂ µ φ∂ µ φ = − ( ˙ φ ) + ( φ (cid:48) ) ≡ X , as usual.We now examine whether these Lagrangians (for various possible potentials V ( φ )) canbe used for generalizations of the Heisenberg model (other than the previously consideredones reviewed in the previous section). In this subsection we review soliton solutions described in a companion paper [18] (afterthe paper was posted, we became aware of [21, 22], which have some overlap with ours,though here we present the solutions in the form we obtain)We have considered T ¯ T and Heisenberg deformations of a canonical scalar with a po-tential V . Starting with the Heisenberg case, the deformed Lagrangian in 1+1 dimensional (Minkowski)space is L M = 1 − (cid:113) ∂ µ φ ) + 2 V , (3.7)and, by use of a modified virial theorem, we have found that the generic static solution iswritten implicitly as x − x = (cid:90) φ ( x ) φ ( x ) dφ (cid:113) (1 + 2 V ) (cid:2) VC − (cid:3) , (3.8)where C is an arbitrary constant.By comparing with the canonical case, we have found that we need to put C = 1 tofind a modified soliton solution, so that the implicit soliton solution is x − x = (cid:90) φ ( x ) φ ( x ) dφ (cid:112) V (1 + 2 V ) . (3.9)For the pure DBI scalar case, with V ( φ ) = 0, the solution is x − x = (cid:90) φ ( x ) φ ( x ) dφ (cid:113) C − K ( φ ( x ) − φ ( x )) , (3.10)and by gluing two such solutions we find the solution to the Poisson equation in onedimension, φ ( x ) = φ ( x ) + K − | x − x | . (3.11)We note that, if we redefine V such that V (0) = 0, this solution is valid also for thecase of nonzero V ( φ ), just that only near x = x . After an infinite, or rather very large,boost, with K − γ ≡ ˜ K , we have the propagating shockwave solution φ ( x − ) = ˜ K | x − | . (3.12)8ut more relevant, perhaps, is the case of the pure DBI action in 3+1 dimensions ,in which case the conservation equation becomes (using the same formalism as in 1+1dimensions, but for the one-dimensional Lagrangian in the radial direction r = x , L ( r ) =4 πr [1 − (cid:112) φ (cid:48) ]) ddx (cid:32) x V (cid:112) V + φ (cid:48) (cid:33) = − x (cid:112) V + φ (cid:48) . (3.13)Here it is harder to write the general solution, but one can find the specific solution(”catenoid”, see later sections) with a delta function source at r = 0, solution of (cid:126) ∇ · (cid:126)C = qδ ( r ), with (cid:126)C = ∂ L /∂ ( (cid:126) ∇ φ ), which is φ (cid:48) = q x − q ⇒ φ ( r ) = q (cid:90) ∞ r dx (cid:112) x − q . (3.14) T ¯ T deformations In the case of T ¯ T deformations of a canonical scalar with a potential V , we can find theimplicit general static solution for φ ( x ) as x − x = (cid:90) φ ( x ) φ ( x ) dφ (cid:112) λ (1 − λV ) (cid:113) λ (1 − λV )( C − ˜ V )] − , (3.15)where again C is an arbitrary integration constant.The relevant case for a soliton, that arises as a deformation of the soliton for thecanonical scalar with potential V is obtained for C = 0, as x − x = (cid:90) φ ( x ) φ ( x ) dφ | − λV |√ V . (3.16)In particular, for the sine-Gordon potential, V sG = − µ β [cos( βφ ) − , (3.17)we find the deformed sine-Gordon soliton, defined implicitly as ± µ (cid:112) β ( x − x ) = (cid:20) ln (cid:18) tan βφ (cid:19) + 4 λµ β cos βφ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) φ ( x ) φ ( x ) , (3.18)and for the Higgs-type potential, V ( φ ) = α ( φ − a ) , (3.19)we find the deformed kink soliton, defined implicitly as x − x = ± a √ α (cid:20) tanh − φa − λαa (cid:12)(cid:12)(cid:12)(cid:12) φ a − φa (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) . (3.20)9ne again finds the same approximate static solution, near a point x = x , the solutionto the Poisson equation in one dimension, φ ( x ) = φ ( x ) + K − | x − x | , (3.21)and as before, after an infinite boost goes over to the same φ ( x − ) = ˜ K | x − | . (3.22)We should also consider the case of soliton solutions of the of the T ¯ T Lagrangian, for a spherically symmetric solution, when we can write a one-dimensional radial Lagrangian for the radius r = x via L ( x ) → πr L ( r ). Then, doingexactly the same steps as before, we obtain the conservation equation ddx (cid:34) x λ (1 − λV ) (cid:112) λ (1 − λV ) φ (cid:48) + x ˜ V (cid:35) = − x (cid:112) λ (1 − λV ) φ (cid:48) λ (1 − λV ) , (3.23)or, formally integrating to an integro-differential equation for φ and φ (cid:48) , x λ (1 − λV ) (cid:112) λ (1 − λV ) φ (cid:48) + x ˜ V = C − (cid:90) dx x (cid:112) λ (1 − λV ) φ (cid:48) λ (1 − λV ) . (3.24)Alternatively, we could obtain an integro-differential equation by defining (cid:126)E ≡ (cid:126) ∇ φ and (cid:126)D ≡ ∂ L ∂ (cid:126)E , ρ ( (cid:126)r ) ≡ ∂ L ∂φ ( (cid:126)r ) , (3.25)so that the equation of motion is (cid:126) ∇ · (cid:126)D ( (cid:126)r ) = ρ ( (cid:126)r ), like for electromagnetism in a medium.Since for ρ ( (cid:126)r ) = δ ( (cid:126)r − (cid:126)r (cid:48) ) we have as solution for (cid:126)D the Green’s function (cid:126)D ( (cid:126)r − (cid:126)r (cid:48) ) = (cid:92) (cid:126)r − (cid:126)r (cid:48) π | (cid:126)r − (cid:126)r (cid:48) | , (3.26)the solution for (cid:126)D is (cid:126)D ( (cid:126)r ) = (cid:90) d (cid:126)r (cid:48) ρ ( (cid:126)r (cid:48) ) D ( (cid:126)r − (cid:126)r (cid:48) ) , (3.27)so we obtain the integro-differential equation for φ and (cid:126) ∇ φ ( φ (cid:48) on the radial ansatz) ∂ L ∂ (cid:126) ∇ φ ( (cid:126)r ) = (cid:90) d (cid:126)r (cid:48) ∂ L ∂φ ( (cid:126)r (cid:48) ) (cid:92) (cid:126)r − (cid:126)r (cid:48) π | (cid:126)r − (cid:126)r (cid:48) | . (3.28)However, this is more complicated than the integro-differential equation obtained be-fore, so we don’t gain anything (in the pure DBI case, on the right-hand side one had only qD ( (cid:126)r ), allowing us to completely solve the equation).10 .2 The shockwave solution In the previous subsection we have considered static solutions K | x − x | near x = x ,infinitely boosted to solutions ˜ K | x − | , representing shockwaves propagating at the speed oflight.However, there are other shockwave solutions, namely ones described by a function ofonly s = t − x = x + x − ( x ± = t ± x ), and not independently on x + , x − . In this case,in [18] we have found that both for the Heisenberg and T ¯ T deformations, we find the sameperturbative shockwave solution as Heisenberg: near s = 0, but s >
0, the solution is φ ( s ) = l − √ s + O ( s / ) , (3.29)while φ ( s <
0) = 0.We have also found that the solution remains true if we change the square root in theaction with another rational power less than 1, p/q < two ultrarelativistic wavesrepresenting high energy nucleons
In the previous section, as well as in [4], the 1+1 dimensional solution was described as ashockwave, and it is, in the sense of being a function of s that has discontinuous derivativesat s = 0. However, from a physical point of view, we will see now that it actually describes two shockwaves colliding, which means it really corresponds to the ultra-relativistic limit ofthe field of colliding moving particles sourcing it, identified with the high energy nucleons.To see this, we first calculate the energy momentum tensor. Including a nontrivialmetric in Heisenberg’s Lagrangian for the pion, of the type of a DBI scalar with a massterm in the square root, L = l − √− g (cid:20) − (cid:113) l ( ∂ µ φ∂ ν φg µν + m φ ) (cid:21) , (3.30)and varying it, gives the Belinfante tensor T µν = − √− g δSδg µν = ∂ µ φ∂ ν φ (cid:112) l ( ∂ µ φ∂ ν φg µν + m φ )+ g µν l (cid:20) − (cid:113) l ( ∂ µ φ∂ ν φg µν + m φ ) (cid:21) . (3.31)But, as we found in [4], for s = t − x →
0, on the solution, the square root Lagrangiandiverges (and we argued that it must be regularized somehow, since divergencies will leadto quantum fluctuations). But, on the solution, with x ± = x ± t , so s = − x + x − , ∂φ∂x + = dφds ∂s∂x + = − x − dφds ∂φ∂x − = − x − dφds . (3.32)11hen, on the solution, near s = 0, the term with g µν l − is finite, the term with g µν times the square root goes to zero, and the leading term diverges, as 1 over the squareroot, so T ++ (cid:39) − ∂ + φ∂ + φ (cid:112) l ( ∂ µ φ∂ ν φg µν + m φ ) ∝ (cid:18) dφ∂s (cid:19) ( x − ) ,T −− (cid:39) − ∂ − φ∂ − φ (cid:112) l ( ∂ µ φ∂ ν φg µν + m φ ) ∝ (cid:18) dφ∂s (cid:19) ( x + ) ,T + − (cid:39) − ∂ + φ∂ − φ (cid:112) l ( ∂ µ φ∂ ν φg µν + m φ ) ∝ (cid:18) dφ∂s (cid:19) s . (3.33)The 1 over the square root diverges, but must be regularized to a finite, yet large value.But then we note that, near s = 0 (which means x + = 0 or x − = 0), we have T + − → x + = 0, T −− →
0, thus only T ++ is nonzero, and is a regularized divergence, and for x − = 0, T ++ →
0, thus only T −− is nonzero, and is a regularized divergence.That means that the shockwave solution of Heisenberg represents two colliding shock-waves (instead of one), one at x + = 0 (going in the negative x direction) with only T ++ and diverging (like the A-S shockwave in gravity), and one at x − = 0 (going in the positive x direction), with only T −− and diverging.This is a solution in 1+1 dimensions, so there is no dependence on the 2 transversedirections, and the collision of the two shockwaves happens at x = t = 0.If we view it as a limit r → φ ( s, r ), then implicitly we assume that thecollision happens at x = r = 0, i.e., head-on collision (no impact parameter b ). Yet it alsomeans that the φ ( s (cid:39) , r (cid:54) = 0) represents the wave function overlap of the two nucleons,as indeed Heisenberg claims in his paper, and this overlap also goes like e − m π r at large r ,as he says also.Moreover, as observed both by him implicitly and by us [4] implicitly, this φ ( s (cid:39) , r (cid:39) (cid:39) φ ( s (cid:39)
0) is consistent with large spatial momenta k , if we consider x small and r small, which one did in order to calculate the emitted spectrum (and moreover, we needlarge t in order for it to be the emission spectrum at large time after the collision, i.e.,emitted radiation). All of these were conditions imposed on the radiation derived from theclassical solution, so their presence is consistent.But we can view the shockwave solution also as part of a solution φ ( x + , x − , r ), at r → only at x − = 0, x + <
0, which means t < x <
