The near-flat-space and BMN limits for strings in AdS 4 CP 3 at one loop

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1 IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 5 ( (3pp doi: / /5//0501 The near-flat-space and BMN limits for strings in AdS CP 3 at one loop Michael C Abbott 1 and Per Sundin 1 Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Rd, Mumbai , India Astrophysics, Cosmology & Gravity Centre and Department of Applied Mathematics, University of Cape Town, Private Bag, Rondebosch, 7700, South Africa abbott@theory.tifr.res.in and nidnus.rep@gmail.com Received 7 August 011, in final form 8 November 011 Published 8 December 011 Online at stacks.iop.org/jphysa/5/0501 Abstract This paper studies type IIA string theory in AdS CP 3 in both the Berenstein Maldacena Nastase (BMN and Maldacena Swanson or near-flat-space limits. We derive the simpler Lagrangian for the latter limit by taking a large worldsheet boost of the BMN theory. We then calculate one-loop corrections to the correlators of the various fields using both theories. In all cases the near-flat-space results agree with a limit of the BMN results, providing evidence for the quantum consistency of this truncation. The corrections can also be compared to an expansion of the exact dispersion relation, known from integrability apart from one interpolating function h(λ. Here we see agreement with the results of McLoughlin et al (008 J. High Energy Phys. JHEP11(008069, and observe that it does not appear to be possible to fully implement the cutoff suggested by Gromov & Mikhaylov (009 J. High Energy Phys. JHEP0(009083, although for some terms we can do so. In both the near-flat-space and BMN calculations there are some extra terms in the mass shifts which break supersymmetry. These terms are extremely sensitive to the cutoff used, and can perhaps be seen as a consequence of using dimensional regularization. PACS numbers: 11.5.Tq, 11.5.Db, Yc 1. Introduction Closed strings in flat spacetime have the property that left- and right-moving excitations decouple, but this is no longer true in curved backgrounds such as AdS 5 S 5 or AdS CP 3. Even after one has taken the Berenstein Maldacena Nastase (BMN, or Penrose limit, focusing on strings very near to a light-like trajectory [5], nontrivial couplings between the left movers and right movers remain. The limit proposed by Maldacena and Swanson in [1] can /1/0501+3$ IOP Publishing Ltd Printed in the UK & the USA 1

2 be viewed as a partial restoration of this decoupling, by virtue of the fact that one light-like worldsheet momentum is taken to be much larger than the other, p p +. Both of these limits are taken using a power of the t Hooft coupling λ as the parameter, thus tying them to the semiclassical limit λ 1. The BMN limit looks at perturbations of order 1/λ 1/ from a null geodesic, allowing us (at a given loop order to truncate the expansion of the Lagrangian in the number of fields. The near-flat-space limit then scales the momenta such that p /p + λ.thiscanberegardedasaworldsheetboostintheσ direction, and the simplification comes from discarding all interaction terms except the leading ones under this boost. It is not obvious that this truncation will be respected quantum mechanically. For this to happen, the contributions from where momenta on internal lines are not large must cancel out, or at least be sufficiently small. In the case of strings in AdS 5 S 5,thisquantumconsistencywascheckedatoneloop in [6] and at two loops in[7]. These papers computed primarily corrections to four-point functions and compared these to the worldsheet S-matrix known exactly from integrability [8]. The second paper [7] alsocomputedcorrectionstothetwo-pointfunctionandcompared these to the dispersion relation E( p. It is useful here to think of the near-flat-space limit as being intermediate between the BMN limit and the full sigma model: p 1/ λ, p λ 1/, p 1, BMN limit near-flat-space giant magnons. We write giant magnons for the third sector as these are the classical string solutions obeying the dispersion relation at finite p [9]. They are of size φ = p along an equator of S 5. For the case of strings in AdS CP 3,relevantforthecomparisonwithABJM s superconformal Chern Simons theory [10, 11], only the bosonic part of the near-flat-space limit has been studied [1]. Fermions are of course essential for one-loop calculations, and so we work out the complete near-flat-space Lagrangian, starting from the BMN case of [13]. We use this to calculate corrections to two-point functions, which we can compare to the exact dispersion relation 1 E = + h(λ sin p. ( The interpolating function h(λ is the only part of this relation not fixed by symmetries. At small values of λ it is h(λ = λ + O(λ 3,butatlargeλ (relevant for semiclassical strings it is instead h(λ = ( λ 1 + c + O λ. (3 The subleading term c has been the subject of some debate in the literature. In one-loop energy corrections to spinning strings [1 16, 3]andtogiantmagnons[17], there is a choice of how to regulate divergent sums over mode numbers, and using the same momentum cutoff for all modes leads to c = log /π. However,ithasbeenarguedthatitisnaturaltousea cutoff of for half of the modes (the heavy modes and this new prescription leads instead to c = 0[3, 18]. The same two prescriptions have been implemented in a recent Hamiltonian analysis [], which argues in favour of the new prescription. Our paper is in some ways the Lagrangian analysis complementary to this. The sigma model in AdS CP 3 has several novel features compared to that in AdS 5 S 5. One is that its excitations do not all have the same mass: half are light (m = 1 in our conventions and half are heavy (m = 1. Another is that the Lagrangian has interactions starting at cubic order, rather than at quartic order [19, 1, 0], greatly expanding the number (1

