Pure spinor indications of ultraviolet finiteness in D=4 maximal supergravity
aa r X i v : . [ h e p - t h ] A ug Gothenburg preprintAugust, 2015
Pure spinor indications of ultraviolet finitenessin D = 4 maximal supergravity Anna Karlsson
Fundamental PhysicsChalmers University of TechnologySE 412 96 Gothenburg, Sweden
Abstract
The ultraviolet divergences of amplitude diagrams in maximal supergravityare characterised by a first possible divergence at seven loops for the 4-pointamplitude (logarithmic) and, in its absence, at eight loops. We revisit thepure spinor superfield theory results of [arXiv:1412.5983], stating the ab-sence of the divergence originating in the 4-point 7-loop amplitude as wellas those of more than seven loops. The analysis, performed in terms of theone-particle irreducible loop structures giving rise to the divergences, is ex-tended, especially with respect to the limits on the dimension for finiteness.The results correspond to those mentioned, known from other approaches,indicating an ultraviolet finiteness of maximal supergravity in D = 4 . email: [email protected] . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ Contents
The ultraviolet divergences of the amplitude diagrams in maximal supergravity[1–4] have long been subject to investigations [5–33]. However, while a divergenttheory in D = 4 initially seemed unavoidable, with increasing UV divergencesthe higher the number of loops present, the explicit 4-graviton calculations of[13, 15, 19, 26] showed a better UV behaviour at four loops than expected. Theresults of [21, 23–25] then showed a first possible divergence at L = 7 for the 4-point diagram: a logarithmic divergence also discussed in [30]. Interestingly, theresults of [23] also stated that, if the 4-point 7-loop would be absent, the 5-point7-loop would be characterised by a slightly better UV behaviour, i.e. not divergentin D = 4 at L = 7 .This is all the more interesting in the light of the recent pure spinor investi-gations of [27, 28]. Unlike other pure spinor approaches [11, 12, 18, 24, 25], theseare performed in a field theory setting, based on the maximal supergravity ac-tion [34, 35] respecting maximal supersymmetry, and where the maximal super-symmetry is kept an inherent property throughout the investigations. Importantly,1 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ the results of [28] showed a cut-off of the possible number of loops in one-particleirreducible loop structures, and because the UV divergences only are caused bysuch substructures, and defined by the worst separate UV behaviour thereof, [28]effectively stated that the UV divergences only depend on the number of loopspresent up to L = 6 for the 4-point amplitude, and L = 7 for the 5-point ampli-tude. By the previous results stated, the possible 4-point 7-loop logarithmic diver-gence would be avoided, and by [23] effectively all UV divergences in D = 4 ,a point not explicitly made in [28]. This indicates a scenario with maximal su-pergravity UV finite in D = 4 , relevant for further investigations, but in partsupported by arguments in e.g. [29].In this article we revisit the arguments in [28] and have a further look atthe UV divergences. For example, the cut-off of the loop behaviour is better inter-preted as a product of the integration over loop momenta, in combination with reg-ularisation properties. The latter which in turn can be used to give an upper limitof the UV behaviour with limits on the dimension, for finiteness, correspondingto those of [21, 23–25], some of which were deduced through U-duality proper-ties. Perhaps the corresponding processes can give some insight into what the purespinor formalism cancellations ought to correspond to in other approaches. It atleast seems to concern U-duality properties in combination with an insensitivityto certain required transformations, in terms of the loop integrations.The article is organised as follows. To begin with, a brief presentation of thepure spinor formalism and the concepts connected to the amplitude diagrams for-mulated in a field theory setting are given. For an extensive review of the first,see [36], and in terms of the latter, we refer to [27, 28] or, in a brief format, [37].We then proceed with a more extensive analysis of the restrictions on the one-particle irreducible loops structures observed in [28]. Especially, the effective op-erators can be limited further, as specified in (3.6), which brings about furtherlimits on the UV divergences: a simple power-counting of the momenta presentyield the correct UV divergences for L ≤ in (4.5). This estimate, assumingan equal division of the momenta in the loop structure, clearly is a bit naive for L > , where the loop configurations begin to play an important role. This is pos-sible to take into account through further observations of the loop regularisationproperties in (4.6): a given loop, be it a part of a diagram or not, does not divergeworse when considered part of a larger loop structure than when figuring on itsown, provided all contributing operator configurations are considered . In this way In our original analysis, also present in [28], we mistakenly ignored the possible contributionof nonzero modes of N mn in the b -ghost. It is possible that contributions from these nonzero modeswill modify our conclusions, and we are currently investigating this. We thank Nathan Berkovitsfor pointing this out. Some equivalences of momenta, part of the effective operators in (3.6), effectively acting out . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ the momenta actually contributing to the UV divergences can be narrowed downto present results in equivalence with [13, 15, 26] for ≤ L ≤ and worst casescenarios in L ≤ corresponding to those of [21, 23–25], which in combinationwith the limits on the one-particle irreducible loop structures in [28] indicate UVfiniteness for maximal supergravity in D = 4 . In [27, 28] a field theory formulation of amplitude diagrams in the pure spinorformalism was set down, benefitting from the maximal supersymmetry respectedby the action [34, 35]: S SUGRA = 1 κ Z [d Z ] (cid:18) ψQψ + 16 ( λγ ab λ ) (cid:16) − T ψ (cid:17) ψR a ψR b ψ (cid:19) R a = η − (¯ λγ ab ¯ λ ) ∂ b − η − L ab,cd (1) ( λγ bcd D )++ 2 η − L ab,cd,ef (2) h ( λγ bcdei λ ) η fj − η f [ b ( λγ cde ] ij λ ) i N ij T =8 η − (¯ λγ ab ¯ λ )(¯ λr )( rr ) N ab (2.1)The pure spinor formulation originates in the linearised D = 11 supergravitytheory, which in flat superspace is possible to formulate in terms of a covariantspinor derivative acting on the 3-form C αβγ , with a structure possible to capture[2,3,34,38,39] through the introduction of a bosonic, pure spinor of ghost numberone [38, 39]: λ α : λγ a λ = 0 . (2.2)In terms of a pure spinor superfield ψ with nothing but λ α λ β λ γ C αβγ at λ ¯ λ r ina series expansion in the variables, the equation of motion and gauge is Qψ = 0 and δψ = Q Λ with { D α , D β } = − γ a ) αβ ∂ a Q = λD + r ¯ ω, ⇒ Q = 0 . (2.3)By this construction, the component supergravity theory is retainable at ghostnumber zero in the minimal formalism ( x a , θ α , λ α ) while the non-minimal vari-ables (¯ λ α , r α ) , counterparts to ( λ α , θ α ) and ¯ λ of ghost number − : (¯ λγ a ¯ λ ) =(¯ λγ a r ) = 0 , allow for the construction of the integral measure [40]. are only valid in the presence of ‘true’ outer legs. The derivatives with respect to ( x, θ, λ, ¯ λ, r ) are ( ∂, D, ω, ¯ ω, s ) , the latter three which oughtonly appear in the gauge invariant 2- and 0-form operators formed out of ( λω, ¯ λ ¯ ω, ¯ λs ) : ( N, ¯ N , S ) . . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ Importantly, Q is a BRST operator and the formulation that of a BRST formal-ism, with an action respecting maximal super-Poincaré symmetry. The Batalin–Vilkovisky formalism [41, 42] therefore presents a consistent extension to interac-tions through the BRST operator being replaced with an action acting nonlinearlyon the superfield through an antibracket [34]: ( A, B ) ∼ Z δAδψ δBδψ [d Z ] , (2.4)The form of which occurs due to the superfield ψ containing all ghosts and anti-fields, effectively representing its own antifield. The subsequent formulation hasthe equation of motion ( S, ψ ) = 0 and is correct provided ( S, S ) = 0 , which ishow the action is set, starting from the BRST action while including the interac-tions stated in the superspace formulation of gravity [2, 3, 43–46].
