Instanton-soliton loops in 5D super-Yang-Mills
aa r X i v : . [ h e p - t h ] S e p QMUL-PH-14-07MIFPA-14-25
Instanton-soliton loops in 5D super-Yang-Mills
Constantinos Papageorgakis and Andrew B. Royston
Abstract.
Soliton contributions to perturbative processes in QFT are con-trolled by a form factor, which depends on the soliton size. We provide ademonstration of this fact in a class of scalar theories with generic modulispaces. We then argue that for instanton-solitons in 5D super-Yang-Millstheory the analogous form factor does not lead to faster-than-any-power sup-pression in the perturbative coupling. We also discuss the implications of suchcontributions for the UV behavior of maximally supersymmetric Yang-Millsin 5D and its relation to the (2,0) CFT in 6D. This is a contribution to theproceedings of the “String Math 2013” conference and is a condensed versionof results appearing in [
1, 2 ].
1. 5D MSYM and the (2,0) SCFT in 6D
Over the last few years, there has been renewed interest in the relation be-tween the maximally supersymmetric Yang-Mills theory (MSYM) in 5D and thenonabelian (2,0) tensor CFT in 6D. In [
3, 4 ] it was conjectured that 5D MSYMwith coupling g YM is exactly the (2,0) theory on a circle of radius(1.1) R = g π . This relies on the observation that Kaluza-Klein momentum along the circle canbe identified with instanton charge in the 5D theory; the latter is a topologicalcharge carried by soliton configurations [
5, 6 ]. Equivalently, in the strong couplinglimit 5D MSYM should define the fully decompactified (2,0) theory, which in turnis expected to describe the low-energy dynamics of multiple M5-branes [
7, 8 ].At the same time, since gauge theories above four dimensions are power-counting nonrenormalizable, one would expect that 5D MSYM should be treatedas an effective theory in the Wilsonian sense, that is only defined up to some cutoffscale. It is then reasonable to wonder what it means to consider such a theoryat strong coupling (and hence high energies) and how that would give rise to thewell-defined (2,0) CFT.This tension naturally leads to revisiting the UV behavior of 5D MSYM inthe context of perturbative renormalization. Despite N = 2 supersymmetry beingresponsible for the absence of UV divergences at low orders [ ], the first logarithmicdivergence was explicitly seen at 6 loops in [ ]. However, instead of immediatelytaking the cutoff to infinity and declaring the theory UV-divergent, one has toalso investigate possible contributions associated with virtual soliton states. The Mathematics Subject Classification. simplest such contribution can be related, via the optical theorem, to the soliton-antisoliton pair production amplitude. The latter is conventionally believed to be“exponentially suppressed” and as a result soliton loops are usually ignored.Here we will argue that instanton-soliton pair production in 5D MSYM doesnot fall faster than any power in the effective dimensionless coupling controllinga given perturbative process [ ]. Motivated by [
3, 4 ], one could then envision amechanism through which soliton loops would lead to exact cancelations againstthe perturbative UV divergences and render the theory well defined. We stress thatwe are not treating 5D MSYM as an effective theory in the Wilsonian sense, andinstead we are supposing that the Lagrangian provides a microscopic definition ofthe theory. This point of view can only make sense if the theory is finite.
