= 4 On the non-planar β-deformed super-Yang-Mills N theory 2 1 0 Q. Jin1 and R. Roiban2 2 n a Department of Physics, The Pennsylvania State University, J University Park, PA 16802, USA 4 2 ] Abstract h t - The β-deformation is one of the two superconformal deformations of the = 4 p N e super-Yang-Mills theory. At the planar level it shares all of its properties except for h [ supersymmetry, which is broken to the minimal amount. The tree-level amplitudes of this theory exhibit new features which depart from the commonly assumed properties 1 v of gauge theories with fields in the adjoint representation. 2 We analyze in detail complete one-amplitudes and a nonplanar two-loop amplitude 1 0 of this theory and show that, despite having only = 1 supersymmetry, the two- 5 N loop amplitudes have a further-improved ultraviolet behavior. This phenomenon is a . 1 counterpartofasimilarimprovementpreviouslyobservedinthedouble-traceamplitude 0 2 of the = 4 super-Yang-Mills theory at three and four loop order and points to the 1 N existence of additional structure in both the deformed and undeformed theories. : v i X r a 1 [email protected] [email protected] 1 Introduction The maximally supersymmetric Yang-Mills (sYM) theory has provided a valuable arena for devising new and powerful methods for perturbative quantum field theory computations. In the planar limit, this effort exposed remarkable properties of the theory, such as dual (super)conformal invariance and integrability, as well as unexpected relations between a priori unrelated quantities, such as the relation between certain null polygonal Wilson loops, scattering amplitudes and correlation functions of gauge invariant operators. Each of these presentations of scattering amplitudes, as a Wilson loop or as a (square root of a) correlation function, manifestly realizes one of the two conformal symmetries of the theory. It has been suggested that, in four dimensions, the symmetries of the theory together with the leading singularities of amplitudes completely determine the planar loop integrand to all orders in perturbation theory [1]. At the nonplanar level the symmetry structure of the theory is expected to be different. While comparatively less studied, explicit calculations of complete amplitudes [2, 3] expose unexpected structure: the color/kinematics duality and the BCJ relations [4], generalizing the U(1) decoupling identities and the Kleiss-Kuijff relations, provide a direct link between the planar and non-planar parts of amplitudes. They also suggest that the kinematic nu- merators of integrands of amplitudes (once organized in a specific way) obey an algebra akin to that of the color factors. The fundamental origin as well as its implications and limi- tations are not completely understood (see however [5] for an interesting discussion in the self-dual sector). The color/kinematic duality has also been discussed in less supersymmetric theories [4]. Apossiblestrategytoidentify theessential elements underlying thefascinatingproperties of the = 4 sYM theory at the non-planar level is to deform this theory in a way that does N not alter its planar properties andthen analyze the resulting theory. Since such deformations necessarily break supersymmetry, this analysis would also probe, in a more general context, other aspects of the undeformed theory: the possibility and subtleties associated to using the Coulomb branch as infrared regulator even at the planar level; the consistency conditions on massless S-matricesdiscussed in[6], etc. Afurther appealofthisstrategyisthat, byreducing the number of supercharges, unexpected phenomena unrelated to maximal supersymmetry but hidden by it may emerge at low loop order. The two deformations of the = 4 sYM theory which preserve conformal invariance to N all orders in perturbation theory at both planar and non-planar level have been identified by Leigh andStrassler in [7]. In bothcases the arguments described there show that there exists a relation between the gauge coupling, the strength of the deformation and the number