The pure spinor formalism has three crucial features in BRST equivalence, gaugefixing, and integration in the presence of general regularisations. The first, BRSTequivalence, originates in calculations only being performed between free, on-shell, external states (obeying Qψ = 0 ), leaving the theory invariant under ↔ { Q, χ } , (2.5)provided a fermion χ of correct ghost number and dimension, with the specialcase of a regulator: e { Q,χ } .Gauge fixing is, in the absence of any antifield other than ψ itself, performedthrough a Siegel gauge [47] in an imitation of string theory: a b -ghost figuring inthe free propagator as b/p is introduced and required to fulfil { Q, b } = ∂ , bψ on-shell = 0 . (2.6)The former of these (in a BRST equivalent sense) gives { b, b } = 0 and sets [28] b = 12 η − (¯ λγ ab ¯ λ )( λγ ab γ i D ) ∂ i ++ η − L (1) ab,cd (cid:16) ( λγ a D )( λγ bcd D ) + 2( λγ abcij λ ) N di ∂ j ++ 23 ( η bp η dq − η bd η pq )( λγ apcij λ ) N ij ∂ q (cid:17) −− η − L (2) ab,cd,ef (cid:16) ( λγ abcij λ )( λγ def D ) N ij −− h ( λγ abcei λ ) η fj − η f [ a ( λγ bce ] ij λ ) i ( λγ d D ) N ij (cid:17) + (2.7)4 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ + 43 η − L (3) ab,cd,ef,gh ( λγ abcij λ ) h ( λγ defgk λ ) η hl − η h [ d ( λγ efg ] kl λ ) i { N ij , N kl } , where L ( p ) denotes L ( p ) a b ,a b ,...,a p b p = (¯ λγ [[ a b ¯ λ )(¯ λγ a b r ) . . . (¯ λγ a p b p ]] r ) , (2.8)antisymmetrises the p + 1 pairs of indices through [[ . . . ]] , and by default obeys L ( p ) L ( q ) ∝ (¯ λγ (2) ¯ λ ) L ( p + q ) [ r ¯ ω, η − ( p +1) L ( p ) a b ,...,a p b p } = 2( p + 2) η − ( p +2) L ( p +1) ab,a b ,...,a p b p ( λγ ab λ ) . (2.9)Moreover, is only non-zero for L ( p ) : p ≤ . [27, 28]A non-degenerate integration measure is given by the non-minimal variablesthrough their properties [34, 48, 49] [d λ ] λ α . . . λ α ∼ ⋆T α ...α β ...β d λ β . . . d λ β [d¯ λ ]¯ λ α . . . ¯ λ α ∼ ⋆T α ...α β ...β d¯ λ β . . . d¯ λ β [d r ] ∼ ¯ λ α . . . ¯ λ α ⋆ ¯ T α ...α β ...β ∂∂r β . . . ∂∂r β , (2.10)where T projects into (02003) , but general regularisations are necessary due tothe bosonic ( λ, ¯ λ ) : in the limit of infinity and on singular subspaces. The firstis remedied by a regulator e − ( rθ + λ ¯ λ ) which also furnishes the required ( θ, r ) forthe fermionic integrations to capture the correct dynamics. The latter is causedby scalars ξ = ( λ ¯ λ ) and η = ( λγ ab λ )(¯ λγ ab ¯ λ ) ∼ ξ σ present in the theory,where σ refers to the 2-form subspace. Only a limited negative power of these ( ξ − , σ − ) [34] can be part of a convergent integrand, often calling for a secondgeneralised regularisation: [12] O reg ( λ, ¯ λ ) = Z [d f ][d ¯ f ][d g ][d¯ g ] e −{ Q, ¯ fg } e iε { Q,gW + ¯ fV } O ( λ, ¯ λ ) . (2.11)Effectively, the introduction of a new set of variables ( f α , ¯ f α , g α , ¯ g α ) counterpartto ( λ α , ¯ λ α , θ α , r α ) , a Q extended akin to from the minimal to the non-minimal for-malism, and a regulator acting on ( λ, ¯ λ ) through gauge invariant operators [27] incombination with the integration, regularises the operator O (in a heat-kernel way)by what was initially allowed for in terms of the singular subspaces. This proce-dure can be performed any number of times and so any integrand built from con-vergent operators (effectively all) can be regularised, but the procedure severelycomplicates analyses, best performed prior to the generalised regularisations. Pro-vided the analysed entity presents a convergent integrand, the results are BRSTequivalent. Otherwise, results vanishing due to the variables subject to a changeunder generalised regularisations (i.e. all) are void, representing × ∞ with apossible non-zero result at the regularisation of the divergence.5 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ The action (2.1) describes the vertices present in the theory: the 3-point and 4-point vertices, the first with two R -operators acting out on two separate fields andthe second in addition containing T acting on a third field. In both cases, differentconfigurations of on which fields the operators act are equivalent, and apart fromthese entities, the tree amplitude diagrams are constructible from the propagatorand external states, with the addition of an overall integration.At the formation of loops, however, the propagator is too local in ( λ, ¯ λ ) to de-scribe loops on its own, necessitating the introduction of something like a gener-alised regularisation. The solution, inspired from string theory, consists of recog-nising the loop momenta as variables in the loop structure: D → P I D I etc. Inaddition, the loop regularisation [12] includes a regulator with exponent k (cid:0) ( λD ) S + ( λγ ab D ) S ab − N ¯ N − N ab ¯ N ab (cid:1) , k > . (2.12)and an integration over the new variables ( ∂ I , D I , N I , ¯ N I , S I ) for each loop I , intotal yielding a formulation where loops structures can be formed and analysed.Effectively, each loop integration demand ( ¯ N , S ) from the regulator, sincethe operators in the loops do not contain those entities. Due to the form of theregulator, this moreover satisfies [d N I ] and brings down λ D , antisymmetrisedwith any r or D due to (illustrated for a regularised r ) { ( λD ¯ λs ) , e { Q,χ } re −{ Q,χ } } ∝ [( λD ) , e { Q,χ } re −{ Q,χ } ](¯ λs ) . (2.13)In fact, all of the D s go into [d D I ] by the loop derivatives (of loop I ) equivalentlybeing positioned on one propagator (not shared between loops) at the loop integra-tion . As that is where D is brought down, and D is fermionic with degreesof freedom, anything but D from the loop structure and D from the regulatoryields zero, all immediately claimed by [d D I ] .During all of this, factors of ( λ, ¯ λ ) are brought down from the regulator through [d N ] λ α . . . λ α ∼ M α ...α a b ...a b d N a b . . . d N a b d N [d ¯ N ]¯ λ α . . . ¯ λ α ∼ M α ...α a b ...a b d ¯ N a b . . . d ¯ N a b d ¯ N [d S ] ∼ ¯ λ α . . . ¯ λ α ¯ M α ...α a b ...a b d S a b . . . d S a b d S (2.14)with M projecting into throughthe overlap between 16 pure spinors and the antisymmetrisation of For each loop, there is at least one propagator carrying only the loop momenta of that loop.This is a given and any propagator can equivalently be considered to fill this function (though notany constellation thereof with