0, so nearone of the shockwaves, before the other one arrives. In that case, we can assume the otherone (the one going in the negative x direction) can be situated at some nonzero impactparameter, r = b (cid:54) = 0. That is why the solution can be also thought of as representing thecollision at some impact parameter b .This explains why the solution represents collision of 2 waves, and why we can get themaximum impact parameter b max by using it.12 .4 On boosting a soliton into a shockwave Before continuing with the analysis of the cross section for high energy scattering, anissue to consider is: how to understand the relation between static soliton solutions andshockwave solutions for the same action. One would expect that by infinitely boosting thesoliton solution, one would get the shockwave, or some limit of it. However, as we just saw,the shockwave solution actually shows the collision of two waves (one at x + = 0, anotherat x − = 0), so that is not what one wants. In fact, we have seen that by boosting a solitonsolution we (rather generically) obtain the solution φ ( x − ) = ˜ K | x − | .Rather, if we are to compare with the shockwave solution, we should try to comparethe energy-momentum tensors, more specifically the one coming out of a single wave, say T −− . Considering the Heisenberg model with a mass term only (neglecting possible higherorders in φ in the potential V ), which is the only thing that matters near the shock at s = 0, the near shock solution is φ (cid:39) √ sl (1 + a s m + O ( s )) , (3.34)where a is an arbitrary constant. Then the square root defining the Lagrangian, and theLorentz-invariant denominator of the Hamiltonian and of T µν , is (cid:113) l ( g µν ∂ µ φ∂ ν φ + m φ ) (cid:39) m √ s √ − a , (3.35)so it diverges as s →
0. We also note the condition a ≤ / T + − (cid:39) √ − a l − m √ s , T −− (cid:39) √ − a l − m √ s x + x − T ++ (cid:39) √ − a l − m √ s x − x + . (3.36)Then, if x + is finite but x − → T ++ → T + − ∝ / √ x − , and T −− ∝ / ( x − ) / .But that means that at the shock T + − → ∞ also, whereas it should really not diverge, atleast in the case of a single wave. So, it is natural to assume that the square root shouldsomehow be cut off neat s = 0 such that it stays finite (in fact, this was implicitly assumedin Heisenberg’s case; for the Hamiltonian density H , but it is the same denominator). Inthat case we would have T ++ → T + − finite, T −− ∝ /x − .But consider the generic soliton solution of φ ( x ) = ˜ K | x − x | near x , which wasinfinitely boosted to φ ( x ) = ˜ Kγ | x − | (for γ → ∞ ), near x − = 0. Moreover, consider theLorentz invariant l g µν ∂ µ φ∂ ν φ which can be calculated on the static solution as l ˜ K . Forthe infinite boost to make sense, we need that ˜ K →
0, ˜ Kγ finite, in which case the invariantis l ˜ K →
0. Then the square root on the soliton solution is (cid:113) l m ˜ K γ ( x − ) (cid:39) ( l m ˜ Kγ ) | x − | , (3.37)where we have assumed that the second term is larger than the 1 (we can have l ˜ Kγ largeenough for that). Then T −− (cid:39) ∂ − φ∂ − φ (cid:112) l ( g µν ∂ µ φ∂ ν φ + m φ ) (cid:39) ˜ Kγl − m | x − | (3.38)13as the same behavior as that expected from the shock near one wave (with the cut-off inthe square root).It would seem like the total energy, (cid:82) dx − T −− , has a log divergence at x − = 0. However,note that for this correct boosted soliton solution, there is a (very small) x − at which T −− remains finite. More precisely, since T −− = ˜ Kγl − m (cid:113) ( x − ) + l m ˜ K γ , (3.39)we have that E ∼ (cid:90) dx − T −− = ˜ Kγl − m sinh − ( x − l m ˜ Kγ ) , (3.40)which doesn’t have a divergence at x − = 0, and the (log) divergence at large x − is spuriousalso, because the original boosted solution was valid only near x = x , or near x − = 0,when boosted. We also note the correct relativistic relation E ∝ γ . Now that we saw that the (1+1-dimensional) T ¯ T action gives the same shockwave near s = 0, and moreover the shockwave solution represents a collision of two ultrarelativisticwaves representing the high energy nucleons, we can trivially generalize the action to 3+1dimensions, by making ( ∂ µ φ ) d x with d x , and use it as Heisenberg did, for the high-energy nucleon-nucleon collision.We want to see if the collision cross section can still saturate the Froissart bound, ifwe use this 3+1 dimensional generalization of the T ¯ T action instead of the action used byHeisenberg (which, in any case, was just a guess, as the simplest , but not necessarily mostuseful or correct; one that gave him the needed properties of the shockwave for high energyscattering of nucleons).In the Heisenberg analysis reviewed in section 2, which we made more precise in [4], thenext step in getting to the cross section of Froissart, is to calculate the radiated energy perunit of k , so first we need the Hamiltonian. At this point, we are still in 1+1 dimensions,so ( ∂ µ φ ) = − ˙ φ + φ (cid:48) . Then H = π ˙ φ − L = ∂ L ∂ ˙ φ ˙ φ − L = − λ ∂∂ ˙ φ (cid:113) λ ( − ˙ φ + φ (cid:48) ) ˙ φ + 12¯ λ (cid:113) λ ( − ˙ φ + φ (cid:48) ) + ˜ V ( φ )= 12¯ λ λφ (cid:48) (cid:113) λ ( − ˙ φ + φ (cid:48) ) + ˜ V ( φ ) . (3.41)But the energy is E = (cid:90) dx H ( x ) = (cid:90) dk H ( k ) , (3.42)14here now the Hamiltonian density is H ( k ) = 12¯ λ λk φ ( k ) (cid:113) λ ( − ˙ φ + k φ ) − ˜ V ( φ ( k )) . (3.43)But, like in the case of [4] (and of Heisenberg), we see that the first term has thesquare root in the denominator, and on the solution near s = 0, the square root vanishes(since V vanishes there, and then the rest is the same square root as for Heisenberg). But,as Heisenberg implicitly noted (and we explained in detail), this is unphysical, and weexpect that some quantum fluctuations or corrections make the square root a small, butnonvanishing constant near s = 0.Then, neglecting the subleading term ˜ V ( φ ( k )), we obtain H ( k ) (cid:39) k φ ( k )const . . (3.44)But, since the same φ ( x, t ) (cid:39) l − √ s solution is valid near s = 0 here, as in the caseof [4], we have the same Fourier transform φ ( k, t ) at large k , eqs. (2.14) and (2.15) there.There is a non-oscillatory piece, which can be dropped as there, since we are focusing inon the radiative solution. Then φ ( k ) ∼ | k | − / , so we have dEdk = H ( k ) ∼ const .k , (3.45)as before. This then leads, as we saw in [4] and in the previous section, after several steps,to the average per pion energy being approximately a constant, (cid:104) k (cid:105) (cid:39) m ln γ , (3.46)where γ is the Lorentz factor for the collision.Only now do we need to consider a fully 3+1 dimensional version of the T ¯ T action.More precisely, we need to use the small field value for it, which is just the canonicalfree massive scalar action. Then, the solution at small field, in terms of transverse radius r = (cid:112) y + z , is φ ∼ e − mr , (3.47)and then the same argument as Heisenberg’s follows: the radiated energy E is proportionalto the total collision energy √ ˜ s , and to the overlap of wave functions, ∼ e − mb , where b is the impact parameter; and at maximum b , we radiate only (cid:104) k (cid:105) , so √ ˜ se − mb max = (cid:104) k (cid:105) ,from which we get the cross section σ tot = πb = πm ln √ ˜ s (cid:104) k (cid:105) . T ¯ T deformedmodels Next, we would like to consider non-Abelian versions of the Abelian actions considered inthe Heisenberg type model. In this section we will consider still 1+1 dimensional models,and in the next one we will consider the full 3+1 dimensional models needed.15or the generalization of the abelian field to non-abelian ones, we can follow two ap-proaches:(a) Straightforwardly utilizing a U ∈ U (2) group element instead of the single realscalar field φ .(b) Using the proposal of [7] for a T ¯ T deformation of the two dimensional WZW model,deformed also by the potential (mass) term. In this paper, the deformation of the WZWmodel was found, and the kinetic term is replaced by a square root term, while the WZWterm is unchanged by the T ¯ T deformation, being topological.We first consider the case (a), of Heisenberg and T ¯ T deformations, with a potential of V = − m (cid:20) cos (cid:18)(cid:114) πN c φ ( t, x ) (cid:19) − (cid:21) , (4.1)where N c is the number of colors. To generalize to a non-abelian case we make the followingreplacements ∂ µ φ ( t, x ) ∂ µ φ ( t, x ) → N c π Tr (cid:104) ∂ µ U ( t, x ) ∂ µ U † ( t, x ) (cid:105) cos (cid:18)(cid:114) πN c φ (cid:19) − →
12 Tr (cid:104) U + U † − (cid:105) . The non-Abelian Lagrangian density corresponding to the Abelian T ¯ T deformation(3.6) (Minkowskian) then reads L M = (cid:20) − (cid:114) N c π λ (cid:16) λm Tr [ U + U † − (cid:17) Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )] (cid:21) λ (1 + λm Tr [ U + U † − m Tr (cid:2) U + U † − (cid:3) λm Tr [ U + U † − . (4.2)Note that whereas the kinetic part of the action (4.2) is invariant under U L (2) × U R (2)symmetries associated with the transformations U → A L U , U → U A R , where A L and A R are constant U (2) matrices, the potential term and hence the whole action is only invariantunder the diagonal U (2) transformation (with A L = A † R = A ) U → AU A † . (4.3)In particular the diagonal U (1) D symmetry should be associated with the baryon num-ber, according to the general theory of QCD. The corresponding Noether currents, for U (2)generators T A = ( σ a , σ a are the Pauli matrices) are given by j Aµ = − i N c π Tr (cid:2) ∂ µ U T A U † − U † T A ∂ µ U † (cid:3)(cid:114) N c π λ (cid:16) λm Tr [ U + U † − (cid:17) Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )] . (4.4)16ote that in the Skyrme model, the presence of the extra Skyrme term means thatthe Noether current has an extra contribution equal to the topological current (for thetopological number associated with the hedgehog ansatz). On a static ansatz ∂ U = 0, theabove zero component of the Noether current, as the corresponding term coming from thekinetic term in the Skyrme model, vanishes, meaning that the Noether charge equals tojust the topological charge, and is associated with baryon number. Here, however, on thestatic ansatz we just get a zero Noether charge.This suggests that perhaps it is better to use the case (b), for the proposal in [7], ofdeforming the WZW model, which will have a nonzero Noether current, thus a nonzeroNoether charge, that will turn out to be equal to the topological charge, and thus can beequated with the baryon number, for the dimensional reduction of an effective model forQCD ( QCD ). The model in [7] will also be deformed by a potential (mass) term.In conclusion, in this section we will analyze the deformations of (i) the WZW model(ii) the massive WZW model and (iii) baryonic sector of bosonized QCD in the strongcoupling limit. It is well known that in 1+1 dimensions the theory of a free massless real scalar describedby the action S = 18 π (cid:90) d x ∂ µ φ∂ µ φ = 14 π (cid:90) d z∂φ ¯ ∂φ , (4.5)where the second form of the action is taken in complex plane, is invariant under trans-formations with parameters which are holomorphic and anti-holomorphic with the corre-sponding U (1) affine Lie algebra conserved currents J = ∂φ , ¯ J = ¯ ∂φ , ∂ ¯ J = ¯ ∂J = 0 , (4.6)and energy momentum tensor T = − ∂φ∂φ , ¯ T = −