of colors which, if satisfied, guarantees conformal invariance of the theory. The so-called β-deformation was extensively studied from several standpoints. The pla- 2 nar scattering amplitudes have been discussed in [8] where it was shown that, for real β, they are inherited from the undeformed theory and thus are dual conformally invariant. From the current perspective this result reinforces the link between integrability and dual conformal invariance. Indeed, the β-deformation preserves integrability both at weak [9] and strong coupling [10] and the construction of its string theory dual [11, 10] suggests that the arguments of [12, 13] relating integrability and dual conformal symmetry should apply in this case as well. While the existence of a line of fixed points is guaranteed by the arguments of [7], the precise relation between parameters is known in general only through two-loop order3. Our calculations of four-point amplitudes at one- and two-loop level will confirm the known constraint. We will argue that this constraint must be modified at three loops. This is similar in spirit to the argument of [8] that, for complex deformation parameter, the planar relation between the deformed theory and = 4 sYM theory breaks down at five loops. N As all massless theories, the β-deformed = 4 sYM theory has infrared divergences and N therefore requires regularization. While in gauge theories IR divergences are conventionally dimensionally regulated, several possible IR regularizations are available. In the planar undeformed theory it has been argued [14] that moving off the origin of the Coulomb branch and partly breaking the gauge symmetry as SU(N+M) SU(N) SU(M) will on the one 7→ × hand regularize IR divergences while on the other will preserve dual conformal symmetry if massesoffields(orvacuumexpectationvaluesofscalars)areassignedsuitabletransformation rules. An essential ingredient in these arguments is that, as a consequence of extended supersymmetry, the relation between vacuum expectation values and field masses does not receivequantumcorrections. Theminimalsupersymmetry oftheβ-deformedtheorydoesnot guarantee the absence of such corrections which, as we will see, appear at the very least due to the corrections to the conformality condition. Therefore, using the Higgs regularization in the β-deformed theory at higher loops requires a certain amount of care as one should isolate the quantum corrections to the regulator. For the same reason, the close relation between the = 4 and the β-deformed planar amplitudes [8], while present in dimensional N regularization, does not appear to hold in a straightforward way in the Higgs regularization. By completely breaking the gauge symmetry, SU(N) U(1)N−1, the Higgs regulariza- → tion may also be used at nonplanar level both in the β-deformed and in the undeformed theory. While we will draw information on the structure of amplitudes in the presence of such a regulator, we will carry out calculations in dimensional regularization. It is interest- ing to note that, unlike in the = 4 theory, dimensional regularization changes the total N number of degrees of freedom of the theory. Indeed, as an = 1 theory that does not N 3In the planar limit this relation is known to all loop orders. There have been in fact two definitions of finiteness, which provide slightly different constraints on parameters. One of them demands that all beta-functions vanish while the other demands that all correlation functions are finite. 