L > ) but the concept is useful at an analysis of the loop structures.Equivalently, that is where loop integration takes place, and for further use the propagator will betermed ‘integration propagator’. . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ entities, compare to [40]. The loop integration then corresponds to λ D − [d ∂ I ] , (2.15)where the remaining integration may cause divergences in terms of ∂ (unpairedmomenta yield zero), and e.g. gives rise to the UV divergences. In particular, theloop regularisation effectively acts only on r .What remains is an analysis of the singularity properties with respect to ( λ, ¯ λ ) .The loop regularisation takes care of a number of singularities through bringingdown λ , in combination with the σ properties of M etc. Invariably, however, somestructures remain divergent and in need of (further) regularisation for a consistentanalysis. Important to remember in this, is that— convergent entities (by integrand standards) such as e.g. low-loop structuresand how two operators act on one another can be examined consistentlywithout regularisation.— vanishing, divergent expressions are primarily avoided by a regularisationof r through the loop regularisation, for L effectively e { Q,χ } r α e −{ Q,χ } : r α → r α + k ( γ ab ¯ λ ) α ( λγ ab D ) (2.16)but, when the entity is non-zero with r x : x = 0 , the expression is BRSTequivalently examined with r remaining unregularised. E.g. L ( p ) is nonzeroonly up to p = 15 , so that r s from L equivalently are regularised down tothat number, and no further [28]. The full regularisation provides an entityas convergent (or divergent) as provided by the term with r .With a restriction to conclusions drawn in these settings, the analysis may proceed. In the pure spinor field theory setting, UV divergences can equivalently be anal-ysed in terms of one-particle irreducible loop structures. This because the diver-gences occur in terms of the loop integrations over the loop momenta correspond-ing to x , and loop momenta are not shared between loop structures merely con-nected by a single propagator. Only the momenta (originally) part of a one-particleirreducible loop structure, inside the loops, share in the loop momenta and, pro-vided a non-zero result, affect the end properties. The overall divergence of anamplitude diagram is set by the constituent one-particle irreducible loop structurediverging the most. 7 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ An important feature of loop structures is that momenta acting along the looppropagators do not act out of the loop(s) or onto anything inside the loop(s) un-less forced to. Furthermore, if remaining in a loop, they must be integrated out bythe loop integration in order not to constitute total derivatives, falling under thefirst point just listed. Considering the loop integration properties mentioned after(2.12), this constrains the parts of the field theory operators (the propagator andthe operators in the 3- and 4-point vertices) that are necessary to consider with re-spect to non-vanishing results. If ∂ cannot form ∂ it must, just like N , be forcedto act out of the one-particle irreducible loop structure, or onto another entity in-side it. In particular, the only cause for this to happen to the bosonic momenta( D is another matter) is by b = 0 , as discussed in [28]. However, the vanish-ing of certain momenta can be specified further than what is done there; furtherrestrictions which also are valid in the [28] discussion on the case of maximallysupersymmetric Yang–Mills theory.To begin with, consider a 4-point vertex as part of a one-particle irreducibleloop structure. If it constitutes an outer vertex, T can equivalently be taken to actinto the loop structure, and otherwise it certainly does, resulting in an N in theloop(s) which invariably will constitute a total derivative. 4-point vertices there-fore are not part of non-zero one-particle irreducible loop structures.Next, consider a b -ghost (containing two derivatives) acting across a vertex,i.e. from one propagator to another: b −→ bb or ψ on-shell (3.1)This is a process equivalently examined with the r s remaining behind: a consid-eration necessary for later regularisations to be valid. Also, any R s may be con-sidered to have acted past the b s next to the vertices. If both derivatives in the b acting across the vertex then act onto the same state, the result is zero either by b or (2.6), as the considered entity is convergent. In this way, b is split onto twopropagators, one of which might be an outer leg. Once split, b = 0 does not occurunless both derivatives sidle up again. Anyhow, for one of the derivatives to actout, this must occur next to an outer leg (by the initial configuration). There, it isequivalent to choose which of the two derivatives acts across the vertex first, andit is only the other one that is forced out. Consequently, if a derivative in b wouldyield zero by staying in the loop, it can be (equivalently) chosen to stay in theloop, with a zero result. Hence, any part of the operators containing an N gives avanishing result, and can be disregarded at examinations.In addition, (¯ λγ mn ¯ λ ) ∂ m cannot pair up into ∂ , so that any term proportionalto it gives a vanishing expression for the same reasons as just stated for N . The8 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ reason for its absence is due to the properties of ¯ λ , see appendix A. With ¯ λγ mn ¯ λ implicit, the operators, apart from ∂ m and in the absence of N , contain ∂ and D : ( λγ mn γ i D ) ∂ i ( λγ m D ) , (3.2)and in the presence of regularised r s, by (2.16) also: (¯ λγ [[ ab ¯ λ )(¯ λγ cd ]] r ) : 4(¯ λγ [[ ab ¯ λ )(¯ λγ cm ¯ λ )( λγ d ]] m D ) , (3.3)effectively presenting entities ( λγ mn D ) ( λγ mj D ) : j = n, (3.4)not acting on the derivatives of the operators originally on the same propagator,as that expression in a BRST equivalent sense contains [ r, D ] (antisymmetrised).However, out of these D s, the only ∂ s possible to form are { ( λγ m D ) , ( λγ nj D ) } ∝ ( λγ mn λ ) ∂ j { ( λγ mi D ) , ( λγ nj D ) } ∝ ( λγ mnijs λ ) ∂ s , (3.5)and ∂ m paired up with either of these four existing ∂ s gives zero.Due to the effective absence of N and (¯ λγ mn ¯ λ ) ∂ m inside the one-particle ir-reducible loop structures, the only operators yielding non-zero results are b eff.loop = 12 η − (¯ λγ ab ¯ λ )( λγ iab D ) ∂ i + η − L (1) ab,cd ( λγ a D )( λγ bcd D )( R a ) eff.loop = − η − L ab,cd (1) ( λγ bcd D ) . (3.6)That said, the parts effectively yielding zero are not irrelevant. They are stillpresent up until loop integration, taking care of properties such as b = 0 .An interesting feature in connection to this discussion on equivalent treat-ments, especially in relation to b being split while acting across an outer vertex,concerns b . Consider the same situation as in (3.1). At an outer vertex, ∂ in b might be equivalently taken to act in. If that yields a zero due to the restrictionson D for loop integration, this means that the term drops out. If it does not, ∂ canequivalently be regarded as acting out, with D in. This presents different classesof equivalence. For example, in the 4-point 1-loop amplitude ( r x : x ≤ ), theonly non-zero element is ( b ) ( R ) with D (one from each of the three outerpropagators) acting out due to b = 0 . The two effective parts of b will be denoted by b n , with n stating the power of r in the absenceof regularisation. Sometimes, this is also used for R . Also note the term j -point: j outer legsconnecting to states beyond the one-particle irreducible loop structure. In j -point, L -loop one-particle irreducible loop structures, j propagators are outer (caused bythe presence of the outer legs) provided L > . For L = 1 , this number instead is j − . . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ Effectively, the one-particle irreducible loop structures consist only of 3-point ver-tices, propagators and outer legs. With L loops and j outer legs, there are L −