3 of Feynman diagrams possible. The cubic interactions always couple two light modes to one heavy mode, and therefore lead to bubble diagrams such as where we draw the heavy mode as a double line. In this diagram, the same loop momentum applies to both the heavy and the light modes, at least in the UV. This points towards the old prescription of integrating up to the same momentum for all modes. In this paper, we use primarily dimensional regularization, but for the simplest case can also obtain exactly the same results using a momentum cutoff. In addition to the term coming from c = log /π, we also find (using either regularization procedure some additional terms. These terms are present in both the BMN and near-flat-space limits, and so do not represent a problem with the consistency of the nearflat-space truncation. These extra terms differ for the various particles, vanishing in the case of corrections to the light fermion propagator. This implies that they break supersymmetry, possibly as a consequence of dimensional regularization. In fact, these extra terms come (primarily from the bubble diagram above, while the term containing c = log /π comes from the tadpole diagrams: + Each loop here contains only one mode, and so for these tadpole diagrams there is no obstruction to changing the heavy-mode cutoff to implement the new sum. This change cancels the log /π term, and thus while slightly unsatisfying (since it does nothing to the bubble diagrams could be said to lead to c = 0. In this paper, we also further the investigation of what one-loop corrections can teach us about the nature of the heavy modes, as initiated by Zarembo in [1] andcontinuedin[13]. There it was argued that the heavy modes dissolve into a multi-particle continuum, under the assumption that c = 0. In our calculation, it is clear that the decay of a heavy mode into two light modes is kinematically allowed. Outline In section,weset upthetheory weconsider,re-writesome resultsof[13]for the BMN limit in more convenient notation and take the near-flat space limit of the Lagrangian. Sections 3 5 study mass corrections to the propagator in the near-flat space limit. The basic light boson calculation is quite simple, but the light fermion case is more involved. We discuss some issues about momentum cutoffs in section before working on the heavy modes in section 5. We then turn to the full near-bmn case in section 6.Allofthesameissuesarisehere,but the calculations are a great deal more complicated and thus less transparent. We can reproduce our near-flat-space results as limits of these results. We conclude and summarize in section 7. Appendix A contains extra notation particularly about fermions. Appendix B contains manipulations to simplify L.AppendixC looks at expansions of integrals used for dimensional regularization.. The string Lagrangian and two strong-coupling limits Our starting point is the gauge-fixed Lagrangian for type IIA strings in the near-bmn limit of AdS CP 3 as derived in [13]. We begin by reviewing this derivation very briefly. 3

4 We are interested in strings moving in the quotient super-space [, 3] OSP(, 6 SO(3, 1 U(3, the bosonic part of which is AdS CP 3.Thesigmamodelisdefinedintermsofthefollowing Lie-algebra-valued flat current: A µ (σ, τ = G 1 µ G, G OSP(, 6 = A µ (0 + A(1 µ + A( µ + A(3 µ. Here, A (k is the component with eigenvalue i k under the generator of the Z automorphism,. Thesubalgebraso(3, 1 u(3 is precisely that fixed by, andthustobeomittedfrom the action. The k = 0, componentsarebosonic,andk = 1, 3componentsarefermionic.The action defined by [] isasfollows: S = g d σ Str [ hh µν A µ ( A( ν + κɛ µν A µ (1 ] A(3 ν. ( The coupling constant is g = λ/ = R /8πα,whichistakentobelarge.Inorderforlocal κ-symmetry to hold we need κ = 1, and we now choose κ = 1. 3 It is necessary to introduce some parameterization of the group. Since we are interested in strings near to the null line φ = t, andwilllaterdiscardthesetwodirectionsduringgauge fixing, it has been found to be convenient to factorize them out from the start. A suitable parameterization is [13] G = (t,φf(χ G (X. The three factors contain the light-cone directions, the fermions and the eight transverse bosonic directions. Explicitly, [ ( ( i 1 (t,φ= exp x + + a x + + i ] x G (X = G AdS G CP = 1 + i z iɣ i 1 z i / exp ( W + W + 1 yt 5 (5 F(χ = χ χ. Here, z i and t are the AdS coordinates, while y, W, W and φ are the CP 3 coordinates. All the fermionic fields are contained in the matrix χ; see appendix A for details. The target-space light-cone coordinates here are non-standard ones, defined as x + = (1 at + aφ, x = φ t, where a [0, 1] is the same constant as appears in the generalized light-cone gauge. The bosonic gauge fixing is simplified by introducing an auxiliary field conjugate to A (. This brings us to a first-order formalism in which the Weyl-invariant worldsheet metric γ µν = det hh µν enters as a set of Lagrange multipliers: [ S = g dσ Str ]. (6 A ɛαβ A α (1 A(3 β 1 ( + ( A ( γ 01 γ A( γ 00 1 By solving for this action is classically equivalent to (; for details see [13]. The gauge we adopt is the generalized light-cone gauge of [, 5], in which we set x + = τ and the density of the momentum p + to be a constant: P + = g dσ p + = (1 aj + a, 3 κ changes the sign under σ σ.

5 where and J are the conserved charges from AdS and CP 3.Notethatthisgaugerelates the length of the string worldsheet to P + and the coupling. As a result of this, the large coupling expansion will decompactify the worldsheet. The light-cone gauge also implies that the physical Hamiltonian is H lc = P = g dσ p = J. The gauge fixing breaks some of the symmetries and the set of charges that commutes with the light-cone Hamiltonian combines into SU ( U(1 [6]. What we have written thus far is exact in the sense that arbitrarily large motions of the string are allowed. We now take the BMN, or PP-wave limit, in which we focus on strings near to the null geodesic φ = t.by near wemeanoforder1/ g,whichmeansthatthislimit is to be taken simultaneously with the semiclassical limit. We implement this by scaling all fields, making the following replacements in L: x 1 g x, π 1 g π, χ 1 g χ, (7 writing x for a generic boson (whose conjugate momentum is πandχ for a fermion. At large g, thelagrangianisthenorganizedasanexpansioninthenumberoffields,whichwewrite as follows: [ S = d σ L (π, x,χ+ 1 L 3 (π, x,χ+ 1 ] g g L (π, x,χ+ O(g 3/. The result we take from [13] asourstartingpointisthelagrangianinthisform.(theonly change is that we generalize it to the gauge a 1. Note that unlike the Hamiltonian analysis of [13, 7], it is not necessary for us to perform complicated redefinitions of the fermions in order to make canonically conjugate pairs (up to some order. We can simply work with them as they stand..1. The BMN Lagrangian The analysis we want to perform in this paper is simplified by eliminating the momentum variables (introduced for the ease of gauge fixing in favour of velocities. The equations of motion for the momenta π are, schematically, L π = 0 π = 0x + 1 L 3 g π + 1 L g π +. The higher order terms on the right mean that the interaction terms L 3 and L will change in nontrivial ways. But the quadratic Lagrangian is essentially given by replacing π 0 x in that of [13] L = 1 +y y + 1 +z i z i + 1 +ω α ω α + 1 ω α + ω α 1 y ( + z 1 i 8 ω αω α + i ( ψ +a ψ+ a + ψ a + ψ a i [ + (s a α +(s α a + (s + a α (s + α ] a 1 ( ψ a ψ a + + ψ +aψ a i (s+ a α (s α a. (8 Here we have introduced worldsheet light-cone coordinates σ ± = τ ± σ, ± = 0 ± 1. We detail some changes of notation in appendix A. 5