12 ¯ ∂φ ¯ ∂φ , ∂ ¯ T = ¯ ∂T = 0 . (4.7)The U ( N ) × U ( N ) non-abelian generalization of this system is the well-known WZWaction given by S = S σ + S W ZW = 12 (cid:90) d x k π Tr[ ∂ µ U ∂ µ U † ] + k π (cid:90) B [Tr U − dU ] , (4.8)where k is the level of the Kac Moody algebras associated with the holomorphic (anti-holomorphic) conserved currents ¯ J ( ¯ J ) given by J = k π ∂U U − , ¯ J k π U − ¯ ∂U , ∂ ¯ J = ¯ ∂J = 0 . (4.9)17he corresponding energy momentum tensor is given by the Sugawara construction T = 1 k T r [ J J ] , ¯ T = 1 k T r [ ¯ J ¯ J ] , ∂ ¯ T = ¯ ∂T = T z ¯ z = 0 . (4.10)The first term in the action, the sigma term, follows from a map ∂ µ φ ( t, x ) ∂ µ φ ( t, x ) → k π Tr (cid:104) ∂ µ U ( t, x ) ∂ µ U † ( t, x ) (cid:105) . (4.11)The T ¯ T deformation of the Euclidean WZW action was worked out in [7]. It takes thefollowing form S = (cid:90) d x (cid:34) − λ + 12 λ (cid:114) N c π λX + N c π N c π λ ˜ X (cid:35) + i π (cid:90) B N c π [Tr U − dU ] , (4.12)where X ≡ Tr[ ∂ µ U ∂ µ U † ] = − Tr[ L µ L µ ] L µ ≡ U − ∂ µ U ˜ X ≡ (cid:15) µρ (cid:15) νσ Tr[ L µ L ν ] Tr[ L ρ L σ ] . (4.13)Notice that the WZW term is undeformed. We have also identified k = N c , we will seein the following subsections why. When expanding in small λ the action takes the form S = N c π (cid:90) d x (cid:34) X λN c π (cid:32) ( ˜ X ) − X (cid:33) + 116 (cid:18) λN c π (cid:19) (cid:16) X − X ( ˜ X ) (cid:17) + O (cid:0) λ (cid:1)(cid:35) + S W Z . (4.14)It is easy to check that both X and ˜ X ij are invariant under the left and right U ( N )transformations U → A L U , A L ∈ U L ( N ) , U → U A R , A R ∈ U R ( N ) , (4.15)and hence these are symmetry transformations of the full deformed action. The corre-sponding currents J µL and J µR are given by J Aµ ( L,R ) = i N c π Tr (cid:2) − ∂ µ U T A U † + U † T A ∂ µ U † (cid:3)(cid:113) λ k π Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )] ± (cid:15) µν Tr (cid:104) ∂ ν U T A U † − U † T A ∂ ν U † (cid:105) , (4.16)where T A are the generators of the U ( N ) group. The left and right currents correspondto the different signs ± , are not conserved holomorphicaly and anti-holomorphically , butrather in the ordinary covariant form ∂ µ J µL = 0 , ∂ µ J µR = 0 . (4.17)Similarly for the energy momentum tensor, and furthermore for the deformed theorywe have T z ¯ z (cid:54) = 0. 18 .2 WZW with a “mass term” So far we discussed the deformation of the WZW theory, the non-abelian generalization ofsingle free massless scalar field. Next we would like to address the same procedure for aninteracting scalar field, namely, a scalar field with a potential, since this is closer to themodel we seek. More specifically we consider a sine-Gordon potential given by V = − m [cos ( βφ ( t, x )) − . (4.18)To map the abelian theory to a non-abelian we generalize the map of (4.11) with thefollowing replacements ∂ µ φ ( t, x ) ∂ µ φ ( t, x ) → k π Tr (cid:104) ∂ µ U ( t, x ) ∂ µ U † ( t, x ) (cid:105) cos ( βφ ) − →
12 Tr (cid:104) U + U † − (cid:105) . The
Minkowski space action that corresponds to the T ¯ T deformation of this system isgiven by S M = (cid:90) d x (cid:20) − (cid:114) k π λ (cid:16) λm Tr [ U + U † − (cid:17) (cid:16) Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )] + k π λ ˜ X (cid:17)(cid:21) λ (1 + λm Tr [ U + U † − m Tr (cid:2) U + U † − (cid:3) λm Tr [ U + U † − (cid:41) + i π (cid:90) B k π [Tr U − dU ] . (4.19) This is the 1+1 dimensional action we consider for the non-Abelian Heisenberg model .In the next section we will see how to uplift it to 3+1 dimensions. Upon expanding insmall λ , and as before identifying k = N c , we find now S M (cid:39) (cid:90) d x (cid:34)(cid:18) − N c π X + Z (cid:19) + 116 λ (cid:32)(cid:18) N c π (cid:19) ( X −
2( ˜ X ) ) − Z − N c π ZX (cid:33) + O (cid:0) λ (cid:1)(cid:3) + S W Z , (4.20)where we denoted by Z = m Tr (cid:2) U + U † − (cid:3) . As was mentioned in the previous subsec-tion, both X and X ij are invariant under the full flavor chiral symmetry U ( N ) L × U ( N ) R transformations. However this is not a symmetry transformation of the term T r [ U + U † ].This term is only invariant under the diagonal U ( N ) transformation (with A L = A † R = A ) U → AU A † . (4.21)The corresponding current is J µA = ik π Tr (cid:2) U T A ∂ µ U † + U † T A ∂ µ U † (cid:3)(cid:114) N c π λ (cid:16) λm Tr [ U + U † − (cid:17) Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )]19 (cid:15) µν Tr (cid:104) U T A ∂ µ U † − U † T A ∂ µ U † (cid:105)(cid:17) . (4.22)In particular the diagonal U (1) ⊂ U ( N ), which will play an important role below, isgiven by J µ = ik π Tr (cid:2) U ∂ µ U † + U † ∂ µ U † (cid:3)(cid:114) N c π λ (cid:16) λm Tr [ U + U † − (cid:17) Tr [ ∂ µ U ( t, x ) ∂ µ U † ( t, x )]+ (cid:15) µν Tr (cid:104) U ∂ µ U † − U † ∂ µ U † (cid:105)(cid:17) . (4.23)Next we would search for soliton solutions of this deformed system. The first point tonotice is that for static configurations X ij vanishes, as does the WZ term, and thereforethe Lagrangian density for static configurations takes the form L M = (cid:20) − (cid:114) k π λ (cid:16) λm Tr [ U + U † − (cid:17) Tr [ ∂ x U ( x ) ∂ x U † ( x )] (cid:21) λ (1 + λm Tr [ U + U † − m Tr (cid:2) U + U † − (cid:3) λm Tr [ U + U † − . (4.24)We now take a particular ansatz for the group element U ( N ) U ( x ) = diag (cid:18) , , ...e − i (cid:113) k π φ ( x ) (cid:19) , (4.25)This ansatz, which takes us back from the non-Abelian setting to an Abelian one, willbe relevant in the next subsection. For this ansatz, we get that k π Tr (cid:104) ∂ µ U ( t, x ) ∂ µ U † ( t, x ) (cid:105) = 12 ∂ µ φ∂ µ φ
12 Tr (cid:104) U + U † − (cid:105) = cos ( βφ ) − . Substituting this into the action (4.19), we end up with the action of the T ¯ T deformationof the scalar field with a sine-Gordon potential. In [18] the soliton solution for this actionwas derived and its properties were analyzed. We will describe some of them in the nextsubsection. QCD In [23] it was shown that the low energy effective action of multiflavor bosonized
QCD ,namely the theory with SU ( N c ) gauge symmetry and U ( N f ) flavor symmetry converges inthe strong limit to Ssc = N c S W ZW ( U ) + m (cid:90) d xT r [ U + U † ] , U ∈ U ( N f ) , (4.26)20here the explicit expression for m is given in [23]. Thus, the action is a U ( N f ) WZW oflevel k = N c , plus a potential term.Solitons of this action should correspond to baryons of QCD . It was shown in [23]that the lowest energy soliton is the one with the following structure of U ( x ): U ( x ) = diag (cid:18) , , ...e − i (cid:113) πNc φ ( x ) (cid:19) . (4.27)If we substitute this ansatz into (4.26) we get an action for static configurations thattakes the form S s = (cid:90) d x (cid:34) − (cid:18) dφdx (cid:19) − m (cid:18) cos (cid:18)(cid:114) πN c (cid:19) − (cid:19)(cid:35) . (4.28)In the terminology of the sine-Gordon model we this action has β = (cid:113) πN c .Alternatively, we could have followed the route taken in the previous subsection, ofdeforming the non-Abelian action, and substituting into it the form of U ( x ) given in (4.27).As was mentioned in the previous subsection, the T ¯ T deformation of the T ¯ T deforma-tion of the sine-Gordon action and its corresponding soliton were analyzed in [18].The soliton solutions for this action were derived in [18]. For our system, the solitontakes the form ± √ mβ ( x − x ) = (cid:20) ln (cid:18) tan (cid:18)(cid:114) π N c φ (cid:19)(cid:19) + 8 λm cos (cid:18)(cid:114) πN c φ (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) φ ( x ) φ ( x ) . (4.29)Using the expression for the baryon number current (4.23) we find that the baryonnumber of this soliton solution is Q B = (cid:90) dxJ B = N c π ( φ ( x → ∞ ) − φ ( x → −∞ )) = N c , (4.30)as it should be in the notation where each quark has unit baryon number.Moreover in [23] it was proven that the classical mass of the soliton of the deformedtheory is exactly the same as in the undeformed one which is given by [23] M B = 4 m (cid:114) N c π = 4 (cid:34) N c cm q (cid:32) g c (cid:112) N f √ π (cid:33)(cid:35) C (cid:114) N c π , (4.31)where ∆ c = N c − N c ( N c + N f ) . The strong coupling effective action of bosonized 2d QCD [17] does not admit a shockwave solution. This follows from the fact that the necessary condition to have such a21olution [4], namely to have an infinite tower of higher derivative terms in the action, isnot fulfilled in that case. However, as was shown in [21] and more explicitly in [18] thedeformed sine-Gordon action does admit a shockwave behavior. Hence, following the thediscussion above also the deformed