1) + j propagators L −
1) + j vertices (3.7) j of the latter which are outer, where one of the R s equivalently might be takento act out, thereby not contributing to the divergences of the loop structure. Thisstructure furthermore needs to provide D to each [d D I ] for a non-zero result; D snot originating in the regularisation of r .The last is a both obvious and subtle feature; at most D can be claimed fromthe regulator, effectively also by regularised r s, as both provide 0- and 2-form λD , of which no more than 24 can be antisymmetrised with a non-zero result.Furhtermore, the λ D in question cannot be paired with the λD in (3.2) to form λ D ; the irreducible representations do not match. Rests then the statementabove. However, this concerns a much larger structure than what has been dealtwith up until this point, so it is best to check for its convergence.The entity λ D is part of a regulator, not a singular operator (possibly) sub-ject to regularisation, and does not encode any singularities. The eight operators λD , do. In the presence of fully regularised r s, the effective operators behave like b ∼ (cid:0) ¯ λξ − , σ − (cid:1) b ∼ (cid:0) ¯ λξ − , σ − (cid:1) ( ξ, σ ) : ( R ) inner ∼ (cid:0) ¯ λ ξ − , σ − (cid:1) ( R ) outer ∼ (cid:0) ξ , σ − (cid:1) , (3.8)where it is taken into consideration that the two operators ( R a , R b ) in a 3-pointvertex are connected by ( λγ ab λ ) , as dictated by (2.1), which sits in the vertex(part of the loop structure, unlike the R s acting out from the outer vertices). Themost striking feature is that while a regularisation of r brings about λ ¯ λ ∼ ξ so thatthere is no difference in behaviour between b and b , the same is not true for σ . ¯ λ pairs up into the required irreducible representation, λ does not. It is more stronglycoupled to the D s, and so in general remains to be analysed in that setting.Effectively and equivalently, three or (if b : b ) four of the eight λD origi-nate in RbR on the integration propagator. With respect to ξ , the worst possiblebehaviour is ξ − , clearly convergent ( ξ x : x > − ). With respect to σ , it is pos-sible include the properties of the integration over the momenta. The effects of [d ¯ N ][d S ] in this respect cancel each other, and the by [d N ] and λ D remain-ing λ ∼ σ . A worst behaviour then is set by eight λD s from ( R b b ) with σ − σ ∼ σ − or from ( R b b ) with σ − σ ∼ σ − (under consideration: oneloop), also convergent ( σ x : x > − ). Consequently, it is equivalent to treat theentity without considering further regularisation.10 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ With the conclusion shown to be valid, it is possible to return to the require-ment of D L (original) going into the loop integrations, claimed from the operatorsin the loop structure. The b -ghost can at most provide two (on outer legs one) D ,and the same goes for the vertices. However, there is a subtlety with respect to thederivatives at L > : no more than ( λγ m D ) can be antisymmetrised and go into D , so that the inner b s at most can contribute with [3( L −
1) + 2 L ] D s. This yieldsa requirement valid for any (sub-) one-particle irreducible loop structure: [28] L = 1 : 1 + 2 j ≥ ⇒ j ≥ .L ≥ L − j ≥ L (3.9)This brings us to the last result of [28]: the limit on L by the shape of the effec-tive operators. Recall (2.13). When considering a loop, RbR (by equivalence) ispresent on the integration propagator. Equivalently, so are λ D from the operators(some from RbR ), where all of the D s and the RbR r s are antisymmetrised . Alsoequivalently, the r s on the propagator (originating in RbR ) are regularised, as in(2.16), with both its parts r + λ ¯ λD fully antisymmetrised with the other ( r, D ) inthe expression. At the integration, λ D is brought down and antisymmetrisedwith these entities, as in (2.13), at which point there is a regularised expression(convergent as examined right above) with non-zero contributions only from theparts not proportional to λ D formed out of 0- and 2-forms, i.e. originating inthe regulator. This draws on the observation right above, confirming D (origi-nal) to be required for the loop integration. Here, instead, the conclusion is thatwhile r is regularised, the integration gives a vanishing result by λ ¯ λD from thatregularisation, provided the original r sits on the integration propagator.Now this is interesting, because since the effective R ∝ r , there is at least r on the integration propagator of a loop. When L > , there is r x : x > on the integration propagators of the loops, and in total the expression vanishes.Moreover, at j = 4 the requirement of D L specifies this further. In such a one-particle irreducible loop structure, all but one of the D s possible to obtain from thestructure are required ( L > ), as specified in (3.9). In specific, (2 L − b s mustbe present on the inner propagators, of which there are L − . Only ( L − maycarry b , and so at least two of the integration propagators must carry b instead of b . A non-zero result then requires L + 2 ≤ ⇒ L ≤ . In total, L ≤ j = 4 L ≤ j ≥ (3.10) By considering the
RbR to be unregularised, an additional σ is present in Rb R and σ in Rb R . However, with j ≥ , the number of λD s originating in R is at least four, decreasing theworst estimate by σ , and so the entity is convergent by integrand standards, in the presence ofloop integration. . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ is the one-particle irreducible loop structure requirements for non-vanishing re-sults. Loop structures with a higher number of loops are of course allowed for, butonly as a product of multiple one-particle irreducible loop structures. Furthermore,as the UV divergences are set by the individual divergences of the one-particle ir-reducible loop structures, no amplitude diagram diverges more than the ones in(3.10).Important to note, is that it is not the regularisation of r that ‘fails’ in this. The r s on the propagators are regularised, the r x : x > are equivalently zero, andthe remaining terms are set to zero by the loop integration. If L > , the loopintegrations give a vanishing result.For a confirmation of the validity of the results, it is possible to look at theconvergence of the entities under consideration at the different points where con-clusions are drawn. When considering the first results yielding zero, those aregiven by ( R ) and ( R ) b on the integration propagators: convergent entitieswhen regularised, the first at worst behaving like ( ξ − , σ − ) and the second like( ξ − , σ − ). Some of the expansions of the regularised r s on the propagators aretherefore cut off with respect to the power of r , equivalently also for any L > .When analysing such an integral, the expression on the integration propagator alsois convergent, as analysed before, but with an extra 0- or 2-form λD which givesa vanishing result at the integration. Since the criteria specified at the end of sec-tion 2.2 have been met, the conclusions are valid regardless of the actual ( λ, ¯ λ ) subspace singularities of the amplitude diagrams. A first, naive estimate of the worst possible UV divergences of a one-particleirreducible loop structure is provided by two procedures: a look at the divergencesin the absence of regularisation of r , and a power-counting of what might combineinto ∂ inside it, when r is regularised.The first is set by the number of free ∂ s in the structure, the same as the num-ber of D s remaining inside after loop integration. At a worst estimate, they cancombine into a number of ∂ s described by L > L/ j − , (4.1)appropriately rounded off, i.e. to the closest (lower) integer. At L = 1 , ∂ s on outerlegs can equivalently be taken to act out, prohibiting ∂ from forming.The second can be termed in r . To begin with, regularisation demands r x : x > , by (3.6) and (3.7): L −