6 We will refer to both fields and derivatives with a + subscript as being left moving, and those with as right moving. The list of fields is as follows: ψ a ± complex light fermion a = 1, (s ± a α real heavy fermion α = 3, (9 ω α complex light boson y, z i real heavy boson i = 1,, 3. The fields transform covariantly under the bosonic part of SU ( U(1, namely SU ( U(1, andwedenotethesu ( from AdS with Latin indices and the SU ( from CP 3 with Greek. Both are raised and lowered with ɛ-tensors: R a = ɛ ab R b = ɛ ab (ɛ bc R c, ɛ 1 = ɛ 3 = 1 ɛ 1 = ɛ 3 = 1 and the action of conjugation is (Ca β = C a β.thelightfieldsalsotransformundertheu(1, with ψ ± and ω α having +1charge. Either by direct inspection or through solving the quadratic equations of motion, one can derive the following Feynman propagators: ω α ω β =δβ α i i p 1, yy = p 1, z i iz j =δ ij p 1. (10 (Note that our light boson has non-standard normalization. For the fermions, ψ a ψ b = δa b D ψψ p 1, where D ψψ = i p + for the case ψ + ψ + (11 = i p, and s a α s β b = δ a b δα β D ss p 1, D ss = i p + for s + s + = 1 +, =+1, + = i p, = i / +, or, +. The cubic Lagrangian is naturally more complicated, and writing ±= ± ± and Z a b = i z i(σ i a b,itisgivenby L 3 = i 8 y(ω α + ω α + ω α ω α + i ψ a + ψ b Z a b i ψ +a ψ b + +Z a b + 1 ( ψ a + ψ b + + +ψ +a ψ b ψ +a ψ b ψ a ψ b + Z a b ɛ ab [( 3i 16 (s α a ψ b (s + α a ψ +b + i (s + α a ψ b 1 (s α a +ψ +b 1 (s α a ψ +b i (s α a +ψ b + i +(s α a ψ b 6 + i +(s + α a ψ b + 1 (s + α a +ψ +b 1 +(s + α a ψ +b ω α ( i 8 (s α a ψ b (s + α a ψ +b + ω α ( i 8 (s α a + ψ b + 1 ] 8 (s + α a + ψ +b ω α

7 + ɛ ab [ 3i 16 (s a α ψ b 3 16 (s + a α ψ b + + i (s + a α ψ b + 1 (s a α +ψ+ b + 1 (s a α ψ + b i (s a α +ψ b + i +(s a α ψ b + i +(s + a α ψ b 1 (s + a α +ψ+ b + 1 +(s + a α ψ+ b ω α ( i 8 (s a α ( i 8 (s a α ψ b 1 8 (s + a α + ψ b 1 8 (s + a α ψ b + + ω α + ψ b + ω α]. (1 The quartic BMN Lagrangian is very involved and we will not present it here... The near-flat-space limit In the near-flat-space limit we focus on those terms which are important at large p.wecan do this by taking a large worldsheet boost (in the σ direction and keeping only the leading terms under this boost. The quadratic Lagrangian is Lorentz invariant and so is unaffected, but the cubic and quartic interaction terms break this symmetry, and are thus simplified in this limit. These simplifications are the main reason for studying this limit. The bosonic fields all behave trivially under worldsheet Lorentz transformations. We have written the fermionic fields in terms of left- and right-moving components, and it is easy to see that these must scale like p ± for L to be invariant. Explicitly, the boost involves the following replacements in L: ± g 1/ ± (i.e. σ ± g ±1/ σ ± ψ ± g 1/ ψ ± and likewise s ± g 1/ s ±. The leading behaviour of L 3 is given by terms that grow like g,andwekeeponlythese terms. Likewise the leading terms in L grow like g.wewritetheresultofthesereplacements as follows: 1 g L g L 1 L NFS 3 + L NFS + O(, g and after discarding the O(1/ g terms are thus studying S = d σ [ L + L NFS 3 + L NFS ]. (1 As noted by [1], this action has no parameters at all. The idea of [6]istoperformasecondboostofthesameform: (13 ± γ 1/ ±, ψ ± γ 1/ ψ ±, s ± γ 1/ s ±, (15 leading to S = d σ [ L + γ L NFS 3 + γ L NFS ]. (16 If γ is kept arbitrary, it can be viewed as just a parameter to keep track of the orders. But if γ = 1/g (as we will assume, then this second transformation is the inverse of the first, except for the fact that we do not recover the interaction terms we discarded. So what has happened is that we are back in the original variables, but have built in the assumption that p p +. 7

8 We now write the interaction terms, starting with the cubic term. Here we simply keep those terms in (1whichgrowas g under the boost (13 (i.e. grow as fast as the right-moving momentum p : L NFS 3 = i 8 yω α ω α + i ψ a + ψ b Z a b 1 ( ψ +a ψ b + ψ a ψ b + Z a b ɛ ab [( 3i 16 (s α a ψ b + i (s + α a ψ b 1 (s α a ψ +b i (s α a +ψ b + i +(s α a ψ b ω α i 8 (s α a ψ b + ω α i 8 (s α a ] + ψ b ω α [( 3i + ɛ ab 16 (s a α ψ b + i (s + a α ψ b + 1 (s a α ψ + b i (s a α +ψ b + i +(s a α ψ b ω α i 8 (s a α ψ b +ω α i 8 (s a α + ψ b ω ]. α We note immediately a distinction from the AdS 5 S 5 case; this leading-order interaction term has both left- and right-moving fields and derivatives, rather than consisting only of rightmoving objects (ψ, s and asin[1]. This is not unexpected from the form of the boost (13; in order for a term with two right-moving fermions to scale as g 1/,itmusthaveeither no derivatives, or both + and.alternatively,itcancontainone with one left-moving and one right-moving fermions. This difference from AdS 5 S 5 will have an important consequence. When we come to drawing Feynman diagrams in the following section, right-moving fields and derivatives will contribute powers of right-moving momenta p, k to the numerator, see (11. But the presence of left-moving objects will introduce also k +,andafactork + k = k will make any loop integral more divergent. Because of this, we will have many quadratically divergent integrals to deal with, while in the AdS 5 S 5 case [7], all of the analogous integrals were finite. For the quartic terms, we write L NFS = L BB + L BF + L FF,andtheall-bosontermisthe simplest: (17 L BB = 1 8 [ z i z i + ( y + ω α ω α ] ( z j z j y 1 ωβ ω β. (18 For the term mixing bosons and fermions (writing ψψ= ψ a ψ a and ωω= ω α ω α wehave L BF = i 8 (s α a (s a α (z i y 1 ωω + i 16 (s α a (s a γ ω α ω γ i 3 ψ ψ ωω + i 8 (ψ ψ ωω + ψ ψ ωω ψ ψ ω ω ψ ψ ωω i 8 ψ aψ b Za c Z c b y[(s aα ψ a ω α + (s aα ψ a ω α ] 1 16 y[(s aα ψ a ωα + (s aα ψ a ω α ] + 1 y(s aα (s α c Zca + i (s α a ψ c ω α Z ca + i (s aα ψ b ωα Z a b + i 8 (s α a ψ cω α Z ca + i 8 (s aα ψ b ωα Z a b i 8 y ψ ψ + i 8 (s aα (s α b Za d Z db 8