2. Soliton pair production and form factors
According to standard QFT lore, soliton production is “exponentially sup-pressed” at small coupling and hence unimportant for perturbative physics. How-ever, upon careful consideration one can formulate a more refined version of thatstatement: the exponential dependence should appear in the dimensionless ratioof the soliton size ( R S ) over its Compton wavelength ( R C ), e − R S /R C [
11, 1 ]. Onthe one hand, when the size is fixed to a value much larger than the Comptonwavelength one recovers the expected suppression, as e.g. for ‘t Hooft–Polyakovmonopoles in Yang–Mills–Higgs theory. On the other, for situations where the sizeis a modulus that ranges over values on the order of the Compton wavelength, onemight expect that small solitons would not be suppressed compared to perturbativeprocesses. As we will see, this is precisely the setup for instanton-solitons in fivedimensions.To that end, it is instructive to revisit the derivation of the soliton form factorfor the simple case of scalar theories [ ]. Consider the following class of Lagrangians(2.1) L = 1 g Z d x (cid:26)
12 ˙Φ · ˙Φ − ∂ x Φ · ∂ x Φ − V (Φ) (cid:27) . We denote by x a ( D − x is shorthandfor d D − x . The field Φ is R n -valued and · denotes the Euclidean dot product.Here we assume that the potential has a dimensionless parameter g controlling theperturbative expansion. Then, in terms of canonically normalized fields ˜Φ = g − Φ,we have ˜ V ( ˜Φ; g ) = g − ˜ V ( g ˜Φ; 1), while we have also set V (Φ) = ˜ V ( g ˜Φ; 1) [ ].We are interested in soliton solutions, classically described by localized, finite-energy field configurations and denoted by φ . Such classical solutions for a fixedtopological sector usually come in a smooth family parameterized by a collectionof moduli, U M , where M = i, m . A subset of these moduli always consist of thecenter-of-mass positions, U i = X ; we will call their conjugate momenta P . U m then parameterize all remaining “centered” moduli. We denote the moduli spaceof soliton solutions for a given fixed topological charge as M ; it represents a localminimum of the energy functional. The sectors are labeled by homotopy equivalence classes π D − ( M vac ), where M vac := { Φ | V (Φ) = 0 } . See e.g. [ ]. Note that although Derrick’s theorem [
14, 13 ] precludes theexistence of soliton solutions for
D > D arbitrary. NSTANTON-SOLITON LOOPS IN 5D SUPER-YANG-MILLS 3
In the presence of a soliton a new sector of the quantum theory opens up,which is orthogonal to the vacuum sector since solitons carry a conserved topo-logical charge [ ]. Single particle states in the soliton sector form a subspace ofthe total single-particle Hilbert space and one can study processes involving bothperturbative particles and solitons as asymptotic states. Such soliton states canbe chosen to be momentum eigenstates, | P i . Note that, apart from the soliton’sactual energy and momenta, these states can carry extra labels corresponding toeigenvalues of additional operators that commute with the Hamiltonian. Thesedepend on the particulars of the theory and will be left implicit for the rest of ourdiscussion.Let us now study the soliton pair-production amplitude, involving a perturba-tive incoming excitation of (off-shell) momentum k and a soliton-antisoliton out-going pair of (on-shell) momenta P f and − P i respectively. Although it is unclearhow one should proceed directly—since there exists no known associated analyticclassical solution and hence no semiclassical expansion scheme—crossing symmetryrelates soliton pair-production to a process where the soliton (baryon) absorbs theperturbative excitation (meson)(2.2) A ( k → P f , − ¯ P i ) = A ( P i , k → P f ) . Note that this is an equality between amplitudes in distinct topological sectors.The RHS is related to the form factor i (2 π ) D δ ( D ) ( k + P i − P f ) A ( P i , k → P f ) == Z d D x e − ik · x h P f | T (cid:8) Φ( x ) e − i R d t ′ H I ( t ′ ) (cid:9) | P i i , (2.3)where H I denotes the interaction Hamiltonian.The Hamiltonian obtained from (2.1) is simply(2.4) H = Z d x (cid:26) g · Π + 1 g (cid:18) ∂ x Φ · ∂ x Φ + V (Φ) (cid:19)(cid:27) . Through a canonical transformation [ ] the original conjugate pair of variables(Φ , Π) can be related to the new pairs ( U M , p N ), ( χ, π ) capturing the collectivecoordinate and massive oscillator dynamics respectively. We haveΦ( x ) = φ ( x ; U ) + g χ ( x ; U )Π( x ) = 12 (cid:0) a M ∂ M φ ( x ; U ) + ∂ M φ ( x ; U )¯ a M (cid:1) + 1 g π ( x ; U ) , (2.5)subject to the constraints(2.6) F ,M := Z d x χ · ∂ M φ = 0 , F ,M := Z d x π · ∂ M φ = 0 , which ensure that the fluctuations χ, π are orthogonal to the zero-modes ∂ M φ .Here we have inserted factors of g so that the fluctuation fields are canonicallynormalized. The functionals a M , ¯ a M are given by(2.7) a N = 1 g (cid:18) p M − Z π · ∂ M χ (cid:19) C MN , ¯ a M = 1 g C MN (cid:18) p M − Z ∂ M χ · π (cid:19) , where C = ( G − g Ξ) − with(2.8) Ξ MN = 1 g Z χ · ∂ M ∂ N φ , G MN = 1 g Z ∂ M φ · ∂ N φ . C. PAPAGEORGAKIS AND A. B. ROYSTON
Here G MN is the metric on moduli space, induced from the flat metric on fieldconfiguration space.In terms of these new variables the Hamiltonian can be written as H = g a M G MN a N + v( U m ) + Z h π · π + g s · χ + 12 χ · ∆ χ + V I ( χ ) i ++ O ( g ) , (2.9)with V I ( χ ) denoting cubic and higher-order interaction terms in the fluctuations χ coming from the original potential. In writing the above, we have ignored operator-ordering ambiguities, such that a M = ¯ a M + O ( g ). These corrections correspondto two-loop effects that will not be important for the rest of our calculation.We have also defined s ( x ; U m ) := 1 g (cid:16) − ∂ x φ + ∂V∂ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ= φ (cid:17) , ∆ := − δ ab ∂ x + δ Vδ Φ δ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ= φ , v( U m ) := 1 g Z d x (cid:18) ∂ x φ · ∂ x φ + V ( φ ) (cid:19) = M cl + δ v( U m ) . (2.10)If φ is an exact solution to the time-independent equations of motion then s ( x ; U m ) =0 and δ v( U m ) = 0. However, in theories with centered moduli it is sometimes con-venient to expand around a configuration that is only an approximate solution.The regime needed to extract information about the pair-creation process throughcrossing symmetry requires large velocity exchange and hence momentum transferof the order of the soliton mass, P ∼ mg with m the meson mass. Therefore, theconventional small-velocity (Manton) approximation usually implemented in theliterature is not sufficient for our purposes.
3. Relativistic scalar form factor
In the case of the two-dimensional kink in Φ theory, seminal work by Ger-vais, Jevicki and Sakita [ ] showed how velocity corrections can be systemat-ically accounted for, to recover the covariant expression for the soliton energy, M cl → p P + M . This answer is to be expected, since the starting point is aLorentz-invariant theory. We will now show how the same techniques can be ap-plied in the more general class of Lorentz-invariant theories considered here. Wewill be interested in evaluating the form factor (2.3) rather than the soliton energy.Fortunately, the techniques of [ ] have been adapted to this context by [ ], themethodology of which we will be following closely.The two qualitative differences between the general case and the kink in Φ theory are: a) lack of an explicit classical soliton solution to work with and b)the possible presence of centered moduli. Both can be taken into account andtheir discussion can be appropriately modified, provided we continue to make thesimplifying assumptions of the Manton (small-velocity and small moduli-space-potential) approximation for the dynamics of the centered moduli. Specifically, wewill impose p m /m ∼ O (1 /g ) and s ( x ; U m ) ∼ O (1), but we will assume p i /m = P /m ∼ O (1 /g ). NSTANTON-SOLITON LOOPS IN 5D SUPER-YANG-MILLS 5