QCD with the particular ansatz of the group element,also admits a shockwave behavior. Thus, the deformed QCD action is a framework inwhich one can discuss the static properties of the baryonic soliton but also its scatteringin the form of shockwave collisions. As we saw in the section 3, the shockwave solution can be used to describe the (saturationlimit for the cross section for the) collision of high-energy nucleons. The nucleons, togetherwith the pion field they generate, when boosted will become first pancake-like, then deltafunction sources surrounded by shockwave pion field, for which the Heisenberg shockwaveis a model.On the other hand, there is one way to deal with nucleons at rest not as sources forthe pion field, but rather as sourceless topological pion field configurations, representingboth the nucleon and the surrounding pion field, as solutions for the low energy expansionin QCD (chiral perturbation theory). That is the Skyrme solution for the nonlinear sigmamodel (for the SU (2)-valued field corresponding to the three physical pions, π ± and π )with a Skyrme term. It is also known that a topological solution still exists when we replacethe Skyrme term with other higher order terms in the action.It is a reasonable hope then that by boosting a Skyrme-like solution we should get ashockwave of the type found by Heisenberg (at least in the region when there is a singlewave), as it was argued also in [5,24–26]. Here we will show that by proposing an action thatis a natural nonabelian generalization of the T ¯ T action we obtain a Skyrme-like solutionthat when boosted looks like Heisenberg’s shockwave, near s = 0 and for t < In this subsection, after reviewing previous results on various solitons, we will argue forthe existence of a soliton of the T ¯ T action that can pass smoothly through the 1 − λV = 0singularity, and then argue about what happens when we infinitely boost such a solution. We start this subsection by reviewing the previous attempt for the Abelian scalar theory,in [5], to realize the above program. As we said, the hope was initially that a single scalaraction would generate both the shockwave-like solution and the Skyrmion-like solution, sothat we can we could infinitely boost the latter into the former.22he first observation is that, by restricting to an abelian analog (a single real scalar),the Skyrme-like solution would be a solution of finite energy, since the nucleon representedas a Skyrmion has a finite energy also. There is one such solution, the BIon solutionfound by Born an Infeld in 1934 in [27], in their attempt to replace the electron solution ofclassical electrodynamics, which has an infinite field energy (the infinite energy shows theneed to go to quantum field theory, i.e., to QED, but Born and Infeld sought a classical replacement).The Born-Infeld Lagrangian (putting the relevant length scale l to 1), L = − (cid:115) − det (cid:18) η µν + F µν √ (cid:19) = − (cid:115) F µν F µν − (cid:18) F µν ∗ F µν (cid:19) = − (cid:113) − (cid:126)E + (cid:126)B − ( (cid:126)E · (cid:126)B ) , (5.1)where in the last form we wrote the electric and magnetic fields, has the equation of motionin the presence of a static source with charge density ρ (so zero magnetic field) of (cid:126) ∇ · (cid:126)D = ρ , (5.2)where as usual we defined (cid:126)D = ∂ L ∂ (cid:126)E . (5.3)The electric field (cid:126)E is then finite ( ≤ (cid:126)E = − (cid:126)D (cid:112) (cid:126)D . (5.4)In terms of (cid:126)E = − (cid:126) ∇ φ , with φ the electric potential, the Lagrangian on the time-independentsolution is L = − (cid:113) − ( (cid:126) ∇ φ ) , (5.5)and the solution for point-like source ρ = qδ ( r ) is φ ( r ) = q (cid:90) ∞ r (cid:112) q + x , (5.6)and has diverging energy density at r = 0 (where | (cid:126)E | = | φ (cid:48) ( r ) | = 1 ), but finite energy , E φ = 4 π (cid:90) ∞ r dr (cid:34) (cid:112) − φ (cid:48) ( r ) − (cid:35) = 4 πq / (cid:90) ∞ dxx + √ x + 1 = 4 πq / Γ[1 / √ π . (5.7)Note that the on-shell Lagrangian looks like one for a scalar DBI action, but with thewrong sign inside the square root (and there is no corresponding time derivative term, sincethe scalar is not a real scalar, but is part of a vector).23n the other hand, the shockwave-like solution can be understood as a solution witha jump in the derivative for a true scalar theory, as Heisenberg did, which (as Heisenbergalso did) restricts us to the DBI action, L = − (cid:113) − det ( η µν + ∂ µ X∂ ν X ) = − (cid:113) ∂ µ X ) . (5.8)If we want a static solution for it, define (similarly to the Born and Infeld case) (cid:126)F = (cid:126) ∇ X ,so the on-shell Lagrangian on this ansatz is L = − (cid:113) (cid:126)F . (5.9)The equation of motion in the static case with a source ρ is (cid:126) ∇ · (cid:126)C = ρ , (5.10)where we have defined (cid:126)C = ∂ L (cid:126)F . (5.11)Then (cid:126)F = (cid:126)C (cid:112) − (cid:126)C . (5.12)The solution for point-like source ρ = qδ ( r ) is the ”catenoid”, X ( r ) = q (cid:90) ∞ r dx (cid:112) x − q , (5.13)which however, as we see, has a ”horizon” at r = √ q , where X ( r ) is finite but X (cid:48) ( r )diverges. When we think of the DBI action as the action for the position of a D-brane,this is understood as one half of a solution of D-brane-anti-D-brane, connected by a funnel(solution obtained by adding a mirror image of the catenoid). The solution also has adiverging energy density, yet a finite energy, but now at this ”horizon”, E X = 4 π (cid:90) ∞ r r dr (cid:104)(cid:112) X (cid:48) − (cid:105) = 4 πq / (cid:90) ∞ x dx (cid:20) x √ x − − (cid:21) . (5.14)Note that both BIon and catenoid solutions become at large r just φ ( r ) (cid:39) qr . (5.15)Perhaps one can find a boosting limit, where one takes q →
0, so as to have no observablehorizon with diverging energy density, but keeping qγ finite in the limit (where γ is therelativistic factor 1 / √ − v ), just like when we infinitely boost the Schwarzschild metricwe obtain the Aichelburg-Sexl shockwave metric [28], by keeping p = M γ finite in the limitwhere M → , γ → ∞ ), but we have not been able to show this. In any case, we will24ee that in the nonabelian case, the generalization with this positive sign inside the squareroot doesn’t have a Skyrmion-like solution, whereas the one with the negative sign does.Next, it was found that in the D-brane action, both X and φ are present, as S DBI = − (cid:90) d x (cid:113) [1 − ( (cid:126) ∇ φ ) ][1 + ( (cid:126) ∇ X ) ] + (cid:126) ∇ φ · (cid:126) ∇ X. (5.16)Therefore this action has both BIon (in φ ) and catenoid (in X ) static solutions. One canhope then that perhaps by infinitely boosting a more general, BIon plus catenoid solution,one can obtain the shockwave solution, but again that is not clear (and we still have theproblem of the nonabelian generalization). T ¯ T action This was the case so far. But with the T ¯ T action replacing the DBI action (with massterm inside the square root, which is irrelevant for the solution near s = 0), one morepossibility arises. For the T ¯ T action, we have a single scalar φ . We can have a BIon-likesolution, namely, a solution of finite energy defined for all r until 0, and yet be defined interms of the same scalar field X (now called φ , but meaning a true scalar, not the electricpotential). And then infinitely boosting it we can obtain the shockwave solution.The catenoid solution has φ (cid:48) ( r ) → ∞ at some r = r , but φ ( r ) finite, but to get to r = 0 with a finite energy like for the BIon, we need the opposite sign inside the squareroot. However, that can happen at large field φ , since the coupling of the ( ∂ µ φ ) term is¯ λ = λ (1 − λV ). Then we expect to have both catenoid-like and BIon-like solutions. Notethat the positive sign of ¯ λ is associated not just to the catenoid solution, but also to theshockwave solution.Both catenoid-like and Bion-like go to q/r at r → ∞ , since then also V (cid:39)
0, so ¯ λ (cid:39) λ .As we go to smaller r , they start to differ. Then φ becomes large, and then so V ( φ ), atleast in the case that we have only a mass term (and maybe also a φ term), so the questionis whether ¯ λ reaches zero or not. If it stays positive, we have a catenoid-like solution (andalso shockwave solution, in the boosted case), which develops a horizon at r (cid:39) √ q , where φ (cid:48) ( r ) = ∞ , φ ( r ) finite. You might think, but the catenoid is singular, so how come forhim, V can stay small enough? The answer is that the singularity is in the derivative , notthe field; the field can stay small. If ¯ λ reaches zero, and changes sign, we have a BIon-likesolution (which stands for the Skyrmion-like one in the nonabelian case).The question we need to answer is whether it is possible to have a continuous solution,for the BIon-like case, that (must!) change sign for ¯ λ from ¯ λ > r → ∞ ( φ small) to¯ λ < r → φ large). If we can, then we should be able to infinitely boost such aBIon-like solution to the shockwave solution, something that seemed impossible before.To see whether we can have a continuous solution for the BIon-like case, we checkwhether the ansatz for having a finite (of order 1, neither zero nor infinity) φ (cid:48) ( φ ) for φ a finite value for which ¯ λ = 0 is consistent for the solutions of the equations of motion.25onsider the value φ = φ of the field for which λV ( φ ) = 1 (so that ¯ λ = 0). Then1 − λV ( φ ) (cid:39) − λV (cid:48) ( φ )( φ − φ ) = − λV (cid:48) ( φ ) δφ , (5.17)where we have defined δφ ≡ φ − φ . Then we find that the Lagrangian on the above ansatzbecomes − L (cid:39) (cid:112) − λ V (cid:48) ( φ ) δφ ( δφ (cid:48) ) − λ V (cid:48) ( φ ) δφ . (5.18)Its equation of motion, 2 λ V (cid:48) ( φ ) δφ × δS/δ ( δφ ), is − δφ (cid:16) (cid:112) − λ V (cid:48) ( φ ) δφ ( δφ (cid:48) ) (cid:17) − λ V (cid:48) ( φ ) δφ (cid:48) (cid:112) − λ V (cid:48) ( φ ) δφ ( δφ (cid:48) ) +2 λ V (cid:48) ( φ ) δφ (cid:34) δφ (cid:48) (cid:112) − λ V (cid:48) ( φ ) δφ ( δφ (cid:48) ) (cid:35) (cid:48) = 0 . (5.19)Multiplying with δφ (cid:112) − λ V (cid:48) ( φ ) δφ ( δφ (cid:48) ) , and after some (longish) algebra, we get − λ V (cid:48) ( φ ) δφδφ (cid:48) + 8 λ V (cid:48) ( φ ) δφ δφ (cid:48)(cid:48) +[1 − λ V (cid:48) ( φ ) δφδφ (cid:48) ] / = 0 . (5.20)The ideal solution is when there is nothing special at this point φ , reached at position x = x , meaning that we can have a regular Taylor expansion of the field φ , δφ ( x ) = Aδx = φ (cid:48) ( x )( x − x ) , (5.21)with A = φ (cid:48) ( x ) a nonzero and finite constant. Substituting this ansatz into the aboveequation, the leading term near x = x , is − δφ ( x ) = Aδx α , with α a power less than 1, and maybe then weobtained consistency, 0=0, for some α ; as it is, we had the correct ansatz).Finally, that means that we can find a BIon-like solution, for which the derivative ofthe field at the point that ¯ λ changes sign stays finite.Note that we have really considered the approximate Lagrangian (5.18) in order to ob-tain the solution. But this Lagrangian is not necessarily obtained from the T ¯ T deformationLagrangian, but can come from any Lagrangian of the type − L = h ( φ ) + (cid:112) λ (1 − λ ˜ g ( φ ))( ∂ µ φ ) λ (1 − λ ˜ g ( φ )) , (5.22)if for 1 − λ ˜ g ( φ ) = 0, then h ( φ ) = 1. The notion of having a potential V ( φ ), and of ˜ g ( φ )to be related to h ( φ ) is not needed.In this subsection, we considered a single scalar, but by the embedding of the singlescalar action into the nonabelian one, we know that the same conclusions can be reached26bout the latter. The BIon-like solution becomes a Skyrmion-like solution, and the