9 1 8 ( i ψ + ψ + ψ + ψ + ψ ψ + [( y + ( z i + ω α ω α ]. At this order, the powers of g under the boost (13 arethesameasintheads 5 S 5 case [1]: terms with two right-moving fermions and scale correctly as g.onthelastlinewedoalso have some terms with left-moving objects. It is possible to remove the terms with left-moving factors from L BF by nonlinear redefinitions of the fermions, in a way which does not change final results. Similar terms were sometimes present in the AdS 5 S 5 case, in certain parameterizations of the coset 5.We write the last term L FF in a form obtained by performing such a redefinition, since the result is much more compact: L FF = 3 8 (ψ ψ 1 ψ ψ ( + ψ ψ + ψ + ψ 3 8 +(ψ ψ (ψ ψ 1 (s β a (s d β (s γ d (s a γ 1 3 (s a α (s α c ψ aψ c + 1 8[ + (s a α (s α c ψ a ψ c + 1 +(s a α (s α c ψ a ψ c + 1 +(s a α (s α c ψ aψ c + 1 (s a α +(s α c ψ a ψ c + (s a α +(s α c ψ a ψ c + 1 (s a α +(s α c ψ aψ c ( (s a α (s α c ψ a + ψ c + 1 (s a α (s α c +ψ a ψ c + 3 (s a α (s α c ψ a + ψ c + 3 (s a α (s α c +ψ a ψ c ]. (0 We discuss details of the redefinitions leading to this form in appendix B. Note however that this is only possible for the highest order interactions. A similar procedure for the cubic interaction L 3 is not allowed. In the following three sections we will use the above Lagrangian to calculate two-point functions on the worldsheet. We return to the full BMN theory in section Corrections to the light propagators For one light mode, the exact dispersion relation is ( 1 E = + h(λ sin p chain, where the interpolating function is h(λ = λ/ + c + O(1/ λ.themomentumhereisthe one canonically normalized for the spin chain, and it is this which scales as p chain λ 1/ in the near-flat-space limit, (1. Taking this into account, we can expand E in λ as follows: E = λp chain [ λcp chain λp chain ( 1 + O λ, = p ( cp + 1 p 1 λ 6λ ] + O ( 1 λ where p 1 = λ/ p chain λ 1/. On the second line we write it in terms of p 1 normalized to match p 0 = E. Thisp 1 is the worldsheet momentum in our uniform light-cone gauge, and its relationship to p chain could in principle be deduced from the gauge but we are content just to read it off here. In terms of the worldsheet momenta, the near-flat-space limit assumes that p p + (or more strictly p λ 1/ and p + λ 1/, and thus we can write (p 1 = 1 (p + p = 5 This is presented in [8]. For similar redefinitions, see [9, 30]. 9

10 1 p + O(1. Doing this in the correction terms, we arrive at p 0 p 1 = 1 ( cp + p ( 1 + O λ. (1 λ 96λ We can now read this as giving a mass correction: p = m + δm,withm = 1,andδm in brackets. The second term there is the same two-loop correction as was calculated by [7] in the AdS 5 S 5 case 6.Weaimtocomputethefirst term of the correction, containing c. The result of [, 17] whichwewouldliketomatchisthat c = log π. Note that in expansion (1 themasssquaredanditsone-andtwo-loopcorrectionsareallof the same order in 1/ λ.thisisapeculiarityofthenear-flat-spacelimit.inthebmncase, p chain λ 1/ and thus p 1 λ 0,restoringthenatural-lookingseparationoforders. Our calculation of the mass shift comes from writing a familiar geometric series involving amputated diagrams we call A: 7 [ ] G ω (p = i p 1 γ A i n p 1 n=0 i = p ( 1 + δm, with δm = iγa. ( We will consider corrections not only to the light boson propagator ωω as written here, but also (in section 3. toallfourofthelightfermionpropagators(11. When correcting for instance ψ + ψ +,thelinesweamputatewillnolongerhavefixednumeratoribut instead the various D +s and D r,so(normalizingtothebosoniccaseweabsorballofthese into A. There are two topologies of diagrams we must compute, both of which contain divergences, and it is simplest to employ dimensional regularization. We show quite a lot of detail partly in order to attach later discussions about other possible cutoffs, in section Bubble diagrams for ωω The loop in each bubble diagram (for a light mode always contains one light and one heavy mode. We begin with the simplest case, in which these are both bosons: ω α (k A B = ω α (p y(q d k = 1 (π, q µ = p µ k µ ( B k 1 (q 1, B = p + k. 8 The factor B includes both factors from the propagators (10 andfromthevertex(17. The decorations ± pull down i k ± from the momentum flowing into the vertex on that line. 6 Of course, the conventional factors of and π in ( are different. 7 Recall that the interaction terms in (16are γ L 3 + γ L,withγ = 1/g = /λ. 10

