Anomaly Mediation from Unbroken Supergravity

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Anomaly mediation from unbroken supergravity

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As Published http://dx.doi.org/10.1007/JHEP09(2013)125

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/116053

Detailed Terms https://creativecommons.org/licenses/by/4.0/ JHEP09(2013)125 Springer July 29, 2013 : August 26, 2013 : September 24, 2013 : Received 10.1007/JHEP09(2013)125 Accepted Published doi: c [email protected] , Published for SISSA by and Zachary Thomas [email protected] c , Jesse Thaler a,b 1307.3251 Supersymmetry Phenomenology When supergravity (SUGRA) is spontaneously broken, it is well known that [email protected] SISSA 2013 Cambridge, MA 02139, U.S.A. E-mail: Department of Physics, UniversityBerkeley, of CA California, 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National CA Laboratory, 94720, U.S.A. Center for Theoretical Physics, Massachusetts Institute of Technology, b c a c

Keywords: ArXiv ePrint: We find that recent claimswere in based the on literature a about Wilsonianinvariant the effective 1PI action non-existence effective with of residual action anomaly gauge contains mediation we dependence, the calculate and expected the the sfermion anomaly-mediated gauge- spectrum spectrum.familiar to two-loop all Finally, soft orders, masses and from uselevel minimal supertrace and anomaly relations one-loop mediation, to goldstino as derive couplings the well consistent as with unfamiliar renormalization tree- group invariance. sarily appears in the regulated action,couplings. yielding soft In masses without this correspondingmasses. paper, goldstino Already we at extend tree-levelzero our we soft analysis encounter masses a of correspond number to anomalymediation of broken appears mediation (AdS) surprises, when to SUSY. regulating including SUGRA At sfermion the in one-loop, soft fact a we way that explain that how preserves anomaly super-Weyl invariance. cently, we showed thatwith one-loop unbroken anomaly-mediated supersymmetry (SUSY). gaugino This massesderlying counterintuitive symmetry should result structure arises be because of associated (broken) thein SUGRA un- anti-de in Sitter flat (AdS) space space.serves is When SUGRA, in quantum the fact corrections underlying (unbroken) are AdS SUSY regulated curvature in (proportional a to way the that gravitino pre- mass) neces- Abstract: anomaly mediation generates sparticle soft masses proportional to the gravitino mass. Re- Anomaly mediation from unbroken supergravity Francesco D’Eramo, JHEP09(2013)125 34 20 27 32 13 16 5 18 8 25 20 23 6 21 9 14 – 1 – 12 37 17 11 15 39 1 30 4.3 Supertraces4.4 and the 1PI effective The action super-Weyl-invariant4.5 d’Alembertian Soft masses and goldstino couplings for chiral multiplets 3.5 1PI effective action and goldstino couplings 4.1 The order4.2 parameter for SUSY Goldstinos breaking in the SUGRA multiplet 3.1 Super-Weyl formalism3.2 for SUGRA Choice of3.3 gauge fixing Counterterms in3.4 the Wilsonian effective Effect action of the regulators 2.1 Derivation from2.2 the SUGRA lagrangian Derivation from2.3 supercurrent conservation Tachyonic scalars2.4 and sequestering Giudice-Masiero terms 2.5 Mass splittings for regulators 1 Introduction If supersymmetry (SUSY)Spontaneously is broken realized SUSY in yields nature,so a in then positive order it contribution to mustsymmetry to achieve structure the be the of nearly cosmological spontaneously zero our constant, broken. cosmological universe constant must we be see SUSY today, in the anti-de underlying Sitter (AdS) space. In the D 1PI gaugino masses A Goldstino couplings from the conformalB compensator Renormalization group invariance of irreducibleC goldstino couplings Super-Weyl transformations 5 Conclusions 4 All-orders sfermion spectrum from anomaly mediation 3 Anomaly mediation and super-Weyl invariance Contents 1 Introduction 2 Invitation: anomaly mediation at tree level JHEP09(2013)125 ] 3 In 2 . As ]. In ]. . For 2 / 2 3 / 13 15 3 , m m 12 , exactly satu- 2 / 2 3 