The transition amplitude from an initial state i described by the functionalΨ i ( U M ( − T ); χ ) to a final state f described by the functional Ψ f ( U M ( T ); χ ) is S fi = Z [ DU DpDχDπ ] δ ( F ) δ ( F ) e i R T − T d tL Ψ ∗ f Ψ i , with L = p M ˙ U M + Z d x π ˙ χ − H . (3.1)An incoming soliton state of momentum P i is defined by taking Ψ i = e i P i · X i ˜Ψ i ( U m ),where X i = X ( − T ), and similarly for outgoing soliton states. The ˜Ψ i,f are wave-functions on the centered moduli space. In general we will denote quantities asso-ciated with the centered part of the moduli space with a tilde. We can considertime-ordered correlators of the meson field between soliton states by inserting ap-propriate factors of Φ( x ) · · · Φ( x n ) under the path integral, and using the relationΦ( x ) = φ ( x − X ( t ); U m ) + gχ ( t, x − X ( t ); U m ).We are interested in the particular case of the 1-point function and hence in h P f , T | Φ( x ) | P i , − T i = Z [ D X D P ] e i ( X i · P i − X f · P f ) Z [ DU m Dp n ] ˜Ψ ∗ f ˜Ψ i ×× Z [ DχDπ ] δ ( F ) δ ( F ) e i R T − T d tL Φ[ U, p ; χ ]( x ) . (3.2)Let us focus first on the internal path integral over χ and π for which we willproceed to compute the leading contribution at small g . This was done in [ ] forthe case of the 0-point function by evaluating the action on the saddle-point solutionfor χ, π corresponding to the moving soliton. One can argue [
16, 1 ] that the samesaddle point solution gives the leading contribution to the 1-point function, eventhough one should now be solving the equations of motion with source. This isa special feature of working with the 1-point function and would not be true forhigher-point functions. We denote this saddle point ( χ cl , π cl ) and expand the fieldsas χ = χ cl + δχ , π = π cl + δπ .Starting with the Hamiltonian (2.9) one can find a saddle-point solution to the χ, π equations of motion perturbatively in g by making use of the above-mentionedscaling assumptions for the coordinate momenta [ ]. One finds(3.3) χ cl = g − φ (Λ( x − X ); U m ) − g − φ (( x − X ); U m ) + O (1) , where(3.4) Λ ij = δ ij + s P M − ! p i p j P is a Lorentz contraction factor. The insertion can then be expressed asΦ = φ (Λ( x − X ); U m ) + O ( g ) ≡ Φ cl + O ( g ) . (3.5)With this solution in hand, we want to evaluate (3.2) in the presence of centeredmoduli. For this, we also need the Lagrangian evaluated on the solution:(3.6) L = P · ˙ X − q P + M + L (0) [ U m , p m ; δχ, δπ ; P ] + L int , where L int starts at O ( g ) and(3.7) L (0) = p m ˙ U m − ˜ H eff [ U m , p m ; P ] C. PAPAGEORGAKIS AND A. B. ROYSTON is an O (1) contribution describing the dynamics of the centered moduli, whoseprecise form we will not require. ˜ H eff includes the 1-loop potential from integratingout the fluctuation fields ( δχ, δπ ). The leading contribution to (3.2) then takes theform h P f | Φ( x ) | P i i = Z [ D X D P ] e i ( X i · P i − X f · P f ) e i R d t ( P · ˙ X − √ P + M ) ×× Z ˜ M d U p ˜ G ˜Ψ ∗ f ( U m ; P )Φ cl [ X , P ; U m ] ˜Ψ i ( U m ; P ) (1 + O ( g )) . (3.8)In the above we have expressed the centered moduli space path integral as aposition-basis matrix element in the quantum mechanics on the centered modulispace with Hamiltonian ˜ H eff . Note that the ( X , P ) path integral is a functionalintegral representation of the quantum mechanics for a relativistic particle. Fromthe point of view of the translational moduli space dynamics, U m are merely pa-rameters, so we can carry out the functional integration over X and P first andthen integrate over the centered moduli space.This was carried out in [ ] employing the techniques of [ ]. Using that result,which is specific to two dimensions, we find that the amplitude (2.3) takes thefollowing form to leading order: A ( P i , k → P f ) ∼ Z ˜ M d U p ˜ G ˜Ψ ∗ f F [ φ ] (cid:18) R S ( U m ) R C ζ ( P f , P i ) (cid:19) ˜Ψ i , (3.9)where F [ φ ]( u ) = R d v e − iuv φ ( v ) is the Fourier transform of