11 where To treat this integral, we introduce a Feynman parameter x as follows: d k 1 B A B = dx (π [ ( ( 0 x k 1 + (1 x (p k 1 ] 1 d l B = dx (π [ l (x ], 0 l µ = k µ (1 xp µ, (x = 1 x x + 1. This effective mass squared (x assumes p = 1,i.e.thatweareveryneartothepoleofthe original propagator. In terms of l µ, (x B = p 3 + (x l p l. 3 We rotate the l 0 contour to the imaginary axis, and in the resulting Euclidean space, the terms l n will vanish8.theradialintegralthengives1/ (x, andfinallyweobtain A B = i 3π p δm = iγa B = 1 16π γ p. (3 Next we treat the case where the heavy and light modes in the loop are both fermions: ψ a (k A F (p = ω α (p s a α(q d k B = (π (k 1 (q 1 Here, B has the same meaning 9,butismuchmorecomplicatedsince,expanding(L 3,there are in all 5 terms, and the propagators (11givevariouspowersofthemomenta.Afterwriting B in terms of l s + ls, terms with s s still vanish, but there are now also divergent terms, with one or two powers of l + l = l in the numerator. As we noted below (17, the fact that left-moving momenta k + and q + appear in B, and produce divergences, is a crucial qualitative difference from the AdS 5 S 5 case. We will treat the resulting divergent integrals using dimensional regularization. Here is the generic integral, including the case n forlateruse: 10 In s ( = d d l (l + l s (π d [l ], d = ɛ n = i π d/ (π d Ɣ ( dl ( 1s L s+d 1 d 0 [L + ] n = i ( ( 1s 1 n s d/ (π d/ Ɣ ( Ɣ ( ( n s d Ɣ s + d. (5 d Ɣ(n 8 We define l 0 = i L 0, l 1 = L 1 and L µ = L (sin θ,cos θ.thenthedenominatoris[l (x],andinthe numerator, any nonzero power l n will lead to π 0 dθ e inθ = 0. 9 Somewhat arbitrarily, the factors from the fermion loop (i.e. re-ordering fields and the a in the loop are part of the prefactor, not B. 10 Here of course π d/ /Ɣ( d is the volume of a unit (d 1-sphere, and γ E = ( 11

12 Expanding in ɛ for n =, I 0 ( = i 1 π I 1 ( = i ( 1 π ɛ + γ E log π log + 1 I ( = i π log + ( 1 ɛ + γ E log π log + 1 log + (See appendix C for comments on these expansions. The coefficients of these integrals are as follows, after using p + p = 1 : B 00 = 3x 6x 3 8x + 0x 8 p 3 B 11 = 19x x p 16 B = 1 p. + O(ɛ + O(ɛ. After integrating on x, thelogdivergences = 1willcancelagainstthequadraticdivergence s =. Then the final result for the fermion bubble is A F = + i 16π p δm = 1 8π γ p. (6 3.. Tadpole diagrams for ωω The tadpoles made with two 3-vertices can be seen to vanish by looking at L 3,(17: =0. But the ones constructed with one -vertex are more interesting. We use (19 to calculate the first two diagrams. The only nonzero terms are those where both decorations act on the external legs, which for ω running in the loop gives A T 1 = ω α (p ω β (k = i 1 d d= ɛ 3 k i p (π k 1 = p 8 Is=0 n=1( = 1. For a heavy mode running in the loop, we obtain A T = y(k or z i (k ω α (p = i 1 8 (3 1 p d k i (π k 1 = p I0 1(1. There are also diagrams of this topology with a light fermion running in the loop. The terms in (19 whichcontributearethoseonthelastline,andtheygivethefollowingtwointegrals 1 (7

13 (including the only quadratically divergent tadpole: A T 3 = ω α (p ψ b (k = i i 8 p d k k + k (π k 1 + i 1 8 p d k i/ (π k 1 = p I1 1( 1 p I0 1( 1. The diagrams with a heavy fermion in the loop can be seen to vanish from (19. All of the tadpole integrals are n = 1casesof(5above,with = 1 or 1. The expansions needed are In=1 s=0 ( = i ( 1 π ɛ + γ E log π log+ log + O(ɛ I s=1 n=1 ( = i π ( ɛ + γ E log π log+ log + O(ɛ. Adding the bosonic and fermionic terms, the divergences cancel leaving a total tadpole contribution of A T = i log p δm T π = 1 γ log p π. (9 ( Total for the light boson Adding up all of these terms, the total amplitude is A B + A F + A T = i p log π + 3i 3π p. As amasscorrectionthisis δm = log γ π p 3 16π γ p. (30 Comparing this to (1, we see that the term with log (from the tadpoles is perfect so long as γ = /λ.theextratermisperhapsunexpected,andwecommentonthisintheconclusions. 3.. Corrections for the light fermion ψψ We now turn to the case of corrections to the light fermion propagators (11. The two bubble diagrams we need are A ψ = = ω α (k + ψ a (p ψ a (p s a α(q d k B (π (k 1 (q 1. ψ b (k Z a b (q The integrals are of course the same as ( above, with different coefficients B = s,s ls + ls B ss.asnotedabove,b now includes the numerators D ψψ of the two amputated propagators (as well as the vertices and internal propagators. We must calculate the coefficients separately for each of the four cases ψ + ψ +, ψ ψ, ψ ψ and ψ + ψ +,andtheresults are all different until the point at which we use p = 1 to get the shift near to the pole of the 13

14 unperturbed propagator. Then they all agree, and we obtain B 00 = 1 18 (6 + x 3x 8x 3 + x + x 5 p B 11 = 1 16 ( 1 7x + x + 6x 3 p B = 1 8 (6x 1p. Then using the expansions of the integrals in (5, we find δm = 1 ( 6π γ ɛ + 1γ E 6 log + 1 log π p. (31 There is an important observation we can make at this stage. In this expansion in 1/ɛ, there is no order 1 term with a rational coefficient (inside the bracket 11.Sincenosuchtermswill appear in any of the tadpole integrals (8, we conclude that there will be no extra term like that seen in (30. A further difference from the ω case above is that the total of the bubble diagrams is not finite. Next we turn to the tadpole diagrams, using (19 and(0. We must compute these for each of the four cases ψ + ψ +, ψ ψ, ψ ψ and ψ + ψ + separately, but again the final coefficients work out to be the same 1.Thosewithalightmodeintheloopare ω β (k ψ b (k A L = + ψ a (p ψ a (p [ 9 = p 16 I0 1( I1 1( 1 ]. With a heavy mode in the loop, we find y(k or z i (k s b β (k A H = + ψ a (p ψ a (p [ = p 3 I0 1(1 + 3 ] 8 I1 1(1. After using the expansions of the integrals in (8, the total is as follows: δm = γ 1 ( 6π ɛ + γ E + log log p. Happily the divergent 1/ɛ term cancels that of (31. Adding the bubble and tadpole contributions gives us that the full mass shift, which for all the ψψ cases is (3 δm = γ log π p. (33 This is exactly what we would expect from c = log in (1. Unlike the bosonic case (30, π there are no extra terms. 11 Such terms do occur in the individual bubble diagrams drawn, which contribute ± 3 6π p, but these cancel out of the total. 1 Note also that whether we use (B.1orthesimplified(0makesnodifferencetothesecoefficients. 1