m 9 4 − is equal to the gravitino . These would appear as 2 1 / ] for additional theoretical − AdS 3 λ 11 m – 4 space. Throughout this paper, we will 4 1 ] for details. There is also a (usually subleading) effect from other anomaly-mediated effects 3 2 / – 2 – 3 . Already at tree-level in AdS space, the com- m ] yields gaugino masses proportional to 2 ]) but come to the conclusion that anomaly mediation , , but we will show that from the point of view of 1 2 14 / 2 3 ] if there are direct couplings of SUSY breaking to the gauginos at , satisfying the Breitenlohner-Freedman bound [ 4 2 m versus SUSY-breaking effects that are only proportional to / 2 3 will have one scalar partner with mass-squared 2 ], these gaugino masses do not break AdS SUSY, and are / 3 m 2 3 2 / 3 m − m 1 2 (i.e. gravitino mediation; see refs. [ ± 2 / 3 ], we will use goldstino couplings as a guide to determine which effects 3 m . Thus, because of the underlying AdS SUSY algebra, there will be effects on 2 / 3 space, anomaly mediation already yields scalar masses at tree level. Following the m ative mass-squared order to have a stableing, theory this after negative AdS mass-squared SUSYAdS is must lifted also SUSY, to be this flat lifted. space requires via Since irreducible SUSY such break- couplings a lifting between must the break SUSY-breaking sector Tree-level tachyons and sequestering ponents of a chiralexample, multiplet if get the SUSY fermionic component mass is splittings massless, proportional then to its scalar partner has a neg- because of the need to fine tune the cosmological constant to zero. Second, we wish Along the way, we will encounter a number of surprises, all ultimately having to do The goal of this paper is twofold. First, we wish to extend the analysis of ref. [ Famously, anomaly mediation [ 4 2 These other effects were dubbed “Kählermediation” since they arise from linear couplings of SUSY A fermion with mass • / 2 1 3 breaking to visibleKählermediation sector and fields gravitino in mediation.anomaly-mediated the effect See Kählerpotential. noted in ref.tree-level. ref. Full [ [ anomaly mediation is simply therating sum the bound. of with the structure of AdS SUSY: m to counter recent claimscontrast, by we de will Alwis use thatanalysis the of anomaly Kaplunovsky same mediation and Louis does logical [ not not starting only exist point exists, [ as but de is necessary Alwis for (which the is preservation based of AdS on SUSY. the mass-squareds proportional to AdS strategy of ref. [ preserve AdS SUSY, allowinggenuinely us proportional to to distinguish between SUSY-preserving effects that are perspectives). Unlike usual SUSY-breakingmasses without effects, accompanying anomaly goldstino couplings, mediation further generatespreserving emphasizing that effect. gaugino this is a SUSY- to the case of sfermions. It is well known that anomaly mediation yields two-loop scalar we recently showed inin ref. fact [ necessary for“gravitino conservation mediation” of to the separatewhich AdS have this supercurrent. nothing to Weuse do called the with this more the familiar phenomenon AdS (butproportional SUSY less to algebra. accurate) name “anomaly mediation” to refer to all effects mass the supersymmetric standard model“SUSY-breaking” (SSM) effects proportional from to theactually point SUSY-preserving of effects when view viewed of from the AdS flat space SUSY algebra, but are context of supergravity (SUGRA), the inverse AdS curvature JHEP09(2013)125 (1.2) (1.1) term µ B terms via ], anomaly depends on µ 8 i B ]) to show how SW 14 F and h µ when the sfermion soft L e G but is rather due to the need to . Here, we will follow the logic of 4 , , 4 ∗ ∗ M . As emphasized in ref. [ χφ 1 3 L e G term preserves AdS SUSY, the − 2 ]. For a chiral multiplet with components / C µ . We will use an ansatz for the SUGRA- 2 3 ] is a way to generate 1 eff F F m 16 2 ≡ – 3 – SW L ⊃ F 3 . In flat space, the harmonic part of the Kählerpotential ] is that bulk counterterms are needed to counteract otherwise 9 ] (based on the analysis of Kaplunovsky and Louis [ + h.c. While