1) + 2 j > , and the variable only needs to be12 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ regularised down to r . Note that regularisation is absent only for L = 1 j ≤ L = 2 j = 4 . (4.2)At regularisation, it is equivalent to consider ∂ ∼ D ∼ r with R ∼ r and b ∼ r , taking into account that b on outer propagators lose at least D ∼ r to theoutside, D L is claimed by the loop integration and full ∂ s must be possible toform. The worst estimate then gives a number of ∂ s: L − j L = 12 L − j L > , (4.3)representing a positive number in the absence of (4.2).Because the momenta ∂ present in the one-particle irreducible loop structureat loop integration(s) are given by the propagators b/∂ and the m entities formedout of the operators ( b, R ) , the UV divergences from the L [d ∂ ] s appear with arequirement for finiteness according to LD − L − − j + 2 m < , (4.4)which with m as specified above (both regularised and non-regularised) restrictsfiniteness to D < L = 1 D < − L L ≥ ⇒ D < L = 2 , (4.5)where the L ≤ properties are set, definitely , by the properties at (4.2). However,the estimate for L ≥ presumes the momenta to be shared equally betweenthe loops, with no restrictions; an unlikely situation, and moreover proven wrongby [13, 15, 26] with L = 3 : D < and L = 4 : D < / . At the discussion on BRST equivalent examinations in connection to (2.16), wenoted that an entity can be examined equivalently in the presence of r providedit is non-zero. For a loop containing r x , this means that the expression for it is ∝ (cid:16) r x + r x − λ ¯ λD + . . . + ( λ ¯ λD ) x (cid:17) ∼ ( ∂ ) y , (4.6)where the last statement does not refer to the ∂ formed out of the r s, but the UVbehaviour of the loop in terms of momenta formed out of ( R, b ) . It is set either by13 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ r x or, if x > , by r . An important point here is that an addition of r z outsidethe loop does not alter the UV behaviour of the loop. In the extreme: at an additionof r outside the loop, equivalently chosen not to be regularised, the last term inthe expansion (4.6) is picked out as the only non-zero contribution. Importantly,this does not alter the UV behaviour of the loop. Plenty of r s get regularised, butthey do not increase the number of ∂ s formed. Especially, if the loop is part ofa one-particle irreducible loop structure and there are regularised r s (with respectto the discussed last term) that neither are shared by another loop, nor possible tofit into the ∂ s formed (originally), those r s cannot go into the ∂ s responsible forthe UV divergences. In an estimate as in (4.3), these ought to be removed.In particular, the structures that require further investigation have L > and by(4.2) require regularisation. The removal of r s, as described, from (4.3) is validprovided the unregularised limit of (4.1) is respected, which will be implicit inmost of the discussion from here on.Also possible to note is that further (general) regularisation changes the upperlimits on the UV divergences as deduced in this section (4) no more for divergententities than for the convergent ones; i.e. not at all. The effective shape of the reg-ulator can be observed in [12], but with respect to the exponent, it only containsthe ( ∂, D, r ) variables in constellations of g ¯ f λD , rs and some additional s . Thelast derivative is however equivalently only claimed by [d s ] from the loop regula-tor, so the two last types of expressions add no further r to the expressions. Thefirst one moreover only has λD in 0- and 2-form constellations, same as the loopregulator, so that it cannot be claimed by [d D ] for the reasons already stated inconnection to regularised r s. Neither can it act on any D , because the fermion g must be claimed by [d g ] , effectively bringing about the same situation as in (2.13). Illustrative example of ‘irrelevant’ r s Consider the two loop (sub-) structure L = 2 , j = 5 : ✫✪✬✩ ✟✟❍❍❍❍✟✟ . . .. . . : L = 1 , j = 4 : L = 1 , j = 5 which by (4.3) has a maximum of ∂ formed by the operators inside. The first loop( ) clearly does not cause any ∂ to form. The same does not go for the second( ). Had it been a one-particle irreducible loop structure, no ∂ would have beenpossible to form by that all but one ∂ (part only of b eff. ) equivalently act out on theouter legs. However, in the loop structure specified above, with loop integrationconsidered to take place on the propagators denoted by ( , ), one propagator isshared between the two loops. b ∝ D∂ then carries momenta split between the14 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ loops, e.g. D ∂ , where the ∂ cannot equivalently be assumed to act out. When itacts into the loop , it can couple up with the ∂ in b on the integration propagator,giving ∂ . Moreover, this gives a non-zero result as the D requirement is metwith b on the three outer legs — moreover the only effective contribution fromthose outer propagators as discussed right after (3.6).Now, the example above is a bit specific as it is a 2-loop structure with twoparts ∼ ( r , r ) , both sporting the same divergence. However, with the structureextended as indicated by the dots (with at least one loop), the ‘ r ’s of loop - which are not shared with the other loop(s) may equivalently be regularised,except for RbR on the integration propagators . This is R and b : in total r ofwhich only r may go into ∂ . However, in this setting there is also the D presentto be considered: D (three inner propagators with b ∝ D ), 18 of which arerequired for the [d D ] s. Effectively, there is a loss of r (any extra D would havebeen added) in the estimate of (4.5).Moreover, if the extended structure (not in any way indicated) require the r on the shared propagator for its maximal formation of ∂ , and we only considerthe part of the - -loop ∝ r , there is a contradiction. Both loops cannot use thesame, regularised r — it ought not be counted twice. One of the formations of ∂ in reality falls short by one r , effectively an entire ∂ . Principles of the extended analysis