the classical solitonprofile, ˜Ψ i,f = ˜Ψ i,f ( U m ; P i,f ) and ζ ( P f , P i ) := 2 ǫ µν P µf P νi ( P f + P i ) . (3.10)The quantity R S ( U m ), inserted on dimensional grounds, characterizes the size ofthe soliton. For example, in Φ theory R S = 1 /m , with m the meson mass. Aswe previously indicated, in the general class of theories considered here it can inprinciple be a function of the centered moduli. R C = 1 /M cl is the soliton Comptonwavelength.Now, given that the classical soliton profile φ is a smooth ( C ∞ ) function of x − X , we can draw a rather strong conclusion about the asymptotic behaviorof the Fourier transform in (3.9). For any values of momenta such that ζ is not O ( g ) or smaller, it is the 2 R S /R C factor that controls the parametric size of theargument of the Fourier transform. Given this, and as long as the soliton size isbounded away from zero, R min S >
0, we will have that (2 R S /R C ) | ζ | → ∞ in thesemiclassical limit. The Riemann–Lebesgue lemma then implies that (3.11) lim g → F [ φ ] (cid:18) R S ( U m ) R C ζ (cid:19) ∼ e − RS ( Um ) RC | ζ | . Let us emphasize that the exponential on the RHS is a typical function exhibiting afaster-than-any-power falloff that we use for concreteness, but the exact expressionwill depend on the details of the theory under consideration. In any case, theimportant property for our purposes is the faster-than-any-power falloff. As stated by the Riemann-Lebesgue lemma, the Fourier transform F [ f ]( p ) of an L -function f ( x ) goes to zero as | p | → ∞ . Accordingly, if f ( x ) is C ∞ , F [ f ( n ) ]( p ) = ( ip ) n F [ f ]( p ) should alsogo to zero as p → ∞ ; i.e. F [ f ]( p ) goes to zero faster than any power. NSTANTON-SOLITON LOOPS IN 5D SUPER-YANG-MILLS 7
This leads to the expression A ( P i , k → P f ) ∼ Z ˜ M d U p ˜ G ˜Ψ ∗ f e − RS ( Um ) RC | ζ | ˜Ψ i (3.12)for the leading contribution to the form factor as g →
0. Note that the centeredmoduli space represents the internal degrees of freedom of the single-particle state.A field theory interpretation requires a single-particle state to have a finite numberof internal degrees of freedom. The eigenvalues labeling these internal degrees offreedom should be discrete eigenvalues of the centered-moduli-space Hamiltonian˜ H eff . Hence the wavefunctions on the centered moduli space ˜Ψ should be L ; thisis automatically the case if ˜ M is compact. Then we have the inequalities Z ˜ M d U p ˜ G ˜Ψ ∗ f e − RS ( Um ) RC | ζ | ˜Ψ i ≤ Z ˜ M d U p ˜ G | ˜Ψ ∗ f ˜Ψ i | e − RS ( Um ) RC | ζ | ≤ e − R min SRC | ζ | || ˜Ψ ∗ f ˜Ψ i || L ≤ e − R min SRC | ζ | || ˜Ψ f || L || ˜Ψ i || L = e − R min SRC | ζ | , (3.13)where in the second-last step we used H¨older’s inequality. Hence we have reachedthe result A ( P i , k → P f ) . e − R min SRC | ζ | . (3.14)In order to use this result to obtain the pair-production amplitude using cross-ing symmetry, we first re-write ζ in terms of k = ( P f − P i ) . Making use of thefact that P i,f are on shell, one can show ζ = s k k − M . (3.15)The above result is consistent with expectations. First, on physical grounds theform factor should be a function of the momentum transfer only; ζ = ζ ( k ). Second,as k → ∞ we expect ζ ( k ) → O (1); otherwise, one would obtain an amplitudewith exponential behavior for large k , in contradiction with the large-momentumbehavior of asymptotically free theories. Finally, it agrees exactly with the pre-scription proposed in [ ] in the context of the Skyrme model, where it was alsoobserved that exponential falloff is inconsistent with asymptotic freedom.This quantity can be analytically continued from spacelike to timelike k andgoes to the same value as k → ∞ in any direction on the complex plane. Thus,via crossing symmetry we arrive at A ( k → − P i , P f ) . e − R min SRC ζ . (3.16)This is in agreement with the original expectation from dimensional analysis. Notethat if R min S is of order R C this does not lead to suppression.