3 have an = 2 extension, IR dimensional regularization will dimensionally-continue vector N multiplets without reducing the number of chiral multiplets. For this reason the notion of critical dimension as the dimension in which the first logarithmic divergence appears at some fixed loop order is not a well-defined concept in the β-deformed theory. One may neverthe- less formally continue to arbitrary dimension the integrand of an amplitude and thus define a quantity which, as we will see, captures the convergence properties of the integrand and (discontinuously) becomes the standard critical dimension as the deformation parameter is set to zero. We will begin in the next section with a brief review of the β-deformed sYM theory, discuss its Coulomb branch and outline the main differences between its scattering ampli- tudes and those of the undeformed theory. In 3 we will discuss the general features of § loop calculations in this theory through the generalized unitarity method, the constraints imposed by supersymmetry and describe non-planar all-loop results that are inherited from the undeformed theory. In 4 we present the explicit expressions for representatives of the § five different classes of four-point amplitudes and discuss the properties of their infrared divergences. In 5 we describe the first nontrivial two-loop amplitude which is sensitive to § the deformation parameter and analyze its properties. We will see that, unlike the one-loop amplitudes which exhibit the standard properties of a finite = 1 theory, the two-loop N amplitudes we evaluate have better UV convergence properties than one might expect based on = 1 supersymmetry alone. We summarize our results in 7 and speculate on the N § relation of the improved UV behavior and the undeformed theory. 2 The β-deformed = 4 sYM theory and its features N The β-deformed =4 theory is one of the two = 1 exactly marginal deformations4 of N N the maximally supersymmetric Yang-Mills theory in four dimensions [7]; it is obtained by adding to the superpotential of the = 4 sYM theory the term δW = Tr[Φ1 Φ2,Φ3 ]. The N { } action may be written in =1 superspace as N S = d4xd4θTr[e−gYMVΦ¯iegYMVΦi] (1) Z 1 + d4xd2θTr[WαW ]+ih d4xd2θTr[qΦ1Φ2Φ3 q−1Φ1Φ3Φ2]+h.c. . 2g2 α − YM Z Z This superspace, inherited from the one used for the = 4 theory, manifestly preserves the N fourth component of the = 4 quartet of supercharges. The parameter q is customarily N parametrized as q = exp(iβ) with β being either real or complex. The two choices have 4The second exactly marginal deformation is given by δW =Tr[Φ3+Φ3+Φ3]. 1 2 3 4 distinct quantum mechanical properties with only the former theory, with β R, being ∈ integrable at the planar level. In the following we will assume that β is real. It has been pointed out in [11] that the addition of δW may be interpreted as a non- commutative deformation of = 4 sYM theory in the R-symmetry directions; instead N of space-time momenta, the relevant Moyal-like product involves the three U(1) charges q = (q ,q ,q ) inherited from the SU(4) symmetry of = 4 sYM theory: 5 1 2 3 N ΦiΦj eiǫabcβaqibqjcΦiΦj eiβbcqibqjcΦiΦj eiqi∧qjΦiΦj , (2) 7→ ≡ ≡ where β = β = β = β. It is not difficult to see that such a phase becomes trivial whenever 1 2 3 at least one of the two fields belongs to the = 1 vector multiplet. In the following we will N denote by [ , ] the commutator in which the product of the two entries has been replaced β • • by the Moyal product (2), i.e. [A,B]β = eiβbcqAbqAc AB eiβbcqBb qAc BA . (3) − The action (1) exhibits three global U(1) symmetries. One of them is the U(1) R- symmetry of the = 1 superspace. The other two simultaneously rephase two of the three N chiral superfields as: (Φ1,Φ2,Φ3) (Φ1,eiα1Φ2,e−iα1Φ3) (Φ1,Φ2,Φ3) (eiα2Φ1,e−iα2Φ2,Φ3) . (4) 7→ 7→ Three linear combinations of these symmetries form the Cartan subalgebra of the SU(4) R-symmetry of = 4 sYM theory under which fermions transform in the fundamental N representation (spinor representation of SO(6)) with charges q1 = (1, 1, 1) , q2 = ( 1, 1, 1) , q3 = ( 1, 1, 1) , q4 = (1, 1, 1) ; (5) 2 −2 −2 −2 2 −2 −2 −2 2 2 2 2 the index ”4” labels the supercharges preserved by the = 1 superspace in eq. (1). The N scalar fields transform in the two-index antisymmetric representation (vector representation of SO(6)) and thus their charges may be obtained in terms of those of fermions as φij , qij = qi +qj , φk4 φij , qk4 = qk +q4 = qi qj , i,j,k = 1,2,3 . (6) ≡ − − This is the same labeling used in the = 4 on-shell superspace. The phase associated to a N product of fields with charges q ,...,q is just 1 n n φ ...φ eiϕ(q(φ1),...,q(φn))φ ...