shock-wave still a shockwave. Thus, we have shown that there is a potential for the Skyrmion-likesolution to be boosted to a shockwave. Exactly how that is done will be examined next,but the existence of a horizon hiding a region with wrong sign inside the square root (whichallows for a topological solution, of finite energy), yet which can be scaled down to zero ina certain limit, is essential. Next, we need to understand what is the condition needed to show that the infinitelyboosted BIon-like solution (which we still don’t know, we just proved it exists) goes tothe shockwave solution. To do that, we consider how we know that an infinitely boostedSchwarzschild black hole is the Aichelburg-Sexl schockwave.In the paper of Aichelburg and Sexl [28], they took a nontrivial limit on the Schwarzschildsolution for a black hole, which has a singularity (for the point mass M ) at r = 0 and ahorizon at r h = 2 M . The horizon exists because the solution is written in terms of the har-monic function (in 3 spatial dimensions) f ( r ) = 1 − M/r . The Einstein equations reduceto the Poisson equation for f ( r ). In the nontrivial limit of M → γ = 1 / √ − v → ∞ ,but p (cid:39) M γ =finite (for v → ds = 2 dx + dx − + dx i dx i + ( dx + ) H ( dx + , x i ) . (5.23)For a shockwave, we have H ( x + , x i ) = δ ( x + )Φ( x i ) . (5.24)For the Aichelburg-Sexl shockwave, we have Φ( x i ) harmonic in the transverse dimen-sions, with Poisson source p , so in D = 4 dimensions,Φ( x i ) = − G N, p ln r , (5.25)where r = x i x i , and in general Φ( x i ) = 16 πG N,d Ω d − ( d − pr d − . (5.26)In previous works, in particular in [25], it was argued that for a similar A-S shockwavein the background of the gravity dual to QCD (say, in cut-off AdS ), Φ( x i ) should beassociated with the pion field profile, at least at large r , which means that what is true forobtaining the A-S shockwave, should also be true for obtaining the pion shockwave, whichis why we are reviewing the gravitational shockwave limiting procedure.The limiting boost procedure in the A-S paper is complicated, and also involves acertain nontrivial change of coordinates, but one can easily get the qualitative picture27ithout calculations: since M →
0, the horizon goes to zero, r h →
0, which is why there isno horizon in the shockwave. Then anything gravitational (curving space) moving at thespeed of light must be a pp wave, and if it is point-like (like the mass in the Schwarzschildmetric is), it is a shockwave (delta function in x + ).But instead of the complicated limiting procedure, one can use a shortcut, and findthe same result in an easier way. Instead of taking the limit on the metric, we take iton the equation it satisfies. We can check (and find also without explicit calculations)that the only nonzero component of the Ricci tensor R µν for the pp wave metric is R ++ .Then this must be proportional to g ++ , and (since R µν has 2 derivatives), we must have R ++ ∝ ∂ i ∂ i g ++ (because of general coordinate invariance, it cannot have ∂ + ∂ i , or d ,the only other possibilities for 2 derivatives acting on g ++ = H ( x + , x i )). In fact, with thecorrect coefficient, R ++ = − ∂ i H ( x + , x i ) , (5.27)which in fact equals the linearized result (it is the only case we know of, when the Einsteinequation is exactly equal to its linearization). The Einstein equation makes this equal to T ++ .Now a point particle moving at the speed of light has only T ++ , and moreover hasmomentum p , as we defined, so T ++ = pδ d − ( x i ) δ ( x + ) , (5.28)leading in turn to the shockwave ansatz H = dδ ( x + )Φ, which finally leads to ∂ i Φ( x i ) = − πG N,d pδ d − ( x i ) , (5.29)with solution the A-S shockwave. Thus after the infinite boost, the Einstein equationsreduced to the equation for the field Φ( x i ) in the pp shockwave ansatz. We see that thepp wave ansatz is a very interesting one: the usually very nonlinear Einstein equationslinearize on the ansatz (the linearized equation is actually exact).We want to mimic the procedure in the case of the abelian pion action. In this case,there is no reduction of the degrees of freedom by the ansatz ( g µν in the general Einsteinequations, Φ on the ansatz). Instead, the variable is a single real scalar φ before and afterthe infinite boost.Then all we need to show is that the equations of motion, or equivalently, the La-grangian, for the BIon-like ansatz, go over to the Lagrangian for the shockwave, under theinifinite boost.First, we need to understand the limiting procedure. To do that, we go back to theshockwave energy-momentum tensor (3.33), to find the total momentum of one of thecolliding shockwaves, for instance x − = 0. For the A-S gravitational shockwave on x + = 0,only T ++ was nonzero, but it was a constant times a delta function, integrating to 1, sofinite total momentum. Yet now, as we said, x − = 0 implies only T −− , which is a large,28et finite constant, times ( dφ/ds ) ( x + ) . Since also ( x + ) is a constant, we have (since φ ∼ √ s ) T −− ∼ (cid:18) dφds (cid:19) ∼ s ∝ √ x + x − . (5.30)By integrating over x − , this means that there is a vanishing momentum located at x − = 0.On the other hand, using ˙ φ ≡ ∂φ∂t = dφds ts , (5.31)the energy is (ignoring the finite and vanishing terms in the energy-momentum tensor) E = (cid:90) dxT (cid:39) (cid:90) dx ˙ φ (cid:112) l ( ∂ µ φ∂ ν φg µν + m φ ) ∝ (cid:90) dx (cid:18) dφds (cid:19) t s ∝ (cid:90) dx s = (cid:90) dx t − x ) / ( t + x ) / , (5.32)and integrating over x at fixed t , around x = t , we get a divergence. So the total energyof the field is divergent, which is consistent with a object of finite mass M (the Skyrmion)boosted with an infinite γ . But that means that now M γ is not kept finite, but is stillinfinite.In fact, now we realize M should be kept fixed in the limit. Indeed, M can only dependon the parameters of the theory, and those are fixed by experiment, and cannot be varied.Now the limit only makes use of Lorentz contraction: the horizon in the x directionhas zero size now. But the action on the BIon-like ansatz included φ (cid:48) ( r ) and φ ( r ), where r was the total radial coordinate, which is now (since v (cid:39) r (cid:39) [( x − t ) γ ] + ˜ r = ( x − ) γ + ˜ r , (5.33)where ˜ r represents now the transverse coordinate. That means that the dependence on ˜ r becomes negligible. Of course, now also ˙ φ becomes nonzero in the ansatz. Thus trivially,we have that the equations of motion on the BIon-like ansatz become the ones on theshockwave ansatz under the infinite boost, therefore the BIon-like solution should becomethe shockwave one (in the single wave region, t < , x <
0, for instance), as wanted.It is just that we can only ”see” the outside horizon solution (with ¯ λ > x direction has become zero by the infinite boost, that is why we don’tsee it. Of course, its extent in the transverse directions has remained the same, but it issituated on the singularity s = 0, so it is impossible to reach. T ¯ T -like actions and chiral perturbation the-ory In this subsection, after reviewing Skyrmion-like solitons of non-Abelian actions and chi-ral perturbation theory, we will propose a 3+1 dimensional, non-Abelian version of the29 ¯ T action, and argue that it consistent with chiral perturbation theory confronted byexperiment. In [5], one of us has considered the possibility of a certain nonabelian generalization of theD-brane action that can have Skyrmion-like solutions. The reason why we considered theD-brane action is that, as we saw, the Skyrmion-like solution is the analog of the BIon-likesolution.Indeed, in a paper by Pavlovsky [29], it was shown that we can have Skyrme-likesolutions for a nonabelian DBI-like scalar action, but with the wrong sign inside the squareroot .For effective actions for QCD in the low energy expansion (note that the various nonlin-ear sigma models used for QCD are, at least for the relevant first order in chiral perturbationtheory, all equivalent through field redefinitions, etc.), one way of defining the theory is byan SU (2) matrix U containing the linearized pions π i σ i in an expansion of U around theidentity, from which we form the quantities L µ = U − ∂ µ U. (5.34)Then the action of [29] is L P = Tr (cid:114) β L µ L µ . (5.35)Considering the spherically symmetric (”hedgehog”) ansatz U = e iF ( r ) (cid:126)n · (cid:126)σ , (5.36)where (cid:126)n = (cid:126)r/r is the radial unit vector and σ i are the Pauli matrices), we find L i = in i (cid:126)n · (cid:126)σ (cid:18) F (cid:48) − sin F cos Fr (cid:19) + iσ i r sin F cos F − i sin Fr (cid:15) ijk n j σ k ⇒ L i = − F (cid:48) − Fr − (cid:126)n · (cid:126)σ sin F cos Fr ⇒ Tr[ L i ] = 2 (cid:20) − F (cid:48) − Fr (cid:21) . (5.37)Now, both in [29] (implicitly, from the equations deduced from it) and in [5] (explicitly),the term with (cid:126)n · (cid:126)σ was erroneously forgotten, so that it was claimed that on the hedgehogansatz, the L µ L µ appearing inside the square root was already proportional to the identity.For all the calculations in these papers to be valid, however, we need then to change theLagrangian to one with square root inside, L (cid:48) P = (cid:114) β Tr[ L µ L µ ] , (5.38)30ince then the Lagrangian on the hedgehog ansatz is unmodified with respect to the resultsthere, and is L (cid:48) P = (cid:115) − β F (cid:48) − sin Fβ r . (5.39)We note then that indeed, the sign inside the square root is wrong (minus, instead ofplus) with respect to the scalar DBI action. But, together with the wrong sign in front ofthe action, this means that at low fields, the action still becomes the same canonical scalar,and only the higher nonlinear terms have different signs.This action [29] has a Skyrmion-like solution, which can be also thought, comparingwith the D-brane action above, as a BIon-like solution (since F ( r ) is like φ ( r ), it has thesame sign inside the square root). But, since the sign inside the square root is wrong, wecan say (and prove rigorously, see [5]) that it doesn’t have a catenoid-like solution (with a“horizon”), and so also not a shockwave solution. The Skyrmion solution has F (0) = N π and F ( ∞ ) = 0, because of the topological constraint, which means that the energy andbaryon number of the solution is quantized.The baryon number density on the Skyrmion solution can be defined. On any solu-tion defined in terms of U = e iF ( r ) (cid:126)n · (cid:126)σ , we have the baryon number (winding number, ortopological number) B = − π (cid:90) d r(cid:15) ijk Tr[ U − ∂ i U U − ∂ j U U − ∂ k U ] , (5.40)so the baryon number density is what is inside (cid:82) d r . Since we get B = − π (cid:90) d r F (cid:48) ( r ) sin F ( r ) r = − π (cid:90) ∞ dr F (cid:48) ( r ) sin F ( r ) , (5.41)it means that the baryon number density is ρ B ( r ) = − π F (cid:48) ( r ) sin F ( r ) r . (5.42)Then, if F (0) = π ), we have a baryon, and if F (0) = − π , we have an anti-baryon.Moreover, we can see in the density, that for the baryon, F (cid:48) ( r ) < F decreases until F = 0 at infinity), so ρ B (0) >