15 . A detour via some delicate cutoffs In this section, we look at some effects of employing explicit energy or momentum cutoffs instead of dimensional regularization. The main reasons for doing so are to try to better understand the extra terms in the mass correction (30, and to explore the prescription dependence of c by looking at old and new cutoffs (in section.3. We will also get extra cross-checks between the bubble (L 3 andthe tadpole (L calculations. Most of this section will be about only the light boson case. We comment briefly on other cases in section...1. Using hard cutoffs The first observation is that if we work with hard momentum cutoffs k <,insteadof dimensional regularization, then the cancellation of infinities is different. Each of A F and A T contains a logarithmic and a quadratic divergence, and above we cancelled these within them, since all are 1/ɛ terms when working in d = ɛ dimensions. However in terms of a hard cutoff, instead these terms cancel between them. This cancellation provides a useful cross-check between the bubble (L 3 andthetadpole(l calculations. The only quadratically divergent terms are the (l + l part of the fermionic bubble integral 13 1 A s= i F = B dx π 0 0 = i [ p π 7 6 and the fermionic tadpole integral L 5 dl [L + (x] log + log O ( ] 1 A T 3 = p i dk K3 π 0 K + 1 p i dk K π 0 K + 1 = i [ p π 1 ] log log + i p [ log + log]+ O π The log divergences similarly cancel. They are 1 A s=1 L 3 F = dxb 11 dl 0 0 [L + (x] = i [ p 7 17 log log ] π A T 1 = p dk K 8 0 K + 1 = i [ p 1 π log + 1 A T = p i dk K π 0 K + 1 = i [ ] p 1 log + π and the finite terms are of course unchanged: A s=0 F = dx B 00 (x = i p [3 log ] π (3 ( 1. (35 ] log + 13 We note in passing the following issue [31]. In four dimensions, the shift from integrating over k µ with k < to integrating over l µ with l < would change the value of the -divergent integral, and thus not be allowed. (The finite and log integrals are unchanged. This is however not a problem in two dimensions, nor is it a problem in dimensional regularization, in any number of dimensions. (36 15

16 A B = i p 1 π 8. Adding these up, we obtain the same finite result as before (30: δm = log γ π p 3 16π γ p. Note in particular that we have exactly the same extra term as we had using dimensional regularization. We now turn to looking for other explanations of this term, for this case... High-precision cutoffs Note that the finite part after the cancellation of quadratic divergences (between bubble and tadpole is sensitive to very small changes in the cutoff. In this section, we investigate the effects of such small changes, replacing with one of these: light = ι, light modes heavy = ϰ, heavy modes. (37 For the next page or so we leave ι and ϰ arbitrary. For the tadpole we can simply write down the result, since the only divergent integral (35involvesonlyalightmode.Theeffectofchangingfrom light is 1 i: ι. (38 For the bubble s s = term(3, we expect some change involving both heavy and light modes, but it is not obvious what mixture of ι and κ should appear. In fact, it is not clear that whether there will be any way of implementing both cutoffs (37 inthesameloopintegral. But we can make an attempt by introducing Pauli Villars regulators [3], replacing in the heavy and light propagators the following: 1 k 1 1 q 1 PV PV 1 1 k 1 k light 1 q 1 1 q. heavy The propagators are unchanged for k,butwhenk they die like 1/k.Defining ageneralizationoftheintegralin(toallowtwoarbitrarymasses d k B J(m, M = ( (π k m (q M the effect of imposing the Pauli Villars regulators is to replace J( 1, 1 with four terms: J ( 1, 1 PV J ( 1, 1 J( light, 1 J ( 1, heavy + J( light, heavy. Each of these terms can be treated exactly as we did before 1,simplyre-usingtheintegrals (5withtheappropriateeffectivemasses (x, m, M.Thusthes = partofeachisgiven by J s= (m, M = 1 0 dx d ɛ l (l + l (π [l (x, m, M], 1 This means that we are regulating each of the four terms with dimensional regularization. We can instead use another hard cutoff for all the terms, and as long as heavy, light,theresultisidentical.(if =, then (0 becomes (ι + ϰ instead. 16

17 where l µ = k µ (1 xp µ = xm + (1 xm + x(1 xp. Note that l µ is defined in the same way in all four terms, so that the identification of the term (l + l in B is the same here as before. (It comes with the same coefficient B. The final result for the s = integral(atp = 1 isthen ( 1 JPV s=, 1 = 7 ( log log ι + ϰ ( 1 + O. (39 Thus, the effect on the bubble diagram of turning on ι, ϰ can be summarized as ι + ϰ. (0 This makes some sense, in that if you change both cutoffs by the same amount ι = κ then this becomes an overall change of. What you cannot predict without calculation is that the effect of ι ϰ should not be some other mixture. The total effect on A of both (38and(0isthenasfollows.Using(3and(35above, we have A A + i ( p 8π ι ι+ϰ, or, in terms of the mass shift, δm = ia δm + (ϰ ι γ p 8π. It remains to argue what tiny shifts ι, ϰ we should use. Ideally, one would like to impose exactly the same cutoff on the physical energy of all the modes. Unlike the worldsheet momentum k 1,orworsetheEuclideanenergyK 0,theenergyis aphysical,gauge-invariantquantity. One argument is this: in the original Lorentzian momentum, E = k0 = k 1 + m on the mass shell. Imposing E < thus reads k1 < m.ifweimposethismomentumcutoff on both of the Euclidean directions, then we are led to K < m.forthelightmodes, this means ι = 1,whilefortheheavymodesϰ = 1. This gives δm δm + 3 p γ 3 π. (1 Sadly this is too small to cancel the extra term in (30, by a factor of. The attentive reader will by this point have smelled something a little fishy. Not only have we made up a fairly strange cutoff, (37, we have also mixed the results of using this in the Pauli Villars bubble integral (39 withthatofusingitinatadpoleintegralcalculatedwitha hard momentum cutoff (35. The reason for doing so is that Pauli Villars is not strong enough to regulate the tadpole integral, and we do not know of any regulator both strong enough and potentially sensitive to independent variations of heavy and light.nevertheless,thecomplaint is well founded; note that one of the finite terms from Pauli Villars differs from that given by the hard cutoff compare (39to(3. What we believe we have shown is that justifiable alterations to the cutoffs used alter δm by terms of the form γ p 1 /π (rational, which is the same shape as the extra term in (30. And also that these do not produce terms of the form p log /π. Thatthecancellationisnot perfect is perhaps a limitation of our implementation of this physical cutoff. 17