the generated is the scale of SUSY breaking. Intriguingly, this coupling is 13 , d 12 there is necessarily a coupling to the goldstino is the scalar auxiliary field. Accounting for the fact that H eff u } F M H , we reproduce the familiar anomaly-mediated spectrum. ], we (erroneously) advocated that the absence of goldstino couplings could be used as a physical component of the compensator (which can be gauge-fixed to zero), but on the 2 3 ⊃ / 3 C φ, χ, F invariant 1PI effective action to extract sfermion soft masses and goldstino couplings. where m Supertraces resolve spectrum ambiguities de Alwis’ calculation (andthe a langauge of similar the issueF Weyl implicit compensator, in anomaly mediation Kaplunovskysuper-Weyl-invariant depends combination and not Louis). just on In the SUSY-breaking effects due to thede boundary Alwis of [ AdS anomaly mediation arises fromSUGRA preserving theory. super-Weyl invariance of Whilecannot a de exist UV-regulated Alwis in (erroneously) such a concluded situation, that we anomaly find mediation that there is residual gauge dependence in breaking effect. Anomaly mediation and super-Weylmediation invariance is not due toadd any local anomaly of counterterms SUSY to itself, preservepresented SUSY in of the ref. 1PI [ effective action. A related story and the Giudice-Masiero mechanismK [ actually breaks AdS SUSY, sincemultiplets and it the secretly goldstino. involves direct Whenthat written couplings Giudice-Masiero in between a arises Higgs more from natural basis, a it combination becomes of clear a SUSY-preserving and SUSY- where renormalization-group invariant, and effectivelythe defines hidden what and it visible means sectors. to sequester Giudice-Masiero in AdS space (i.e. the chiral plus anti-chiral part) is unphysical. This is not the case in AdS space, den and visible sectors{ are sequestered [ mass is zero in flat space: (“hidden sector”) and the SSM (“visible sector”), even in theories where the hid- In ref. [ Of course, the name “anomaly mediation” is still justified since it generates effects proportional to beta • • • 3 4 definition of sequestering.needed Because to of have a this stable tree-level theory. tachyon subtlety,function though, coefficients. this goldstino coupling is JHEP09(2013)125 , R F (1.5) (1.3) (1.4) (containing the ∗ ]. The complete χ, 1 M 20 − – µ 17 D µ σ † χ . Using an ansatz for the all- f C i . f 2 + / C -component of the chiral curvature 2 3 2 F F m 1 − − . a C 1 ∗ R − . The key confusion surrounding anomaly + F 1 a s 12 vanishes for unbroken SUSY in AdS, but takes C – 4 – ≡ R − C = F R φ S F ∗ φ s −C = )). As a cross check of our calculation, both the two-loop soft once the cosmological constant has been tuned to zero. Thus in 1.1 2 / corrections to the self-energies. We will find that while the coeffi- soft mass 2 3 2 L m in SUGRA frame) does not break SUSY. Instead, the SUSY-breaking /p − 2 : 2 ] for a related story). Not surprisingly, a similar supertrace is needed to / m 3 R ) in the 1PI effective action: is the d’Alembertian appropriate to curved space. The first term is the are indeed ambiguous (since they depend the precise form of the ansatz), 2 17 / m i , and we will have to tease these two effects apart by carefully considering 2 3 3 2 C / m − 2 3 ( m O couplings proportional to anomalouscoupling dimensions from (on eq. ( topmass of and the the tree-level one-loop goldstino quantities, goldstino coupling as are expected renormalization-groupsfermion (RG) from spectrum invariant the is summarized general in analysis table of refs. [ arising from terms inof the motion. SUGRA multiplet proportional to the gravitinoTwo-loop equations soft masses andorders one-loop SUGRA-invariant goldstino 1PI couplings effectivesoft action, masses we from anomaly will mediation. recover But the in familiar addition, we two-loop will find one-loop goldstino After using the Einstein equation, on the value flat space, we will find bothto SUSY-breaking and SUSY-preserving effects proportional AdS SUSY. We will also find corresponding goldstino couplings proportional to term order