0, whereas for the antibaryon F (cid:48) ( r ) > F increases until F = 0 at infinity), so ρ B (0) < fields .As we saw, the DBI D-brane action is Abelian, whereas the Skyrme-type Pavlovsky oneis non-Abelian, which is essential in order to obtain a topological Skyrme-type solution,31o we need to write a non-Abelian generalization. In [5] an attempt at generalization wasmade only for the space dependence (without the time dependence), and only for | (cid:126) ∇ φ | (cid:28) ¯ M (energies below some cut-off), so the fields are A = φ , or more precisely (cid:126)E = (cid:126) ∇ φ , and L i = U − ∂ i U replacing X . The action was L const . = 1 − Tr (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) − (cid:126)E M (cid:33) (cid:18) − L i M (cid:19) − ( E i L i ) M M + L M + m π M ( U + U † −
2) + M A φ , (5.43)but it is not very satisfying. It was found that it has a BIon-like solution in terms of φ , andat φ = 0 it has a catenoid-like solution (with a “horizon”) in terms of F , but no Skyrmion-like solution for F ( r ), since it has the wrong sign with respect to [29] (the correct sign isfor φ , but that cannot have the Skyrmion radial ansatz, since it is not in the fundamentalrepresentation of SU (2)).But as we said, one should rather have an action that corresponds to a particular formof chiral perturbation theory. To see what that means, we review a few relevant pointsabout the latter.The Lagrangian for low-energy QCD contains the u and d quarks (and maybe the s quark), and some mass terms for the quarks ( u and d certainly, s if we include it), and in themassless quark limit is invariant under global U (2) L × U (2) R (extended to U (3) L × U (3) R if s is included). The masses break the symmetry, so the Goldstone bosons for the symmetrybreaking, the pions π a = ( π + , π − , π ) become pseudo-Goldstone bosons.The diagonal part of the U (1) L × U (1) R component is baryon number, as used in section4, and the other U (1) is a chiral symmetry broken by anomalies. The SU (2) L × SU (2) R isbroken spontaneously to the diagonal SU (2) V , and the other SU (2) is a chiral symmetry,that must be broken spontaneously, but its mechanism in QCD is unknown.To deal with this spontaneous breaking phenomenologically, one constructs a low energyLagrangian in terms of the low energy physical (gauge invariant) mesonic states π a and σ ,put together into the matrix ( τ a are the Pauli matrices)Σ = σ
1l + iτ a π a , (5.44)and constructs the linear sigma model Lagrangian for it, similar to the Higgs Lagrangian.This restricts Σ to its vacuum value v , if we ignore the Higgs-like fluctuation s ( x ),Σ = ( v + s ( x )) U ( x ) , U ( x ) = e iτ a π (cid:48) a ( x ) /v . (5.45)This results in the NonLinear Sigma Model for the unitary matrix U , U † U = 1l, L NL = − v ∂ µ U ∂ µ U † ] . (5.46)This is however only the lowest term (second order) in a low energy expansion inderivatives ∂ µ , or momenta p µ , which is usually taken together with pion masses m π ,32ecause of the on-shell relation p = m π . The next term would be a p term, so L = L ( p ) + L ( p ) + L ( p ) + ... (5.47)The NLSM Lagrangian (5.46) is the most general chiral invariant effective Lagrangianwith 2 derivatives written in terms of the unitary U , however, there are many way tore-express it via field redefinitions, for instance as an SO (4) vector model, etc.Since in this work we have dealt only with unitary matrices U , we consider it to be thebasic first term in the expansion. Indeed, it is the first term in the expansion in derivativesof the non-Abelian version of the Heisenberg or T ¯ T deformation, see for instance (4.19).At the next order ( p ) however, there are many possible terms. If we also considerthe coupling to external fields (that makes the global symmetries local, by considering acovariant derivative), adding the terms¯ qγ µ ( V µ + γ A µ ) q − ¯ q ( s − iγ p ) q (5.48)to the QCD low energy action, we find corresponding terms that can be added to theeffective action, now written in terms of U , χ = s + ip , and also in terms of a Lµ and a Rµ ,equal to V µ ± A µ , with their corresponding field strengths, f Lµν and f Rµν . We now will write a nonabelian version of the T ¯ T action extended to 3+1 dimensions, fromthe point of view of chiral perturbation theory, and then we will compare with the generalexpectations about the latter discussed in the review [30]. That means that we want towrite an action in terms of the matrix U above, and the associated L µ = U − ∂ µ U .To do that, we note the standard replacement, when going from one scalar to the SU (2)-valued scalars U , ( ∂ µ φ ) → − L µ L µ , (5.49)that leads one from the free massless scalar to the nonlinear sigma model action L NLSM = f π L µ L µ ] = − f π ∂ µ U ∂ µ U − ] . (5.50)But this is the starting point of the T ¯ T deformed action in 1+1 dimensions consideredin section 4, trivially extended to 3+1 dimensions. Moreover, as we argued there, we needto also add a WZW term, in order for the Noether charge to match the topological charge,and be equated to the baryon charge. That means that we extend to 3+1 dimensions theanalysis of section 4.Then at zero potential ( V = 0), the Abelian T ¯ T action (now equal to DBI) extended to3+1 dimensions and generalized to a nonabelian action relevant for QCD in the low energy33xpansion is L DBI,NLSM = − f π β (cid:34)(cid:115) − β N c π (cid:18) Tr[ L µ L µ ] − β N c π ˜ X (cid:19) − (cid:35) + L W ZW , (5.51)which has the opposite sign inside the square root with respect to the Pavlovsky case (thathas a Skyrmion solution, as we saw), but otherwise it is the same. We need to define ˜ X and L W ZW in 3+1 dimensions. The latter is standard, and the former can be extendedmost naturally (so as to go back to the 1+1 dimensional result upon dimensional reduction)as ˜ X = 12 (cid:15) µνρσ (cid:15) λτρσ Tr[ L µ L λ ] Tr[ L ν L τ ] , (5.52)which then gives a nonzero contribution on the static ansatz L = 0,˜ X = (cid:15) imp (cid:15) jnp Tr[ L i L j ] Tr[ L m L n ]= 8 sin Fr (cid:26) F (cid:48) + sin Fr + sin F cos Fr (cid:27) . (5.53)With this choice, the Lagrangian on the ansatz is therefore modified by ˜ X . However,another possibility would be˜ X = (cid:15) µνρσ (cid:15) µ (cid:48) ν (cid:48) ρ (cid:48) σ (cid:48) Tr[ L µ L µ (cid:48) ] Tr[ L ν L ν (cid:48) ] Tr[ L ρ L ρ (cid:48) ] Tr[ L σ L σ (cid:48) ] , (5.54)which would vanish on the static ansatz L = 0. With this extension to 3+1 dimensionsthen, the Lagrangian on the ansatz would be unmodified by this term.It remains to add the potential (mass) term. In chiral perturbation theory, a mass termfor one scalar is generalized in the SU (2) case to m π φ → − f π m π
12 Tr[ U + U † − , (5.55)the same that we used in section 4 (the 1/2 is there to cancel the factor of 2 from thetrace). Then the T ¯ T action in the 3+1 dimensional nonabelian case for V (cid:54) = 0 will be (fora potential that is just a mass term, and for λ = 1 / ( f π β ), but note that F = φ/f π fordimensions to work, where φ is the usual dimension 1 field) L λDBI,NLSM = + f π m π Tr( U + U † − (cid:16) m π β Tr( U + U † − (cid:17) − f π β (cid:20)(cid:114) β N c π (cid:16) − Tr[ L µ L µ ] + N c π β ˜ X (cid:17) (cid:16) m π β Tr( U + U † − (cid:17) − (cid:21)(cid:16) m π β Tr( U + U † − (cid:17) Note that we can perhaps be more general, and also replace a generic potential, V ( φ ) = m π f ( φ ) → − m π f (cid:18)
12 Tr[ U + U † − (cid:19) . (5.56) L W ZW , (5.57)which is basically the non-Abelian action (4.19) rewritten in 3+1 dimensions. We refer tothis action as a T ¯ T -like deformation of the Skyrme action. It does not fit to the proposalof [12] for the deformation in dimensions higher than 1+1.In analogy to the expansion in terms of small λ in eqn (4.19) we find now an approximateexpression for the Lagrangian density L (cid:39) (cid:18) f π
16 Tr[ L µ L µ ] + f π m π U + U † − (cid:19) + 1 f π β (cid:40) (cid:18) f π (cid:19) (cid:18) N c π (cid:19) (cid:104) (Tr[ L µ L µ ]) −
2( ˜ X ) (cid:105) − (cid:18) f π m π U + U † − (cid:19) + 14 (cid:18) f π (cid:19) (cid:18) N c π (cid:19) (cid:18) f π m π L µ L µ ] Tr[ U + U † − (cid:19)(cid:27) + O (cid:16)(cid:0) / ( f π β ) (cid:1) (cid:17) . (5.58)With respect to the Abelian pion case, we therefore identify the mass potential as V ↔ − m π f π U + U † −
2) = m π f π ( F/ . (5.59)However, note that λV = ( m π /β )2 sin ( F/ m π = 134 M eV (for π ) or 139 M eV (for π ± ), f π = 130 M eV (or, depending on convention with respect to multiplication ordivision by √ β can only be ∼ Λ QCD . Now it dependshow this quantity is defined, but if it is defined as the string tension, we would expect itto be around 0.4 GeV, or perhaps even a factor of 2 smaller. In that case, the factor of( m π /β )2 sin F/ We can make a comparison with the chiral perturbation theory at order p (first non-trivial order). However, if one would take as β ∼ Λ QCD understood as the glueball mass, ∼ GeV , then 1 − λV ∼ − (1 / F/
2, which clearly cannot be =0. That means this choice doesn’t lead to a horizon. Infact, even if we choose V ∼ m π f π F / m π φ /
2, that still leads to 1 − λV ∼ − (1 / F , and atopological solution without a horizon has F (0) = Nπ (with N = 1 for one Skyrmion) and F ( ∞ ) = 0, so∆ F = π < λ φ = λ f π F potential, with λλ f π ∼ / F (0) − F ( ∞ )) = π >
10, giving a horizon. Then we replacein (5.57), V = − m π f π U + U † − → − m π f π (cid:20) log UU (cid:21) + λ f π Tr (cid:20) log UU (cid:21) . (5.60)In terms of the Skyrmion ansatz, we have V → m π φ − φ ) + λ ( φ − φ ) , (5.61)so 1 − λV = 1 − m π β ( F − F ) − λ f π β ( F − F ) , (5.62)allowing for the possibility of a horizon. Thus in this case, we would need to consider a rather nontrivialpotential, but it is also possible to obtain one that has a horizon.