18 .3. Old and new prescriptions We have discussed above very small changes to the cutoffs, and now we turn to a large one. First of all, recall how c = log(/π was found in [, 17]. There, the energy corrections coming from light modes or heavy modes alone diverge as follows: δe log heavy log light. If we use the same cutoff for both of them, they cancel leaving a factor of log. This we call the old sum [33, 17] whichiswhatwasusedin[1 16, ]. If however we use for heavy and for light, then they cancel precisely, leaving c = 0. This is called the new sum 15,and was originally suggested by [3], see also [33, 17, ]. What we are doing above is equivalent to the old sum; we treat all loop momenta k alike. In order to implement the new sum, we should change the heavy tadpole integral (36 as follows: A T = y(k or z i (k ω α (p new ÃT = p i π Λ 0 dk K K 1 = A T i log p π. ( This change precisely cancels the final finite tadpole total, (9, leaving A T = 0. This can perhaps be viewed as taking us to c = 0, as expected for the new sum. To make this change in dimensional regularization, we can write the integral (7 intermsof k = k/, and then treat k in the same way as k for the light mode integrals, obtaining the same effect: I1 0 (1 = d ( k 1 new 1 I 1 d k 1 (π k 1 = 1 (π k 1 = I1 0 (1 + i { log 1 log 1 } π A T = p I0 1 (1 new A T i log p π. However, this tadpole diagram is not the only place in which the heavy modes play a role; each bubble diagram also has one heavy propagator. But there is no obvious way to impose a very different cutoff compared to the light mode in the same bubble. (Attempting to use the Pauli Villars procedure above with heavy = + does not work; the most divergent term is then I s= 3 log, which no longer cancels that from the tadpole. Thus we see no 3 way to implement something like the new sum for all terms, tadpole and bubble. This argument is in some sense the inverse of that made by [], from much the same data. Their cubic interaction H 3 is essentially our L 3 written in momentum space, and there is of course a delta function conserving momentum at the vertex. If both light modes (say ω and ω carrymomentum, thentheheavymode(y cancarrymomentumupto. It is our understanding that this observation is the heart of that paper s argument in favour of the new prescription. While instead we use the same vertex to draw the bubble diagram, in which it is difficult to adjust the heavy and light cutoffs independently, leading us to the old prescription. 15 The new sum is very simple from the algebraic curve perspective, and thus sometimes goes by this name. Likewise the old sum is a cutoff on the worldsheet mode number, which is simple to implement in that formalism. But both sums can be implemented using any technology. 18

19 .. Modes other than the light boson So far we have discussed in section only the ω particle. Let us first comment on how the issues of section.3 translate to other cases. For the heavy bosons which we treat in the following section, the observation that the new sum sets the tadpoles to zero carries over trivially; the calculation of the tadpole contribution is identical. However, the bubble diagram contains two light modes; thus our objection to the new sum does not hold for these. For the light fermion ψ, thetadpole(3 alsocontainsthe -divergent integral I1 1 (1, for which I1 1 new (1 I1 1 ( 1 = I1 1 (1 + i log 1 π. This change is identical to that for I1 0 (1 above. Then it is easy to see that while we do not create a divergence, the finite change (from trying to go to the new sum is not so simple in this case, i.e. it does not cancel the log term in (33. The fact that we cannot cancel the log like this for ψ is possibly related to another issue. If we attempt to do the whole calculation with a momentum cutoff (generalizing section.1, then we do not get a finite result. The reason for this is not entirely clear to us, but let us observe here that the cancellations of 1/ɛ terms which make the result using dimensional regularization finite are highly nontrivial see (33, and (7below. This is true also for the heavy bosons of the function section, where for instance it is clear that the bubble A y is finite (, but the tadpoles A T are as for the ω case, and thus divergent ( Correction to the heavy boson propagators For a heavy mode, we should start with the following dispersion relation: E heavy = h(λ sin p ( chain pchain = E light. This is simply the energy of two superimposed light modes, written in terms of their total momentum. This relation was confirmed to hold at one loop in the giant magnon regime p chain 1in[17]. 16 Expanding E in 1/ λ exactly as for the light mode case, and in particular using the same normalization p 1 = λ/ p chain,weobtain ( cp p 0 p 1 = 1 + p ( 1 + O λ. (3 λ 38λ The mass correction δm is again the term in brackets. Its first term, which we aim to compute here, should be identical to that for the light modes. The calculation of course uses a similar geometric series to (, although this time δm = i γ A,sincey and z i (unlike ω α arecanonicallynormalized. After working out the corrections for fields y and z i we discuss the issue of their stability in section This giant magnon, in an RP subspace, is a nonlinear superposition of two elementary magnons, as was shown by [3]. When discussing giant magnons it is often convenient to write E = 1 + 8λ sin(p /, wherep is the momentum of one of the constituent magnons, not the total. (See for instance section.3 of [35]. 19

20 5.1. Diagrams correcting yy Recall that for the light boson, the bosonic tadpoles came from (18 and the fermionic ones from the last line of (19, which is L tad = 1 8 [ z i z i + ( y + ω α ω α ](i ψ b + ψ b + ψ bψ+ b + ψ +bψ b. Both of these terms share the same first factor, and by looking at this factor it is clear that we can re-use exactly the same diagrams when the external particle is y. Theonlychangeis afactorofbecausey is real, while ω α was complex, but since we now have δm = i γ A (without a, the correction δm is unchanged: A T = ω α or ψ a + y = i log p π y or z i y =AT δm T = 1 log γp π the same as (9. There is only one bubble diagram to draw, which has two ω α particles in the loop. The contribution is as follows: A y = =i = 1 8 y(p 1 0 d k (π = 1 i 8 p π dx ω α (k ω α (q 1 0 i i (k 1 (q 1, now with q µ = k µ p µ ( k + q 8 d l (l + (1 xp (π [ l (x ] (1 x dx (x = i 8π p. ( where (x = x x + 1 = 1 (x and (as before l = k (1 xp. Thus the total mass shift for the y particle is δm = 1 γ log p π 1 8π γ p. As for the light boson ω in (30, there is an extra term not expected from the dispersion relation (3. This now comes from the only bubble diagram possible, rather than the addition of a bosonic and a fermionic bubble, and thus it is difficult to imagine a cancellation of this term. 5.. Diagrams correcting z i z j The tadpole corrections are completely identical to those for the yy case above, by the same argument used there. 0