There are a few rules to observe in this, best listed in general. However, keep theillustrative example above in mind. The first point made is that— for a general use, i.e. when part of a one-particle irreducible loop structureis considered, (4.5) and updated limits for L ≥ can be used with (4.4) toobtain the maximal number of ∂ s possibly formed in a sub-loop structure.For the specific behaviour, depending on the structure (as we soon shallillustrate), further analysis is required. E.g. a 1-loop structure at most has [( j − / ∂ s, possible to reduce down to none only if no propagator isshared: the behaviour depends on the number of ∂ s not equivalently actingout of the loop.Important to remember in the continued analysis is that the set behaviour is notrestricted to certain configurations of operator terms in the loop(s), such as ( ) and( , ). The expansion in (4.6) merely states the worst behaviour, consequently alsovalid for configurations which in a 1- or 2-loop setting does not give rise to thespecified formation of ∂ . For example, in ( ) two b were required for ∂ to form. It is desirable to keep the restrictions on the loop structures observed by
RbR on the integra-tion propagators, and so is is necessary to keep that configuration in the analysis of the loops. . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ In ( , ) this is not true. By regularisation, ∂ is formed in the absence of b . Inthe amplitude diagram above, this term exists also, but the principle is crucial inhigher loop structures where (4.1) falls short of capturing the full divergence. Inthis way,— the formation of ∂ in the substructures set an upper limit on the total ∂ .Often, however, the actual situation can be specified further. There is typically anover-counting of r , which can be corrected in two ways:1. The sub-loop structures can be analysed in terms of whether or not all r sand D s by regularisation go into ∂ and [d D ] . If loop-specific variables ∼ r cannot be fitted into these structures, they are equivalently lost in thecounting of (4.3), by equivalently being subject to further regularisation inthe larger structures.2. In the presence of many shared propagators, the actual division of r betweenthe connected loop structures is of relevance, as those r s equivalently canbe considered to be regularised. Such an r cannot belong to more than oneloop, with the ∂ of b in this setting equivalently represented by rD . D s onthe other hand are naturally split by b = 0 .In the presence of many shared propagators, the latter most often is the most ef-ficient approach. When r is shared between two subset loop structures in a reg-ularised expression, each can be considered to at most diverge by what is set bythe separate j -point L -loop structures. However, that estimate counts the r , equiv-alently regularised, that are shared between the loops twice , which cannot be.Either one of the substructures yield no ∂ in the process or the r x shared removesthe worth of x/ — rounded towards the (closest) larger integer — in terms of ∂ from the estimate, whichever removes the least ∂ s. E.g. two 2-loop structuressharing r and sporting j = (5 , in this at most show a ( ∂ ) , by that the 5-point 2-loop is considered to yield what r s are shared to the more divergent entity.On the other hand, if both loop structures are 6-point (or more), ( ∂ ) should beremoved from the subset estimate.However, it is important to note that free (inner) ∂ s (not required in terms of D for [d D I ] ) still may combine into ∂ , limited by (4.1) both in terms of the part andthe whole, in the ‘part absence’ of regularisation just discussed. When that is anissue, it is equally practical to note that there is an over-counting of D in the loopsubstructures, by R effectively being divided between the loops, on the sharedpropagators, in the same way as just discussed in terms of r . Since D ∼ r , thisalso limits the the combinations in the different substructures, e.g. a ∂ formed outof the free ∂ s in one substructure effectively removes some power of r from theconnecting loop(s). 16 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ By these procedures, it is possible to analyse the actual L ≤ limits on finite-ness. Moreover,— the overall limits are set by the minimal j .The fact that a higher number of outer legs by no means cause a worse divergenceis e.g. possible to discern in an iterative manner from j min . If the UV behaviourat j is known, the worst possible UV behaviour of j + 1 is obtainable through j → j + 1 with the extra outer leg equivalently added to the integration propagator(equivalently put anywhere in the diagram). As such the addition of the leg at mostbrings ∼ r into that loop (one R and b less a D or ∂ , forced out by b = 0 ) whichat most might add one extra ∂ , countered by the /p of the additional propagator.We will now proceed with a re-evaluation of (4.5) for ≤ L ≤ . Subsequentto that, we will use the described principles and the further observations madeto provide new estimates of the worst behaviour sported by 4- and 5-point one-particle irreducible loop structures of L ≤ . The 4-point L = 3 There are two structurally different 3-loop diagrams. The first structure is depictedin fig. 1a); it has a 4-point configuration limited by all substructures requiring j = 4 . There are two additional legs as denoted by +1 , but the loops ( , ) arerestricted to represent 1-loop 4-point structures, contributing ( ∂ ) to the loopdivergence. By that, at least r on each outer leg in ( , ) is lost to the generalregularisation, and there is at least one such on each of the two loops. In addition,the arrow marks a propagator unique to ( ) by the distribution of the integrationpropagators. It cannot contribute to the divergences of ( , ), yet one derivative ofthe b residing there is lost to ( ) by b = 0 : all of the b on the marked propagatorcannot transfer to the loop integration propagator, giving a further loss of r . As atotal of r is lost, the entire possible divergence by (4.3): ( ∂ ) is avoided, and thelimit on finiteness is D < / by (4.4).The second structure, in fig. 1b), is a bit more intricate since an inner vertexis shared by all of the three loops. It is more compact, and we will see that thecompact structures give rise to the worst UV divergences. Quite simply, it allows r to be divided to the greatest extent between the loops, in a situation as closeto (4.5) as is possible to obtain. In fig. 1b), it is equivalent to put one outer legon the propagator indicated by a : the basic structure is completely symmetric.Furthermore, at most two outer legs can be added to the same propagator sinceall substructures require j = 4 and the basic structure has j = 3 for the separateloops. The configurations possible to distribute as such are (2 , , (2 , , or allouter legs on separate propagators. In the first scenario, the one pair is equivalentlyconnected to a , at which point the second only can be placed on c : is a 5-point17 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ ✫✪✬✩ a) n n +1 +1 x b) ✫✪✬✩ (cid:0)(cid:0) ❅❅❍ a bc1 (cid:26) - +2 c) ✫✪✬✩ n n +1 +1 •• d) ✫✪✬✩ n n +1 +2 •• x e) ✫✪✬✩ (cid:26) - (cid:26) -