parameter in SUGRAsuperfield comes from the SUSY-breaking in the SUGRAmediation multiplet is that theresets are two the different underlying order AdSwhich parameters are curvature in only and SUGRA related — oneIn to one which particular, each which a accounts other non-vanishing vacuum for after expectation tuning SUSY value (vev) the breaking for cosmological — constant to zero. is unambiguous and gives a(see useful ref. measure [ of the “softdefine mass-squared” unambiguous for “goldstino a couplings”. sfermion familiar sfermion soft mass-squaredappear as term, but thecients two non-local termsthe necessarily supertrace in the resulting sfermion spectrum.at For example, there are three terms that show up where Because there are many such ansätzeconsistent with SUGRA, there is an ambiguity • • JHEP09(2013)125 . . 5 2 2 2 4.5 , we / / / (2.1) 2 3 2 3 2 3 3 ˙ ˙ ˙ — — γm γm γm , 1 4 1 4 1 4 − − ; these boundary + h.c. , we review the 4 R L 1 2 e with Ricci curvature 12 G 2 2 j / / 4 R − . 2 3 2 3 χ − γ i 2 1 φ 1 / 12 γm γm 2 3 ij eff m b F ]. We conclude in section 3 γ + is the anomalous dimension of the L , which in turn depends on the γ e 2 G R / j 3 1 χ 12 ı 2 2 ¯ m ∗ / / 2 3 2 3 − φ R -terms, which are described in section 1 6 2 m m ————— B ıj / ¯ eff 2 2 2 3 − a F to SUSY breaking. The fine-tuning of the − − = 0). Here, m Tree-level One-loop Two-loop i + i 2 j K – 5 – − χ h i prior χ 2 -terms and ij / ], so 3 A 3 M m 1 2 = . Starting with unbroken SUSY in AdS − ), and the loop level refers to the order at which the effect starts. µ j 1 φ i − AdS log 4.17 ]) necessary to preserve the SUSY algebra in AdS φ λ 9 ij B dγ/d 1 2 ) and ( Soft mass-squared Soft mass-squared Soft mass-squared Goldstino coupling Goldstino coupling Goldstino coupling ≡ , we show how the spectrum evolves as SUSY breaking is tuned to achieve 0. In this table, “soft mass-squared” and “goldstino coupling” refer to the − 2 γ / j 2 3 4.14 φ ) ı ¯ m 2 4 ∗ / R → 2 3 φ 2 ıj ]. Less well known is that those mass splittings have an impact on the phe- ¯ m = 12 m 21 2 , we discuss anomaly mediation for sfermions up to two-loop order, completing , − AdS = 12 . Sfermion soft masses and goldstino couplings from minimal anomaly mediation (i.e. “grav- λ 4 15 Flat space SUSY AdS R Curved space L ⊃ − Considering only chiral multiplets, we can write the fermion and sfermion masses and The remainder of this paper is organized as follows. In section ( (broken SUSY) (broken SUSY) = 12 cosmological constant to achieve flat space is summarized insfermion-fermion-goldstino figure couplings as nomenology of SUGRA, evenIn if the particular, geometry the (after couplingsof SUSY of breaking) the the is gravitino) goldstino that canserve (eaten of be SUSY. flat to used space. Crucially, form these to theunderlying couplings track longitudinal AdS which depends components curvature effects on break SUSY and which effects pre- 2 Invitation: anomaly mediationIt at is tree well level known thatticles rigid [ AdS SUSY requires mass splittings between particles and spar- to SUSY-preserving massdiscuss splittings super-Weyl between invariance fermions inhow and UV-regulated anomaly sfermions. mediation SUGRA arises theories assection In a at super-Weyl-preserving section one and loop, SUSY-preservingthe effect. analysis and In of show goldstino couplings that was initiated in ref. [ ary terms (analogous toterms ref. are [ irrelevant in flat space. structure of SUGRA at tree-level, and show how the underlying AdS algebra gives rise chiral multiplet and ˙ R flat space with supertraces in eqs. ( Minimal anomaly mediation alsoThis yields table only includes the contributions from bulk terms and not from one- and two-loop bound- Table 1 itino mediation” in the language of ref. [ JHEP09(2013)125 (2.3) (2.4) (2.5) (2.2) is the scale eff , F ij M ]. Starting with the 2 / 22 3 m SUSY) 4 + ), with a third derivation 4 , ij ij 2.3 B ) (plus modifications to those B = 2.3 = . † flat ij , AdS ij W . We then discuss the phenomenolog- is the goldstino, and Flat Space (broken AdS 3) L A 5 e − G SUSY in AdS k + log , b , b