4. Instanton-soliton loops in 5D MSYM
Let us now apply the above result to the case of interest, i.e. instanton-solitonsin 5D MSYM. Instanton-solitons are finite-energy -BPS configurations, obtainedby solving the selfduality equation for the gauge field strength in the four spatial C. PAPAGEORGAKIS AND A. B. ROYSTON directions, F = ⋆ F , and have mass M cl ∝ /g . As such, they are described bystandard 4D instanton solutions, which for topological charge c ( F ) = 1 and SU(2)gauge group, correspond to classical gauge fields given by(4.1) A i = U ( ~θ ) − (cid:16) η aij ( x − X ) j (( x − X ) + ρ ) T a (cid:17) U ( ~θ ) , A = 0 , with a = 1 , , i = 1 , ..., η aij the ‘t Hooft symbols. This solution has eightmoduli: four center-of-mass collective coordinates X , a size modulus ρ and threeEuler angles ~θ parameterizing global gauge transformations. The associated modulispace is a hyperk¨ahler manifold(4.2) M = R × R + × S / Z , with metric(4.3) d s = 4 π g h δ ij d X i d X j + 2( d ρ + ρ ˜ G αβ d θ α d θ β ) i , where ˜ G αβ is the metric on SO(3) ∼ = S / Z , the group of effective global gaugetransformations.The existence of the noncompact size modulus ρ means that we can have arbi-trarily small or large soliton sizes. However, it is also responsible for the absence of L -normalizable wavefunctions on the centered moduli space ˜ M . This renders theinterpretation of instanton-solitons as asymptotic states confusing, since they wouldcorrespond to particles with a continuous infinity of internal degrees of freedom.Moreover, the semiclassical expansion parameter in this theory is in fact g = g /ρ , which coincides with R C /R S . In particular, note that g ( ρ ) is moduli de-pendent. In the context of finding the saddle-point solution (3.3) we can imagine afixed ρ , such that g ( ρ ) is small. However, when evaluating amplitudes, where onemust integrate over all sizes, the semiclassical approximation breaks down. Conse-quently, the small-sized instanton-solitons invalidate our argument for exponentialsuppression.One can attempt to circumvent this conclusion by turning on a scalar VEV, h Φ i 6 = 0, and going out onto the Coulomb branch. It is known that in this casefinding instanton-soliton solutions requires turning on an electric field, which sta-bilizes the classical size [
18, 19 ]. From the point of view of the quantum theory,turning on an electric field generates a potential on the centered moduli space,(4.4) δ v( U m ) = 2 π g h Φ i ρ , and lifts the flat direction associated with the instanton-soliton size. Although ρ isno longer a true modulus, the VEV provides an additional dimensionless param-eter, ǫ := g h Φ i , that can be adjusted so that we remain in the small-potentialapproximation, where it is still appropriate to represent states as L -wavefunctionson ˜ M . In order to determine the precise form of the resulting L -wavefunctions,one would need to compute the centered-moduli-space Hamiltonian ˜ H eff , appearingin (3.6) and (3.7)Our formalism has been general enough to accommodate such potentials onmoduli space. Thus, despite the classical stabilization, one must still integrate Here Φ is one of the five adjoint scalars of 5D SYM and should not be confused with thescalar fields for the linear sigma models considered in the previous sections.
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