21 The only bubble diagram has ψ fields in the loop: ψ a (k A z = z i (p ψ b (q Since the terms in L 3 are written with Zb a instead of z i, it is easiest to work in terms of these, and so we write A z = A and calculate the latter. Here is the integral: A = i d k B (π (k 1 (q 1 1 d l B 00 + l + l B 11 + (l + l B = dx, = x x + 1 (π [l (x], 0 where as before B = s,s ls + ls B ss,andweusethesame (x as for the yy case above. After using p + p 1, the coefficients needed are B 00 = 1 8 (x 1 (3x 3x p B 11 = x + x 3 B = p. p Expanding, using (5, once again the logarithmic and quadratic divergences cancels within the bubble term. The final result is ( A z = i p. 8π The total mass shift for the z particles is thus δm = 1 γ log p π + 1 π γ p Breaking of mass coincidence The heavy modes have m = 1, precisely twice the light modes m = 1.Andthequantum numbers work out that there is a pair of light modes which carries all the same indices as each heavy mode 17.Onecanaskhowthisgetsmodifiedatoneloop,andwebeginwithsomerather simple observations as follows. (i By expanding the dispersion relation we had that, for both heavy and light modes, δm = c p λ + two loops. This implies m light = δm and m heavy = 1 δm.usingc = log /π 0.11 < 0, we conclude that after these corrections the heavy mode is more massive than apairoflightmodes.thus,itiskinematicallyallowedtodecayintotwolightmodes. 17 As can be seen from the interaction terms in (17. The decomposition of course matches that used in the algebraic curve for the construction of off-shell frequencies [36, 33, 17]. 1

22 (ii By direct calculation, we find in addition to this some other terms. These are not large enough to alter the conclusion. Here are the numbers after including these terms: and for heavy modes, m ω = 0.3 γ p m ψ = 0.11 γ p m y = 0.13 γ p m z = γ p. The decays y ω + ω and z ψ + ψ are thus still allowed. (iii If c = 0, then we expect that δm = 0, and so at this order the coincidence of masses remains unbroken. This issue was investigated in a more subtle way by Zarembo in [1], and this was continued in [13]. The idea is this: instead of working exactly at the pole of the original propagator, p = 1foraheavymode,wecanmakeanexpansioninp 1andlookatthe effect of the second term. If we assume that p < 1, then the following integral is real: 1 (1 x dx 1 0 x(1 = xp p 1 p arcsin(p p 3 = π 1 p + 8(1 p + O(1 p 3/. This integral is the generalization away from p = 1ofthatappearinginthe yy bubble (. It comes with a factor i /π in A and a further factor i γ in δm in all a minus sign. Thus the term here contributes to δm < 0, ensuring p < 1self-consistently.Inourfullcalculation there is also the tadpole contribution (proportional to log, but this too is negative, as noted above. The conclusion is that even including the expansion in p 1, the mass corrections are all real, and so the pole of the corrected propagator G y (p remains on the real axis. While this integral diverges for real p > 1, the expansion given does hold for complex p.onemightwonderwhether,ifsomethingcancelledtheleadingterm,thesecondterm might amount to a complex mass correction. One way to investigate this is as follows. Define some coefficients B, C, D, F like p = 1 + δm, δm = γ [B + C 1 p + D(1 p + O(1 p 3/ ] + γ F + O(γ 3. Then B > 0isthesimplecasealreadydiscussed.IfB = 0, then C > 0lookslikethatitwill guarantee a negative mass shift, thus keeping 1 p real, while C < 0isnotobvious.But we can solve this equation explicitly for p,andexpandinginγ,weobtain p = 1 Bγ ± BC γ 3/ + ( C BD + F γ + O(γ 5/. Now it seems clear what happens if B = 0; regardless of the sign of C the correction is still a real number, but it enters only at two loops. 6. Corrections in the BMN Limit In this section we repeat our calculations above using the BMN Lagrangian of [13]. The physical reason for doing so is that we want to check that the near-flat-space limit and the diagrammatic calculation of δm commute.

23 When taking the near-flat-space limit of our BMN results, in fact we recover not only the term arising from c = log /π in the dispersion relation, but also the extra terms seen above (for each particular mode. This indicates that these extra terms cannot be artefacts of the simpler limit. In the BMN calculation, we also find some constant terms which are not visible in the near-flat-space limit, but which also break supersymmetry. The approach and the diagrams needed are identical to those for the near-flat-space limit, but the number of terms involved is much greater. Consequently, we show much less detail here. But this also means that the cancellation of divergences is more delicate than it was, and this provides further checks on our work The dispersion relation We can expand the dispersion relation (inthesamewayasforthenear-flat-space case(1. The normalization of the worldsheet momentum is exactly the same p chain λ/ = p1,asthis is a statement only about the gauge we are in. But now we have p chain λ 1/ from (1, which changes the result to E = 1 + p 1 + ( 1 cp 1 + O. (5 λ λ This reads p 0 p 1 = 1 + δm,andthusweexpecttosee δm = λ cp 1. This time, the correction term δm is of order 1/ λ,whilep 0 and p 1 are of order 1. (The two-loop term we wrote in (1 isnowoforder1/λ. For a heavy mode, E heavy (p = E(p/, weagainobtainp 0 p 1 = 1 + δm with the same δm at this order. 6.. Results for light modes For the light bosonic propagator we find ωω : δm /γ = 9 16πɛ 3 π p 1 + from bubble diagrams 9 16πɛ log π p 1 + from tadpoles = 5 96π 3 π p 1 log π p 1 in total. (6 The dots for bubble and tadpole contributions are constant (and non-divergent terms. If we now write p 1 = 1 (p + p and perform the worldsheet boost, then we end up with the result obtained starting from the near-flat-space Lagrangian. Thus the limit commutes with the calculation. However, for the BMN case, we see that there is one additional nontrivial 1/ɛ cancellation between the bubbles and the tadpoles which is invisible in the near-flat-space calculation (where all 1/ɛ cancelled within each diagram. This provides a further consistency check. For the light fermionic propagator we have ψ ψ + : δm /γ = 7 6πɛ + 1 8π 7 6πɛ 1 8π ( 1 ɛ 1γ E + 13 log ( 1 ɛ 1γ E + 1 log 1 log π 1 log π p 1 + bubbles p 1 + tadpoles. 3