G ıj ¯ kj k W δ 2 G M / ( – 6 – k 2 3 ı ¯ G ff 2 e m e is given by + log M F , SUSY-breaking effects lead to flat space with broken = G − 2 K + 2 + / 2 -dependent mass splittings between multiplets without Pl 3 V ıj 2 ¯ 2 ≡ kj / m M m 3 , the interactions of the longitudinal components of the M G 2 2 k / m ) is to consider the SUGRA lagrangian directly. The scalar / = = ı ¯ 2 3 3 M m 2.3 m 1 ıj flat ] ¯ is its fermion partner, 3 − AdS a − λ i ] implies a physical difference between the Kählerpotential and the superpo- 23 − 2 ıj ¯ ψ 7 , = =0 6 E m = V V AdS ıj ¯ a is a sfermion, . Fine-tuning of the cosmological constant, adapted from ref. [ i φ In this section, we give two different derivations of eq. ( The Kähleranomaly [ 5 where the potential Kähler-invariant tential, but it does not enter at tree level. 2.1 Derivation from theThe SUGRA first lagrangian way to derivepotential eq. for ( SUGRA is [ of the goldstino areregime still physically with relevant. energies Indeed,gravitino in are the captured goldstino by equivalence thegoldstino theorem goldstino couplings couplings that in appear eq. at ( higher-loop order). corresponding goldstino couplings (i.e. without breaking AdS SUSY). using the conformal compensator givenical in appendix implications of these goldstinoregulator couplings fields. for Though sequestering, the Giudice-Masiero goldstino terms, and is eaten by the gravitino in SUGRA, the couplings mass splittings (i.e. when flatwe space will SUSY show is that broken). In AdS space at tree-level, however, which shows that one can have of SUSY breaking. Assuming the flat space SUSY algebra, one can show that which emphasizes that goldstino couplings arise when sfermions and fermions have non-zero Figure 1 underlying AdS curvature (AdS) SUSY. where JHEP09(2013)125 . 2 / 3 (2.9) (2.6) (2.7) (2.8) m , it is (2.14) (2.11) (2.12) (2.13) (2.10) 1 = 1 − AdS λ -terms for the D . c . + h , j . χ i χ ) . As shown in figure j Pl G i M , G 2 ]. Here, subscripts represent deriva- Pl . + , . , j 23 k M Pl i 2 ) yields no goldstino couplings, as is to χ E G G / i M , i 2 2 3 k ¯  2 i ), and indices are raised and lowered with G / ∇ 2.3 G δ m G/ k 3 ( (unbroken SUGRA) 3 2 e 3 G 2 1 / (unbroken SUGRA) m e D √ 2 3 3 − G/ = 0), the scalar mass-squared and holomorphic − p m e – 7 – = √ 2 i 2 eff 1 2 i i 2 = ≡ ∂G/∂φ j / F = V as ij − 3 h − L G ¯  = eff = i i ij e M m eff k G k F 2 χ i F = 0), then we have a negative cosmological constant / M µ h∇ M G 3 V 2 i h D ik i / m µ 3 G M − σ h ¯  m † = = χ = ¯  ¯  i 2 i ij ij B m iG = 0 by choosing M and its inverse. The gravitino mass is given by ¯  V i ), so the spacetime background is AdS, with curvature G 2 Pl are derivatives the with Kähler-covariant respect to spacetime and scalar L ⊃ − i M 2 ∇ / 2 3 m and 3 µ − D = If SUGRA is broken, then there are a few important effects. Defining the SUSY- If SUGRA is unbroken ( i V h In addition, SUSY breaking givesgauge rise multiplets to for a simplicity) goldstino, points which in (assuming the no direction where we have restoredpossible factors to of fine-tune the Planck constant breaking scale as we find the cosmological constant is modified to be Note that inserting thesebe mass expected values into since eq. there ( is no goldstino when SUGRA is unbroken. and at the extremum ofmass the can potential be ( expressed in terms of ( The fermion mass matrix is where fields, respectively. tives with respect to scalarthe fields Kählermetric ( and the quadratic fermion interactions in SUGRA are Throughout the text, we use the conventions of ref. [ JHEP09(2013)125 and 2 (2.22) (2.17) (2.18) (2.19) (2.20) (2.21) (2.23) (2.15) (2.16) / 3 m . , j φ . k 6 c χ ]: . i → ∞ χ 23 i + h Pl k µ M † G j = 0 [ j ): , † µ i G . E i i ¯ ψ  µ 2.7 i V G h ψ Pl G , = 0). µ 1 ) below. + i , M σ E + i j 2 l ¯ ). E V Pl 2 ¯ ,  G h eff G 2.26 i G i k − F E M k G 2.17 j i ∇ ∇ ¯ G = 0 and  j l ¯ G ∗ − i i G + 3 ¯ φ k + 3 i l ¯ ∇ k V + h.c. k h + , R χ µ G † ν i i j j G k j + 2 ψ µ − χ j G G i σ G k k l ¯ E ∇ µν i 2 ¯  i ∇ G / ¯ k G σ k i G ¯ 3  j k 7 ∇ † – 8 – µ G k R i ∇ ∇ G ψ + i im 2 k − G