35f we use ˜ X , this term is of order p , so doesn’t contribute here. If we use the morelikely ˜ X , we find ˜ X = (Tr[ L µ L µ ]) − Tr[ L µ L ν ] Tr[ L µ L ν ] ⇒ (Tr[ L µ L µ ]) − X = − (Tr[ L µ L µ ]) + 2 Tr[ L µ L ν ] Tr[ L µ L ν ]= − (cid:16) Tr[ ∂ µ U ∂ µ U † ] (cid:17) + 2 Tr[ ∂ µ U ∂ ν U † ] Tr[ ∂ µ U ∂ ν U † ] . (5.63)In [30], section 4.7, is written the most general Lagrangian at order p (like the firstnontrivial order in the expansion of the square root for us), in the case m π = 0 (thus withsome difference with respect to our case) −L = L (cid:110) Tr[ D µ U ( D µ U ) † ] (cid:111) + L Tr (cid:104) D µ U ( D ν U ) † (cid:105) Tr (cid:104) D µ U ( D ν U ) † (cid:105) + L Tr (cid:104) D µ U ( D µ U ) † D ν U ( D ν U ) † (cid:105) + L Tr (cid:104) D µ U ( D µ U ) † (cid:105) Tr (cid:16) χU † + U χ † (cid:17) + L (cid:104) Tr( χU † − U χ † ) (cid:105) + L Tr (cid:16) U χ † U χ † + χU † χU † (cid:17) − iL Tr (cid:104) f Rµν D µ U ( D ν U ) † + f Lµν ( D µ U ) † D ν U (cid:105) + L Tr (cid:16) U f
Lµν U † f µνR (cid:17) + H Tr (cid:0) f Rµν f µνR + f Lµν f µνL (cid:1) + H Tr (cid:16) χχ † (cid:17) . (5.64)The covariant derivatives D µ U are related to turning on external a Lµ , a Rµ flavor gaugefields. Needless to say that we can also uplift the ordinary derivatives in (5.58) to covariantones. Besides the terms with scalar sources χ and external gauge fields a Lµ , a Rµ , there are10 terms, and comparison with experiment is done in their Table 4.3. In it, the coefficientsthat are best understood as nonzero (experimentally, at more than 3 σ away from zero)are 2,3 and 9,10. However, the only terms without external fields are 1, 2 and 3, with L = 0 . ± . L = 1 . ± . L = − . ± .
1. We see that expanding the square rootin our Lagrangian (5.57) we obtain the terms first two terms above, with coefficients either1 and 0 (for ˜ X ), or -1 and +2 (for ˜ X ), the last one consistent with the experimentalresults above for about 2 σ in L and 2 σ in L (re-normalizing the coefficient of L tomatch), however, the third term would also have zero coefficient, which is within slightlymore than 3 σ .Thus, at this point it is not yet clear if this is consistent with our model.More importantly it is premature to make any claims about the relations between theprediction of our model and the experimental data. We now consider the boost of the Skyrmion-like solution for the non-Abelian action, andshow that we get the shockwave solution (in the region where there is only one wave). Wealready showed this in the Abelian analogue case, but now we generalize to the real case(non-Abelian).We ignore the potential, since as we saw, the horizon of the Skyrme-like solution willshrink onto the singularity s = 0, so we can safely ignore V , and consider only the non-36belian DBI of the type considered by Pavlovsky. On the Skyrmion ansatz, this Lagrangianbecomes (cid:115) (cid:18) β F (cid:48) + sin Fβ r (cid:19) (1 − λV ( F − F )) . (5.65)Under the boost, F (cid:48) goes back to F (cid:48) − ˙ F and, since r → ∞ under γ → ∞ (at finite x − ,so finite s , meaning for our φ ( s ) shockwave), the term with sin F/ ( β r ) vanishing.That means that on the ansatz, the Lagrangian goes over the the Heisenberg La-grangian, the Abelian scalar DBI one, with f π ( F − F ) = f π ( F − F ( r = 0)) identified withthe scalar φ of the abelian action. Then φ increases away from r = 0, so x − = 0 (with˜ r = 0), as does the shockwave solution.Then by the logic explained in the case of the Abelian analogue, the Skyrmion-likesolution should go over to the shockwave one under the infinite boost.We can also ask whether in the shockwave solution we have a baryon-baryon or baryon-antibaryon collision. We saw that the baryon density ρ B ( r ) can be calculated, and Then,if F (0) = π ), we have a baryon, and if F (0) = − π , we have an antibaryon. Moreover, wecan see in the density, that for the baryon, F (cid:48) ( r ) < F decreases until F = 0 at infinity),so ρ B (0) >
0, whereas for the antibaryon F (cid:48) ( r ) > F increases until F = 0 at infinity),so ρ B (0) < F ( r ), or,in the DBI picture, field φ ( s ), we have a baryon-baryon collision, not a baryon-antibaryoncollision, for which φ (which stands in for F ( r ), up to an additive constant, a we saw)would have to be discontinuous at the collision point x = 0 , t = 0. But we could considerit also to be a baryon-antibaryon solution if we could define different φ ( x = + ∞ ) and φ ( x = −∞ ) at fixed t . But we can’t, since the solution depends on s = − x + x − only, notindependently on x and t (or x + and x − ). This means that the solution does not probethe region x = ±∞ at fixed t , in this region the solution would depend on both variables. Finally, we come to the possible interpretation of our results for QCD. T ¯ T ? The first thing to ask is: why can we use the T ¯ T action in the context of Heisenberg’smodel? The simplest answer is that Heisenberg himself just guessed an action with theright properties namely, a jump in ( ∂ µ φ ) for the shockwave solution and saturation of theFroissart bound, so we can choose another one that has the same properties and, like weshowed, has the added property that it can have both a shockwave and a soliton solutions,in such a way that the solitons can be boosted into shockwaves.37 second point is that the T ¯ T deformation action has unique properties, as beingthe only deformation of simple models that can be completely solved, like in the case ofintegrable models. We could think perhaps of the T ¯ T deformation of the 3+1 dimensionalcanonical scalar action (instead of the deformation in 1+1 dimensions). After all, thesystem we are describing is 3+1 dimensional. But we show in Appendix A that this is notpossible. We could also think perhaps of the T ¯ T deformation of the Maxwell or Yang-Millsaction for a vector (be it the fundamental gluon, or a low energy vector particle). But weshow in Appendix B that that is also not possible.Accepting this apparent “uniqueness property”, we are still left with the question how torelate this construction to the fundamental QCD description of the colliding nucleons. Aswe mentioned at the beginning, in the defining relation of the T ¯ T deformation Lagrangian(3.1) all objects are quantum renormalized and UV finite, and on the right-hand side wehave an operator regularized by point-splitting. Yet we treated the resulting Lagrangianas a classical one, giving rise to classical fields, which is in accordance with the reasoningof Heisenberg for his model.The simplest possibility of relating the T ¯ T deformed Lagrangian to QCD then is to thinkof it as being a quantum effective action in the Wilsonian sense for the pions (or rather,a single (pseudo)scalar version of it, that we tried to extend to SU (2)-valued fields). Theparameter λ of the deformation (corresponding to l for the usual Heisenberg model (1.2)) must be related to the QCD string tension .The quantum theory for this Wilsonian effective theory, now reduced (the Wilsoniantheory, not its quantum theory) to 1+1 dimensions (the direction of propagation and time)should give the defining equation of the T ¯ T deformation (3.1), as a function of λ . Notethat this would not mean that the same equation (3.1) is satisfied for the full quantumtheory of the 3+1 dimensional Wilsonian effective theory (in fact, we saw in Appendix Athat the T ¯ T deformed 3+1 dimensional theory doesn’t have the needed classical solutions).The defining relation (3.1) can be rewritten, with λ = the string tension σ , ∂ L σ ∂σ = det T ( σ ) µν . (6.1)One possible interpretation of this relation is that as we vary the string tension, thevariation of this Wilson effective Lagrangian is proportional to the density square of thepion field, since the pion field breaks up into a quark at one end and an anti-quark at theother, but both are proportional to the density at that point (note that this is how weunderstand the defining relation anyway: by point splitting). And then the Lagrangianchanges by the variation of this pion particle energy density.The Wilsonian effective theory reduced to 1+1 dimensions admits a classical limit aswell, which is when the value of its field is large (many pions in the same state), so we cantalk about a classical profile. For 1+1 dimensions, we have concentrated on the shockwave,which is a statement about the field around a nucleus, moving at high speed. There is alegitimate question as to why the Wilsonian theory is still valid, if we are at high energies,38ut perhaps the answer is simply that we consider the field around a nucleus, and thenucleus has high energy, yet the pions do not, since they are virtual (part of the field).Indeed, the DBI action is used for the saturation of the Froissart bound for σ tot , which bythe optical theorem is ∼ Im M (˜ s, t = 0) < ∼ ˜ s ln ˜ s , (see also around eq. 28 in section 4of [31]) which are both dominated by emission of large number of soft (low energy) pions,which would make a classical (many particles), but low energy field. Like in [4], we can ask what is the interpretation from a gravity dual point of view? Asusual, the φ scalar is the position of some D-brane in the fifth dimension. But we havealready analyzed this in [4], even for the nonlinear sigma model g ij ( φ ) ∂ µ φ i ∂ µ φ j , in eq.(4.15) there. Because of eq. (4.18) there, we have a single g ( φ ), and by comparison withour case (3.6), we have g ( φ ) = ¯ λ ( φ ) = λ (1 − λV ( φ )) . (6.2)We had analyzed in detail the case g ( φ ) = 1 /φ , which corresponds to motion in AdSspace in the gravity dual. But here, we see that the metric in the fifth dimension isdependent of V ( φ ), and moreover has a kind of “horizon”, but only in the sense that themetric g ( φ ) changes sign at φ .Until now, we have compared with the our holographic model, eq. (4.15) in [4], only inthe metric g ( φ ). But the T ¯ T action has more terms. In particular, there is an e − ϕ factor(where ϕ is the dilaton) multiplying the square root, just like 1 / ¯ λ in (3.6), so we identifythose two. Then there is another term, outside the square root, − ˜ V , that is just like theWZ term A e ϕ , so again we must identify the two, leading to g ( φ ) = ¯ λ = λ (1 − λV ( φ )) T p e ϕ ( φ ) = ¯ λ = λ (1 − λV ( φ )) µ p A e ϕ ( φ ) = − ˜ V ( φ ) = 1 − λV ( φ )2 λ (1 − λV ( φ )) . (6.3)Note that since the dilaton ϕ ( φ ) must be positive, and since at the ”horizon” 1 − λV = 0, T p e − ϕ ( φ ) changes sign, it follows that the tension T p changes sign, from D-brane type, toorientifold-type. However, a orientifold does not fluctuate like a D-brane, so it is a negativetension brane in the sense of Randall-Sundrum (which however doesn’t have a string theoryinterpretation). This is then a truly phenomenological interpretation of AdS/CFT: justuse the map, without worrying about string theory rules and derivations. Yet, note thatthe since at this ”horizon”, the brane becomes tensionless, so states are massless, our probeinterpretation for the D-brane becomes meaningless, hence the singularity is a fake one,that needs to be resolved. What we do have instead is a phase transition in the string-likedescription (between D-branes and ”orientifold-like objects”). In terms of the field theoryside, the Wilsonian description in terms of only pions is no longer complete, we have tointroduce the nucleons as well. We ”cheat it in” by using the Skyrmion-like solution forthe pion field to describe the nucleons. 39 Conclusions and open questions