φ ϕ(q(φ ),...,q(φ )) = q(φ ) q(φ ) . (7) 1 n 1 n 1 n i j 7→ ∧ i<j=1 X 5For the present choice of superspace, the product of component fields is directly inherited from the product of superfields because the charge vector of the manifestly-realized supercharge does not affect the phase in eq. (2). 5 This expression may be further simplified if the charge vectors have additional properties, such as a vanishing total charge n q(φ ) = 0. i=1 i Based on the properties of noncommutative Feynman graphs [15] it has been argued P that, in dimensional regularization and for real β, all planar scattering amplitudes of the β- deformed = 4 theory arethe same –up to constant β-dependent phases – asthe scattering N amplitudes. For complex β this equivalence appears to break down at five loop order [8]; it was moreover shown [16] that finiteness of the planar propagator corrections require that β be real. The deformed theory distinguishes between an U(N) and an SU(N) gauge group. In the former case, the scalar fields valued in the diagonal U(1) factor do not decouple and their coupling constant flows for all values of the parameters of the theory. They decouple in the infrared, where their coupling constants reach zero and the Lagrangian becomes that of the SU(N) theory. In a trace-based presentation, the component Lagrangian for this gauge group is = + (8) 1 2 L L L = Tr 1F 2 + 1ψ¯4D/ψ4 +D φ D φi + 1ψ¯D/ψi L1 4 µν 2 µ i µ 2 i h i +√2g (ψα4L[φ¯ , ψi] ψ¯α˙R[φi, ψ4])+ hǫ ψαiL[φj, ψk] +h¯ǫijkψ¯α˙R[φ¯ , ψ¯ ] YM i α − i α˙ √2 ijk b α β i j α˙k β (cid:0) (cid:1) +1g2([φ¯, φi])2 1 h 2ǫ ǫilm[φj, φk] [φ¯, φ¯ ] 2 i − 2| | ijk β l m β = + 1 h2 ǫ ǫilmTr[[φj, φk] ]Tr[[φ¯, φ¯ ] ] i (9) 2 ijk β l m β L 2N| | where L and R are chiral projectors and appears upon integrating out the auxiliary fields 2 L due to a tracelessness condition. In the component Lagrangian one may further generalize the Moyal product (2) by replacing β with a generic 3 3 antisymmetric matrix [10, 9]. Such × a generalization completely breaks supersymmetry and the resulting theory appears to be unstable [17]. The original arguments of Leigh and Strassler [7] imply that the theory with an SU(N) gauge group is conformally invariant if the three parameters g , h and q obey one relation, YM γ(g ,h,q) = 0. The complete functional form of γ is not known; through two-loop order YM and for q = 1, vanishing of the β-function as well as finiteness of correlation functions | | require that [18, 19] 1 g2 = h 2 1 q q−1 2 . (10) YM | | − N2| − | (cid:18) (cid:19) This relation is expected to receive higher-loop corrections as well as corrections depending on higher powers of 1/N2. 6 2.1 Coulomb branch Thestructure oftheCoulomb branch oftheβ-deformed = 4theory wasdiscussed indetail N in [20, 21] and certain terms in the effective action for the light fields were evaluated through two loops in [22]. As usual, the classical vacuum structure of the theory is determined by the F- and D-term equations [φ23,φ31] = 0 = [φ31,φ12] = 0 = [φ12,φ23] (11) β β β ǫ [φij,φk4] = 0 . ijk As it is well-known, for β = 0 the solution to these equations is given by generic diagonal unitary matrices of unit determinant. At a generic point on the Coulomb branch the SU(N) gauge group is broken to U(1)N−1. A single scalar field with non-trivial vacuum expectation value – e.g. φ23 = diag(λ ,...,λ ) – is sufficient to break completely the gauge symmetry. 1 N All fields except those valued in the Cartan subalgebra U(1)N−1 become massive with masses given by: mW = λ λ , mΨ = λ λ , mφ = λ λ . (12) ij | i − j| | ij| | i − j| | ij| | i − j| Duetothe extended supersymmetry these expressions donot receive corrections to any order in perturbation theory. For nonvanishing and generic6 value of β the number of vacua is smaller; they are all inherited from the vacua of the undeformed theory. The vacuum described above, in which only one scalar field has nontrivial vacuum expectation value, continues to exist. The masses of the W-bosons is unaffected by the deformation, while the masses of the charged fields become mW = mλ4 = λ λ mφ23 = mλ1 = λ λ (13) ij ij | i − j| ij ij | i − j| 1 mφ13 = mφ12 = mλ2 = mλ3 = h(g ,N,q) qλ q−1λ (14) ij ij ij ij g YM | i− j| YM Since the = 1 supersymmetry algebra does not have a central charge, the relation between N vacuum expectation values and masses of fields may receive quantum corrections. The IR divergences ofscattering amplitudes of theundeformed theory may beregularized – at least in the planar limit – by evaluating them at a generic point on the Coulomb branch where the vacuum expectation value of scalar field(s) acts as a regulator. The nonrenormal- izationofeqs. (12)implies theextractionofthefinitepartofamplitudes throughthesubtrac- tionoftheknownformofIRdivergences[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] is straightforward. This is no longer so in the presence of the deformation. Indeed, since 6Interesting features emerge if β/(2π) is rational. 7 (14) receive corrections at least through the condition that the theory is finite (eq. (10) at one- and two-loop orders)), the universal IR divergences that should be subtracted will in fact depend on these corrected masses. It is moreover in principle possible that masses of the vector and chiral multiplets receive different finite renormalization. These new effects will modify the finite part of some L-loop amplitude by terms proportional to lower-loop ampli- tudes and, for β R, are expected to appear at least at (1/N) since conformal invariance ∈ O requires h 2 = g2 [38, 39] to all orders in planar perturbation theory. | | YM 2.2 The tree-level amplitudes of the β-deformed = 4 sYM theory N The interpretation of the β-deformed theory as a non-commutative deformation implies that most of the tree-level amplitudes of the deformed theory are inherited from those of the = 4 sYM theory. The presence of the double-trace terms in the component Lagrangian N signals however that there exist additional amplitudes, not present inthe undeformed theory. To all-loop orders, the color decomposition of an SU(N) gauge theory is [40] β(k ,...,k ) = Tr[Taσ(1) ...Taσ(n)]Aβ (k ,...,k ) (15) A 1 n n;1 σ(1) σ(n) SXn/Zn + Tr[Taσ(1) ...Taσ(n1)]Tr[Taσ(n1+1) ...Taσ(n)]Aβn;2(kσ(1),...,kσ(n)) Sn/ZnX1×Zn−n1 + terms with m 3 traces+ (1/N) . ≥ O In the following we will sometimes use the shorthand notation Tr[Ta1...Tan] Tr . The 1...n ≡ tree-level single-trace terms to leading order in 1/N are inherited from the = 4 theory7: N Aβ,(0)(k ,...,k ) = eiϕ(qσ(1),...,qσ(n))AN=4,(0)(k ,...,k ) (16) n;1 σ(1) σ(n) n;1 σ(1) σ(n) where the phase ϕ is defined in eq. (7): n n n n ϕ(q ,...,q ) = β qaqb = q q . (17) 1 n ab i j i ∧ j i=1 j=i+1 i=1 j=i+1 X X X X Clearly, the phase ϕ depends on the color ordering. The antisymmetry of β ensures that ϕ(q ,...,q ) = ϕ(q ,...,q ) ; (18) 1 n n 1 − this property is crucial for the finiteness of the theory. Tree-level 1/N2 terms in the single- trace sector appears because the coefficient of the superpotential differs from the gauge coupling at this order (10). 7The sameis true for the leadingterms inthe 1/N expansionofthe single-tracepartialamplitudes to all orders in perturbation theory [8]. 8 While the noncommutative interpretation of the β-deformation is transparent in the trace-based color decomposition, a color decomposition based on the structure constants is possible. Such a decomposition also makes contact with color/kinematics duality [4] and allows us to examine its fate once β = 0. From this perspective, the main consequence of 6 the deformation is the appearance of the symmetric structure constants d in the coupling abc of scalar fields, either among themselves or with fermions. The structure constant in the three-point vertices with all changed fields are then replaced with fabc = Tr[Ta[Tb,Tc] ] = eiϕ(a,b,c)Tr[TaTbTc] eiϕ(a,c,b)Tr[TaTcTb] ϕ β − = cos(ϕ(a,b,c))fabc +isin(ϕ(a,b,c))dabc (19) where the phase ϕ is determined by the fields attached to the vertex. These modified structure constants are antisymmetric if, when interchanging two color indices, one also interchanges the R-charges of the corresponding fields; this is possible because the R-charge is conserved at each three-point vertex. It is not difficult to see that multi-trace tree-level amplitudes contain an even number of such modified structure constants. 