j G / 1 2 + ∇ D D 3 G G ) uses conservation of the AdS supercurrent. The i k k + i 2 2 + m to matter are determined by the supercurrent alone: / / ∇ G k G 2 3 2 3 2.3 µ j D h∇ D µ − † G m m ¯  2 2 2 j ψ ∇ τ i ) (at least for the case of / / / i µ 2 3 2 3 3 ψ = = G G ρ D 2.3 m m m ıj ij ¯ + D + b a l ν ¯ = = = k σ ] which omits the first term in eq. ( ¯ G  2 i l ¯ † 2 ij ij µ G 0 = j 23 ¯ ψ k m m i M ∇ i R is negligible for visible-sector fields. This then yields the goldstino µνρτ − i h∇ i D 2 2 / G = / 3 h )) yields the relation 3 = 0, the goldstino couplings retain information about the structure of the L m m is the Kählercurvature tensor. i 1 2 1 2 l ¯ 2.12 V ¯ k − h i R fixed), in which the last term vanishes and the spacetime background is AdS (with Thus, despite the fact that SUGRA is broken and the cosmological constant is lifted The Yukawa couplings can similarly be extracted from eq. ( L ⊃ − There is a typoHere, in and ref. [ throughout the text, we do not choose any gauge fixing for the gravitino, so there is also 6 7 eff This relation can beF most naturally interpreted in the rigid limit ( quadratic mixing between the goldstino and the gravitino. See eq. ( Appropriate manipulation of the gravitinogiven equation eq. of ( motion (and the Einstein equation, An alternative derivation ofsupercurrent is eq. the ( Noetherlinear current of couplings (rigid) of SUSY the transformations, gravitino and in SUGRA, the to yield underlying AdS SUSY, and not the structure of2.2 flat space SUSY. Derivation from supercurrent conservation recalling that couplings anticipated in eq. ( One can read off thegoldstino couplings direction: of the goldstino to visible-sector fields after picking out the where SUSY breaking, and their form is well-known for The fermion and sfermion mass matrices are generically deformed due to the presence of JHEP09(2013)125 8 (2.28) (2.29) (2.24) (2.25) (2.26) (2.27) ) contains the ) can then be arise from the + h.c. 2 / µ 2.26 3 2.23 ψ µ m σ † L e G ]. It also implies that the Pl eff 27 , F M , ¯  2 26 i , [ i = 0 but rather the rigid limit of δ , √ 2 2 . µ ) = 0. µ / / † ν j † 2 3 3 , j µ e ]. Among other things, this implies σ † µ m + h.c. ∂ m µ † + µ j σ L terms necessary for AdS supercurrent 25 i , σ L e + 2 G , is irrelevant for our discussions since it χ 2 + e L ¯ ¯   + h.c. + G / 24 ij ∗ 3 k e µ µ G µ L . φ µ σ σ M µ j m e ν ı i M ¯ σ G 4 2 ( σ † i 2 ∂ / µ L µ ik ¯  µ 3 eff † χ – 9 – i † e − ∗ j e D G g F j e M m 2 ) 2 2 2 / 2 = W / 3 / − ı + ¯ 3 √ √ 3 ¯  = 0, whereas in AdS space 2 i µ D ij im ı ⊃ = ¯ m im † D 1 2 m B )). In order to have a (meta)stable vacuum after SUSY µ µ µ (2 χ eff 1 † j ∂ 1 2 F = = − j 2.9 ¯  µ i ij − j e b a µ L e ). Note that the terms proportional to D G µ 2.3 ∇ (see eq. ( eff µ 2 F σ 1 / 2 † L 2 3 e √ G m i 2 − − − ). In the rigid limit, we see clearly that conservation of the supercurrent is satisfies the criteria = 2 / is the (gravity) covariant derivative [ L 3 is the remaining “matter” part of the supercurrent. eq. ( ), as Noether’s theorem requires µ m µ j e D = 2.23 In both flat space and AdS space, the supercurrent for chiral multiplets contains When SUSY is broken, the supercurrent contains the goldstino This assumes that the SUGRA action only contains a Kählerpotential and a superpotential without 1 8 − AdS so in the absence of AdSmass-squared SUSY breaking, the sfermions would be tachyonic, with a common additional higher-derivative interactions.account, The giving supercurrent rise to is new modified effects when detailed in loop section effects are taken into additional goldstino massconservation. and 2.3 Tachyonic scalars andThe fermions sequestering in the standard model are massless (prior to electroweak symmetry breaking), goldstino couplings as expected from eq. ( The other termvanishes proportional on to thematter goldstino fields equation and of the motion. goldstino equation of Using motion, the we equations find of that motion eq. ( for the where the last term is necessary for conservation of the AdS supercurrent. where interpreted as the goldstino equation of motion