4a b • • ❞ ❞ a / b : +1+2 f) ✫✪✬✩ (cid:0)(cid:0) ❅❅❍ ( - ( - ❞ •• • Figure 1: Illustration to go with the re-evaluations of the limits for finiteness set by the4-point 3- and 4-loop one-particle irreducible loop structures. a) and b) show the 3-loopstructures, while the rest show the 4-loop equivalents. Integration propagators are markedby numbers, the presence of additional legs are marked by ‘ + x ’, either at a specific loca-tion or in general. Divisions into substructures ( - , - ) are indicated by dots and circles,a process which ought to be self-explanatory with the integration propagators numberedand indicated. Note that the 2-loop substructure - in d) can be turned about the axis de-scribed by the dots. With the loop integration propagators moved also, configurations canbe made equivalent by what will be referred to as ‘symmetry and renumbering’. More-over, e) and f) are similar with respect to UV divergences, and here described in terms ofthe same features. For the 4-point amplitude, one more outer leg needs to be added to thepropagator a or b , in addition to two more legs, distributed at will. Note, however, that thepropagator b in f) describes a non-planar structure, effectively passing above one of theother propagators; it is not describing a 4-point vertex. - is a 4-point 2-loop structure. In the second scenario, the two lastouter legs also must be added to - . By symmetry, it is equivalent to put the firstof those on either b or c , so that the situation either falls within the first scenario, orat least, the 4-point 2-loop by the outer leg on b causes a loss of r in comparisonwith (4.3). Moreover, this last observation is equally true in the third scenario, withall outer legs on different propagators, since the loop integrations can be placedon three of those with at most one outer leg shared between two loops, so thatthere is at least one 4-point 1-loop with an outer leg. Either way, at most ∂ canbe formed, and the structure is finite in D < .In conclusion, the closer look at how r might be divided yields a limit onfiniteness different from (4.5), as already known from [13]: D < L = 3 . (4.7)18 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ The 4-point L = 4 There are four structurally different 4-loop diagrams. The first structure is depictedin fig. 1c). The configuration is limited by all substructures requiring j = 4 . Thereare two additional legs as denoted by +1 , but the loops are restricted to representa pair of 2-loop 4-point structures, visible by considering the vertices marked bydots as outer vertices, in total contributing ( ∂ ) to the loop divergence. Hence,the limit on finiteness is D < / by (4.4).The second structure, in fig. 1d), similarly as for the structure a ) causes an r loss by the 4-point loop and an r loss by the loop configuration of . Theloops - make up a 4-point 2-loop substructure giving ( ∂ ) with (by symmetryand renumbering) at most one outer leg on the propagators shared with loop :three propagators, carrying a total of r , are unique to that loop structure, whileexcepting the integration propagators. A total of (at least) r is lost: ( ∂ ) is lostin (4.3), so at most one such can be formed and the relevant limit is D < .The third and fourth structures in fig. 1e) and f) are different in that the secondis non-planar. However, it is possible to analyse them both at once. One propa-gator equivalently (by symmetry of the basic structure) has one leg attached asshown, in e) for the leftmost loop to be 4-point. Also in e), the rightmost loophas an outer leg attached to either one of the propagators denoted by a and b . Bysymmetry and renumbering, this also goes for f), where both a and b can be con-sidered to be part of a different subset of loops than the first attached leg. In total,this gives the possibility of looking at two loop structures: - and - , sharingthree propagators and two inner R s with at least R b ∼ r D shared by the twosubstructures, which are 4-point 2-loop structures to which two more outer legshave to be added. In fact, with i shared outer legs, the shared r s and D s amountto at least r i D i , by the D s from the added b i also effectively being shared.Anyhow, regardless of where those additional legs are placed, the shared entitiescannot be counted twice, which in combination with the 4-point 2-loop structurein total corresponds to a loss of ( ∂ ) in the estimate of (4.5). Consequently, thelimit on finiteness is D < / .In conclusion, the closer look at how r might be divided yields a limit onfiniteness different from (4.5): D < / L = 4 , (4.8)as already known from [19]. Further limits on ≤ L ≤
4- & 5-point diagrams
It is possible to note that all 4- and 5-point L -loop one-particle irreducible loopstructure configurations with L > are possible to form from the 4- and 5-point19 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ L − propagators connecting to the original structures. For example by starting fromthe 5-point 7-loop diagram, with five outer legs distributed on the inner propaga-tors, it is possible to in an iterative manner choose a propagator with no outer legattached, cut it and create a ( j + 2) -point ( L − -loop one-particle irreducibleloop structure, all the way down to L = 4 with j = 11 . The -loop structure has j = 9 on inner propagators, making the last step feasible, and representingthe last given, feasible step. Moreover, with the 5-point 7-loop given, the 4-point6-loop naturally is accommodated for as well.This is interesting, because L → L + 1 in this setting corresponds to an ad-dition of two inner vertices and three propagators, in total R b ∼ r , i.e. with D claimed by the additional loop integration, the introduced, extra components ∼ ( ∂ ) . Based on the overall limit on the 4-point 4-loop given by (4.8), (4.4) canthen be rewritten as D < /L ≤ L ≤ j = 4 , (4.9)corresponding to the results of [21, 23]. The remaining question is if further ∂ can be formed in the new loop configuration.By equivalently, at each L → L + 1 , considering the alteration of the one-particle irreducible loop structure as consisting of two steps, a more detailed lookinto the situation can be provided. Firstly, two additional outer vertices are added,which by j → j + 2 at most can provide two new ∂ . Then, all but the two outer R s and D s forced out are equivalently claimed for the formation of ( ∂ ) to bepossible. After that point, the two outer legs may be considered to be connectedthrough a new integration propagator. The added structure then describes ( D , D )on the integration propagator, by ( b , b ), in addition to the r s of the RbR whichcannot be regularised ( r , r ). However, since the latter forces an additional regu-larisation of the r s on the j -point ( L − sub-loop structure, the calculation reallyis a zero-sum game: the ( ∂ ) can be accommodated for (in the case of b by one ∂ being moved to the introduced loop).On the other hand, any additional ∂ would require ∼ r to be claimed fromthe ( L − sub-loop structure, with L > . As demonstrated during the discus-sion on the 4-point 4-loop structure, the base in 1e) and f), used for this estimate,is highly compact and therefore contributing with the most divergent result. Inaddition, the structures are the most limited by shared ∼ r s, minimal in their con-figurations. A saturation of the ∼ r s possible to share have already been observed,so there is no possibility of withdrawing r without affecting the other ∂ s