2.3 Examples: four-point amplitudes at tree-level As we will discuss in detail in 3.3, four-point amplitudes in the β-deformed = 4 theory § N can be classified following the number and position of fields in the vector multiplet. At tree-level the situation is more constrained, as most amplitudes are closely related to those of the undeformed theory and, as such, superficially enjoy all the constraints imposed by = 4 supersymmetry. N By considering the R-charges of fields it is not difficult to see that only single-trace amplitudes with at most one field in the vector multiplet, (0)(1g+,2φ23,3ψ134,4ψ124) and (0)(1φ23,2φ14,3φ13,4φ24) , (20) A4 A4 receive β-dependent corrections. For all other field configurations the antisymmetry of the phase (17) implies that any potential β-dependent phase factor is absent; in general at least three different nontrivial charge vectors are necessary for a single-trace tree-level amplitude to be affected by the deformation. At higher loops differences appear for other external field configurations. All color-orderedamplitudes included inthefirstamplitude ineq. (20)aremodifiedeither by a factor of q = exp(iβ) or its inverse – for example β,(0) ˜ N=4,(0) A (1234) = h(g ,N,q)qA (1234) 4;1 YM 4;1 Aβ,(0)(1243) = h˜(g ,N,q)q−1AN=4,(0)(1243) , (21) 4;1 YM 4;1 9 ˜ ˜ where h is defined as h = h/g . For this field configuration there is no double-trace YM tree-level amplitude. For the second amplitude in (20) all single-trace color-ordered amplitudes are the same as in the = 4 theory except for N Aβ,(0)(1324) = h˜(g ,N,q)2q2AN=4,(0)(1324) (22) 4;1 YM 4;1 Aβ,(0)(1342) = AN=4,(0)(1342)+(1 h˜(g ,N,q)2)AN=4,(0)(1324) . (23) 4;1 4;1 − YM 4;1 In the first case the complete amplitude comes from a single β-dependent four-scalar vertex. In the second, the amplitude receives two different contributions: from a β-independent 4-scalar vertex and from two three-point gluon scalar vertices, the former being equal to h˜(g ,N,q)2AN=4,(0)(1324). This amplitude also contains a double-trace component: − YM 4;1 1 Aβ,(0)(13;24) = h˜(g ,N,q)2(q q−1)2 . (24) 4;2 −N YM − Thistermisgeneratedbythedouble-traceLagrangian ineq.(8). Supersymmetry requires 2 L that there exist similar two-trace amplitudes with two fermions and two scalars as well as two-trace amplitudes with four external fermions. They may be generated from (24) though supersymmetry Ward identities or by direct evaluation starting from the Lagrangian (8) and focusing on the terms containing the symmetric structure constants. Higher-point multi- trace amplitudes can be found though e.g. color-dressed on-shell recursion relations or color-dressed MHV diagrams in a trace basis. There exist, of course, other presentations of color-dressed amplitudes of this theory. In particular, to examine the fate of the color/kinematics duality for the β-deformed theory it is usefultoexpressthecolorfactorsintermsofstructureconstants. Boththemodifiedstructure constants (19) as well as the standard antisymmetric structure constants are necessary, the former coming from the interaction terms depending solely on the chiral multiplets and the latter from the vector multiplet interactions. Using the modified structure constants (19) it is not difficult to see that the representa- tives (20) of the two classes of amplitudes which receive β-dependent deformations may be written as n n n β,(0)(1g+,2φ23,3ψ134,4ψ124) = h˜ 12f12af34 +h˜ 23f23af14 +h˜ 13f31af24 (25) A4 s β a s β a s β a 12 23 13 n′ n′ n′ β,(0)(1φ23,2φ14,3φ13,4φ24) = 12f12af34 + 23f23af14 + h˜ 2 13f31af24 , (26) A4 s a s a | | s −β β a 12 23 13 ˜ ˜ where h stands for h(g ,N,q) and f is f for ϕ = β. The third term in the second YM β ϕ equation above includes the double-trace partial amplitude in (24). In eqs. (25) and (26) the various numerator factors n have a Feynman diagram interpretation, being determined by ij 10