In this paper we have analyzed actions of T ¯ T deformations of the actions of canonicalscalars. The purpose was to use them as effective actions, both in the Heisenberg modelfor high energy nucleon scattering, and in a Skyrme-like action that admits (topological)solitons that can be identified with the nucleons themselves.We have first found that for the 1+1 dimensional T ¯ T deformation of a canonical scalar,which we have found in a companion paper [18] (see also [21, 22]) that it has solitonsolutions, as well as shockwave solutions, the latter can be used to describe nucleons andtheir scatterings that saturate of the Froissart bound. Moreover, we can understand theshockwave solution as corresponding to the collision of two boosted solitons.We have found then that the same action, when extended to 3+1 dimensions, has bothshockwave solutions and BIon-like solutions, with finite radial derivative, while the Heisen-berg action (1.2) has catenoid solutions, with horizons where the radial derivative blowsup. We have argued that the boosted BIon-like solutions should describe the shockwavesolutions.The 3+1 dimensional nonabelian scalar version of the T ¯ T deformation action, wasfound to have Skyrmion-like solutions and shockwave solutions describing the collision oftwo boosted Skyrmion-like solutions. It was also found to be possibly consistent with thephenomenological fit of chiral perturbation theory to experiment.We interpreted these results as saying that the T ¯ T deformation action should be un-derstood as a quantum Wilsonian effective action for QCD, reduced to 1+1 dimensions,though why do we have this equality is unclear. We also attempted an interpretation fromthe point of view of a gravity dual, but the interpretation could not be completed.There are plenty of open questions that are awaiting further investigation. Here we listsome of them: • The T ¯ T deformation at higher than 1+1 dimensions is not unique. In [12] it wasproposed to use | detT | α with α = d − α . An interesting question is what is the deformation conditionthat yields the T ¯ T -like deformation that we are using (5.51). • The relation between T ¯ T deformed actions and QCD. We view the current paper asjust a starting point for this analysis. • The standard theory of Skyrmions has been plagued with several significant problems.In particular (i) The results show that if the pion mass is set to its experimentalvalue then the nucleon and delta masses can not be reproduced for any values of theSkyrme parameters; the commonly used Skyrme parameters are simply an artifactof the rigid body approximation [32]. (ii) It yields nuclear binding energies that arean order of magnitude larger than experimental nuclear data 15% instead of 1% [33].(iii) It predicts intrinsic shapes for nuclei that fail to match the clustering structure40f light nuclei. It will be very interesting to explore the whether these problems canbe circumvented in the deformed Skyrme theory. • Expanding the deformed Skyrme acion in terms of the deformation parameter yieldsthe Skyrme action plus additional terms that are higher order powers in the derivativeof the fields. Such terms are part of the chiral Lagrangian. Thus, it will be veryinteresting to perform a comprehensive comparison between the coefficients of theseterms that determined by the experimental data and those that follow from theexpansion of the deformed action.
Acknowledgements
We thank Aki Hashimoto for useful discussions. The work of HN is supported in partby CNPq grant 301491/2019-4 and FAPESP grants 2019/21281-4 and 2019/13231-7. HNwould also like to thank the ICTP-SAIFR for their support through FAPESP grant 2016/01343-7. The work of J.S was supported in part by a center of excellence funded by the IsraelScience Foundation (grant number 2289/18).
A 3+1 dimensional T ¯ T deformation and the Heisenberg model In this Appendix we will consider one possible T ¯ T deformation of the free scalar in 3+1dimensions, defined in [7], and we will analyze its applicability to the Heisenberg model.The deformed Lagrangian in the presence of a general potential V was defined implicitly,as a solution of an equation, which is impossible to solve in a (simple) closed form in thegeneral case. This would be needed for V = m φ /
2, for instance to analyze the radiationarising from a possible shockwave solution, as seen in the text.However, we will content ourselves with the case of V = 0 ( m = 0), since this sufficesin order to find the leading order shockwave solution near s = 0, where any mass scaleis irrelevant. In that case, the equation for the Lagrangian can be solved exactly for anyspacetime dimension D , with result − L D, ( t, X ) = t (cid:115) t (cid:20) ( ∂ µ φ ) (cid:21) D − − D − . (A.1)Consider, as in the text, a solution φ ( s ) depending only on s = t − x , and independentalso of y, z . If the solution is a shockwave, then moreover φ ( s ) = 0 for s <
0. Then X ≡ ( ∂ µ φ ) = − s (cid:18) dφds (cid:19) , (A.2)41o the Lagrangian on the shockwave solution is − L D, ( t, X ) = t (cid:118)(cid:117)(cid:117)(cid:116) t (cid:34) − s (cid:18) dφds (cid:19) (cid:35) D − − D − . (A.3)The equations of motion are dds t (cid:118)(cid:117)(cid:117)(cid:116) t (cid:34) − s (cid:18) dφds (cid:19) (cid:35) D − − − DD − t t ( D − (cid:115) t (cid:20) − s (cid:16) dφds (cid:17) (cid:21) D − ×× (cid:34) − s (cid:18) dφds (cid:19) (cid:35) D − (cid:18) − s dφds (cid:19) = 0 . (A.4)Define 4 t D − ≡ l D − , (A.5)so that the Lagrangian on the shockwave solution reads − L D, ( t, X ) = t (cid:118)(cid:117)(cid:117)(cid:116) (cid:34) − l s (cid:18) dφds (cid:19) (cid:35) D − − D − , (A.6)and the equations of motion read dds (cid:118)(cid:117)(cid:117)(cid:116) (cid:34) − l s (cid:18) dφds (cid:19) (cid:35) D − − − DD − ( D − (cid:115) (cid:20) − l s (cid:16) dφds (cid:17) (cid:21) D − ×× (cid:34) − s (cid:18) dφds (cid:19) (cid:35) D − (cid:18) − s dφds (cid:19) = 0 . (A.7)The we check whether Heisenberg’s near s = 0 solution φ ( s ) = l − √ s (A.8)is still valid for the equations of motion (A.7) for the D dimensional Lagrangian. We firstnote that, if it is, then the square root in the Lagrangian and equations of motion stillvanishes.One can check explicitly, that in the Heisenberg case, corresponding to D = 2, thesolution comes from the leading terms, coming from derivatives of the square root, and inthe end, the leading terms in the equation of motion on the solution reduce to dds (cid:34) s (cid:18) dφds (cid:19) (cid:35) = 0 , (A.9)42hich is still true in the D dimensional case. Moreover, like in the D = 2 case, the leadingterms, coming from the derivatives of the square root, will also result in the above equation.It is then clear that (A.8) continues to be a solution in the D dimensional case.However, in that case, the on-shell Lagrangian for the shockwave solution is L on − shell = ( − D − = e πiD ∈ C . (A.10)This is only real in the D = 2 case, but the Lagrangian is supposed to be real on classicalsolutions. The only possible exceptions are quantum solutions like instantons, describingsome transition from the quantum path integral.Since the Lagrangian is supposed to be an effective one, describing a regime of lowenergy, it cannot be used to describe quantum transitions, so there is no possibility to usefor for D (cid:54) = 2, in particular cannot be used for D = 4. B On the Abelian and non-Abelian T ¯ T deformations of Maxwelland Yang-Mills theories In this Appendix we consider the Abelian and non-Abelian T ¯ T deformations of Maxwelland Yang-Mills theories in [15], and find that the BIon solution of Born and Infeld is nota solution, and moreover, there is no analogue of this soliton solution for these theories.The Lagrangian is given in an implicit form, in terms of χ ≡ λ Tr F µν F µν , λ L ≡ h ( χ ) (B.1)by the equation χ = 1256 (1 − (cid:112) − h ( χ ))(3 + (cid:112) − h ( χ )) . (B.2)By applying d/dχ on the above equation, we obtain1 ≤ dhdχ = 16(3 + √ − h ) ≤ . (B.3)On the other hand, it is easy to see that the equation of motion (with respect to thegauge field A µ ) is D µ (cid:18) dhdχ F µν (cid:19) = D µ (cid:18) √ − h ) F µν (cid:19) = 0 . (B.4)Another way to think about this is to defined, as in the main text (and as in a medium), ∂h∂E i ≡ D i , (B.5)such that the equation of motion for a static solution is ∂ i D i = (cid:126) ∇ · (cid:126)D = 0 , (B.6)43r rather, qδ ( r ) on the right-hand side, for a delta function source. Here (cid:126)D = dhdχ (cid:126)E. (B.7)On the other hand, (cid:126)E remains finite, just like in the usual Born-Infeld theory. Indeed,in 2 dimensions, [15] show that, defining F = F µν F µν = − F = − E in Minkowskispace, the Born-Infeld Lagrangian is L = (cid:112) λF = (cid:113) − λF , (B.8)so that E = F ≤ √ λ . (B.9)For the T ¯ T theory, maximizing the right-hand side of (B.2), which happens at L =1 / (8 λ ), where the right-hand side takes the value 27/256, so that F ≤ λ ⇒ E = F ≤ (cid:114) λ , (B.10)which is just very slightly smaller than the maximum value for Born-Infeld.Coming back to our issue, we see that D i remains finite, since both terms on the right-hand side of (B.7) remain finite. This is unlike the case of the Born-Infeld soliton, where (cid:126)E ≤ E max = 1 / √ λ here), but (cid:126)D = − (cid:126)E (cid:112) − (cid:126)E → , (B.11)the singularity coming from it being the singular solution of the (cid:126) ∇· (cid:126)D = δ ( r ) equation. Thiswas found first in 3+1 dimensions, but the same continues to be valid in 1+1 dimensions,where for Born-Infeld we have D = ∂ L ∂E = − E √ − λE , (B.12)and again the singularity comes from it being a solution of the ∂D = δ ( r ) equation.In conclusion, since (cid:126)D is non-singular for the case of the T ¯ T action, we cannot haveBI-type soliton solutions for it. References [1] M. Froissart, “Asymptotic behavior and subtractions in the Mandelstamrepresentation,”
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