of thestructure. The only possibility of acquiring new ∂ s are by extending the c) and d)structures; a scenario by equivalence falling under the extension of e) or f), withone of the free propagators in c) and d) equivalently added.20 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ Hence, by the compactness of the e) and f) diagrams, the overall behaviour islimited by (4.9). Important to note is that it is merely a rough lower limit, whichpossibly might be further specified to something more allowing. For example, asat the transition between L = 3 and L = 4 , the full ( ∂ ) is not always captured.The 5-point setting can be analysed in the same way. There, the question of con-cern is a possible finiteness in D = 4 , i.e. at L = 7 , as indicated in [23]. What wethereby wish to know is if D < holds for the 5-point 4-loop one-particle irre-ducible substructures, but only in a 5-loop equivalent sense. That is, not the actual5-point 4-loop behaviour, but what effectively, as above for the 4-point structure,gives the 5-point 5-loop structures. In this setting, it is equivalent to consider ageneral 7-point version of the 4-loop structures, with any two outer vertices re-moved; the effective behaviour generated at the L → L + 1 is the same. Also,naturally, the 5-point 4-loops constructed out of the diagrams e) and f) are just ascompact as the 4-point versions with respect to the formation of ∂ , so with aneffective D < , in the equivalent setting, the overall limit would correspond to D < /L ≤ L ≤ j = 5 . (4.10)We will skip the finer points of the 5-point diagrams as well as trying to find outtheir strict divergences; as long as the equivalent 5-point structures correspond to D < , i.e. at most ( ∂ ) formed, the worst possible divergence of the one-particleirreducible loop structures is given by (4.10).Consider the discussion on the 4-point 4-loop one-particle irreducible loopstructures right above. The 5-point versions just have one extra outer leg, and asalready stated, the diagrams behave no worse than what is true for the minimal j ,so diagram c) and d) by default fall under D < . The real issue is the diagram e)and f). However, the 5-loop structures caused by an extension of f) are equivalentlycaused by an extension of e): any non-planar 5-loop diagram can be reduced toplanar by at least six different cuts, compare e.g. to the illustration in [25], and inthe 5-point setting at least one of those is free from outer legs.What then remains is the diagram of fig. 1e) in a 5-point setting. It is possi-ble to note that the structures with no shared outer legs at most result in the totalformation of ( ∂ ) . The combination of a 7-point 2-loop and a 4-point 2-loop ischaracterised by the latter contributing with ( ∂ ) while requiring D from theshared R s, reducing the former by ∼ r to at most form ( ∂ ) . The other combi-nation of a 6-point 2-loop and a 5-point 2-loop is similarly restricted: if the lattercontributes with ∂ through regularisation or free ∂ s, the former cannot contributewith ( ∂ ) .Interestingly, in a 7-point equivalent setting, this sets the overall behaviour to D < , because regardless of the distribution of 7 outer legs on the diagram of21 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ fig. 1e), it can by symmetry and renumbering in combination with two outer legsequivalently being removed, present a 5-point 4-loop, divided as in e), with noshared outer legs. In this article, the pure spinor field theory results of [27, 28] with respect to max-imal supergravity have been revisited. The observation in [28] of the UV diver-gences only depending on one-particle irreducible loop structures constructed outof propagators and 3-point vertices in terms of certain effective operators, in totalconstraining the non-zero j -point L -loop structures to j =4 : L ≤ j ≥ L ≤ , (5.1)has been extended to a confirmation of the limits on the dimension for finiteness:[13, 19] L = 1 : D < ≤ L ≤ D < /L (5.2)and [21, 23] ≤ L ≤ , j = 4 : D < /L. (5.3)An additional, crucial result — possible to note for e.g. L = 7 , j = 5 in [23],which seems to discuss the very same one-particle irreducible limit — is ≤ L ≤ , j = 5 : D < /L, (5.4)where both (5.3) and (5.4) constitute rough lower limits, possibly subject to furtherconstraints at a more detailed analysis.In this setting, all amplitude diagrams in maximal supergravity (where
L > only is possible as a product of several one-particle irreducible structures) areconcluded to be finite in D ≤ (5.5)by the workings of the pure spinor formalism, i.e. a formulation with both on-and off-shell maximal supersymmetry. In particular, the restrictions on the UVdivergences which usually are discerned in terms of U-duality, in the pure spinorformulation show in terms of the loop regularisation r ↔ λ ¯ λD (variable/momentaequivalence) and how far those momenta can be shared within the loop structures.Furthermore, the limit on L occurs not so much due to this equivalence (although22 . K ARLSSON : ‘P
URE SPINOR INDICATIONS OF ULTRAVIOLET FINITENESS IN D = 4 . . . ’ partly a product thereof) as due to the insensitivity of the integration over theloop momenta to the r ↔ λ ¯ λD conversion, thereby (through the number of r spresent being restricted) limiting L . What this corresponds to in terms of otherapproaches to the UV divergences in maximal supergravity is, however, difficultto tell, although it would be highly interesting to see a corresponding analysis ina different setting.The result is intriguing with respect to the UV finiteness of maximal super-gravity in four dimensions. This is a scenario traditionally regarded as highly un-likely. Still, the analysis in a pure spinor field theory setting indicate preciselythat. Admittedly, the pure spinor formalism is difficult to interpret in terms of thecomponent fields of ordinary maximal supergravity, but despite sometimes beingregarded as somewhat obscure, it encodes the same physics. In total, a confir-mation of the results in a different setting would be most welcome. There is apossibility of an overall behaviour of maximal supergravity (disregarding the spe-cific dependence on L ) just as in maximally supersymmetric Yang–Mills theory:UV finiteness in D = 4 . Ackowledgements
I would like to thank M. Cederwall for helpful discussions.
A Properties of the pure spinor
The spinors in D = 11 supergravity are symplectic. To (in the flat setting of thepure spinor formalism) capture the relevance of ordering, all spinor indices arechosen to be lower and contracted through ε αβ = ε [ αβ ] : ( λγ a ) γ = λ α ε αβ ( γ a ) βγ .The Fierz identity is ( AB )( CD ) = X p =0 p ! ( Cγ a ...a p B )( Aγ a p ...a D ) , (A.1)assuming bosonic entities ( A, B, C, D ) ; if the statistics differ from this, the addi-tion of an appropriate sign suffices for the correct expression to be obtained. Inthe presence of two pure spinors, this furthermore reduces to ( Aλ )( λB ) = −
164 ( λγ ab λ )( Aγ ab B ) + 13840 ( λγ abcde λ )( Aγ abcde B ) . (A.2)In total, there are several useful identities for the pure spinor, compare to [27]23 . K ARLSSON : ‘P
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