arising from the lagrangian that the goldstino in rigidcondition AdS for space conservation has of a theeq. mass supercurrent ( of is 2 not different in flat space versusparameter AdS space. In flat space, the fermionic SUSY transformation where λ JHEP09(2013)125 ], 1 (2.30) (2.31) ), this 2.3 , one way to 2 . term in eq. ( ]), and therefore decou- ¯  hid i a 29 , W 28 + vis W L = . G ∗ AdS tachyonic mass balanced against 2 χφ / L 2 3 Hidden Sector e G m 2 2 ,W / 2 3 , you necessarily obtain zero scalar masses but − eff hid – 10 – F m W 2 ) shows that there are direct connections between + Ω 2.3 and vis L ⊃ K = Ω 3 Visible Sector K/ − e 3 − SUSY-breaking mass to yield the physical tree-level sfermion mass of zero couplings. ¯  2 i / a 2 3 . An extra-dimensional realization of the sequestered limit, where SUSY is broken only m There are two potential ways to interpret this result. One interpretation is to con- This result is rather surprising from the point of view of strictly sequestered theories [ simistic, though, since the tachyonic upliftingcosmological is constant. an Concretely, automatic this consequence(and uplifted of adjusting the mass the corresponding arises from goldstinoyou the couplings have scalar arise the auxiliary from sequestered mixing field form withnon-zero of the gravitino), so once clude that sequestering correspondslimit at to tree-level, a one fine-tunedthe has limit. +2 the underlying Afteronce all, the cosmological in constant the is sequestered tuned to zero. This interpretation is probably too pes- the visible and hiddenan sectors irreducible necessary coupling for to stability the of goldstino the when theory. the In sfermion particular, soft mass there is is zero in flat space: sequestered form of the effective four-dimensional Kählerpotential and superpotential: Naively, one would think thathidden the sector goldstino (assuming the from SSM SUSY-breaking itselfpled must does from be not the break localized visible SUSY in sector. [ the But eq. ( where anomaly mediation is the onlyachieve source the of sequestered soft limit masses. is As totor shown have (i.e. in the SUSY-breaking figure visible dynamics) sector live in (i.e.no different the light parts SSM) of degrees and an of the extra-dimensional hidden freedom space sec- with connecting the two apart from gravity. This implies a special the goldstino and chiralin multiplets flat in space the after visible SUSY sector breaking. in order tobreaking, have a these stable tachyonic tree-level massesimplies theory must an be irreducible lifted, coupling between but the from goldstino the and the matter fields. Figure 2 in a hidden sector.visible Naively, the sector goldstino fields. is But localized due in to the mixing hidden with sector the and gravitino, would there not are couple irreducible to couplings between JHEP09(2013)125 , ). 9 B 2.32 (2.32) (2.33) (2.35) (2.36) (2.34) term pro- µ B . 2 / . 3 2 / 3 m m is a nice realization of − = + ) (and corresponding loop term and a 2 ] = µ ) gives a concrete definition µ µ 35 B µ 2.31 – µ B 2 2.31 , 33 ], the visible and hidden sectors , L ⇒ 32 e G ⇒ – 27 , + h.c. X , 30 d F d 26 1 2 H H √ u u + h.c. d + H + h.c. H h µ θ from the superpotential does not break SUSY d u – 11 – h h ⊃ µ ⊃ u 2 X / B K 2 3 F µh 3 W 2 ] is a way to generate a − / = m e 3 16 3 − m NL − d + ψ X d u ψ ψ u 2 ) is the only coupling between the visible and hidden sectors. / 3 µψ 4.5 m ], the sequestered limit implies that gaugino-gauge boson-goldstino couplings are ), while generating 3 µ without (apparently) requiring couplings between the visible and hidden L ⊃ − 2 2 / here is crucial, since if instead one had the superpotential L ⊃ − / 3 3 ), we see that the Giudice-Masiero mechanism actually does break SUSY µ m m B 2.3 = 2 ) experimentally as a way to gain access to the underlying AdS curvature. ) is actually RG invariant, so one might conjecture that it corresponds to precisely = 0). Written in this language, it is confusing how a goldstino coupling could ij b ij b 2.31 2.31 We can do a Kählertransformation to make the physics manifest. To model SUSY Regardless of how one interprets this result, the irreducible goldstino coupling is an A second, more optimistic, interpretation is that eq. ( As shown in ref. [ 9 breaking, we use a non-linear goldstino multiplet [ zero. From eq. ( (with (i.e. appear in the Giudice-Masiero mechanism since there is no goldstino present in eq. ( the fermion and scalar mass terms would be one generates the fermion and scalar mass terms The sign of portional to sectors. Via aphasize holomorphic that piece these in are the superfields) Kählerpotential (written using boldface to em- tachyonic scalars to haveeq. a ( stable theory in flat space.2.4 One might even Giudice-Masiero hope terms toThe measure Giudice-Masiero mechanism [ eq. ( the (attractive) IR fixedgenerally, one point can identify when needed a to theorycorrections, is see have sequestered if section a eq. ( conformally sequestered theory.unavoidable More consequence of AdS SUSY lifted to flat space, since something needs to lift the of sequestering. Whilesequestering, the the extra-dimensional sequestered picturedimensional limit in models can figure with beeffectively conformal decouple achieved sequestering under in RG [ flow more toin the general infrared, the assuming theories. hidden all composite sector vector multiplets have In mass four- dimension greater than 2. As we explain in appendix JHEP09(2013)125 2 ), / 2 3 . If into 2.37 m (2.42) (2.38) (2.39) (2.40) (2.41) (2.37) UV , and a UV 2 / 3 m = . But there are .... 2 µ / ..., is due to a partial + 2 3 , 2 d 2 / / ] or higher-dimension m 3 ..., 2 3 H , which equals +2 − u m 2 37 | , m = f H , | + h.c.) + 36 2 = + NL in both the Kählerpotential / d = 2 W 3 X ij Ω /µ H +1) + h.c.) + NL µ u − 1 3 µm f d e B X − − H + ( H → u d ]. of , b of ( 6 NL 2 H / µ H µ W ( 2 3 X u B B and fine-tuning the cosmological constant ..., H † NL m + 2 / f ]. However, already at tree-level, we can see + X , 3 † NL − 3 14 = + – 12 – NL Ω m X 2 − = / X + 2 3 + † NL f ∗ X NL Ω X m F NL + X + 2 X / + 2 † NL 3 3 + f K 2 m − / X . At tree-level, the physics is invariant to doing a Kähler + 2 3 2 → 2 / = = / 3 , we have 3 m 3 3 + K d / m − W − m terms from direct couplings to 3 K H µ − ). = √ u = = e B µ 3 3 H / 2.3 = Thus, we conclude that the relation B W = 0. In a theory where the visible Higgs multiplets are sequestered from − K − f − 11 2 NL e 10 = 3 X − Ω Consider a Pauli-Villars chiral regulator field with a SUSY-preserving mass Λ Despite the fact that Giudice-Masiero can be written in a sequestered form in eq. ( At loop level, oneOf must account course, for the the physics Kähleranomaly [ is invariant to Kählertransformations at tree-level; all we have done here is 10 11 this regulator does not break AdS SUSY, then it must have anchoose additional a scalar convenient Kählerbasis negative to make the physics more clear. In order to setsection, the we stage want to forgences discuss talking in a about SUGRA. bit There anomaly about areSUGRA, mediation various the for at ways physics example to loop that by introduceoperators introducing level regulates an that Pauli-Villars logarithmic in effective regulators regulate cut-off UV the [ scale the diver- next the Λ UV consequences behavior of [ having a physical regulator in AdS SUSY. cancellation between a SUSY-preserving andtuning a SUSY-breaking between effect, (otherwise) and independent corresponds parameters.only to irreducible In a goldstino the couplings strict are sequestered limit allowed, Giudice-Masiero where terms2.5 must be absent. Mass splittings for regulators as required by eq. ( there is secretlysector a goldstino. coupling between the visible sector Higgs multiplets and the hidden also SUSY-breaking and superpotential. This yields aafter contribution to tuning the cosmological constant to zero. Therefore, we have We see immediately thatcorresponding the Higgs SUSY-preserving multiplets contribution have to a SUSY-preserving to zero requires transformation so choosing SUSY-breaking, the relevant pieces of the Kählerpotential and superpotential are where the equations of motion set that satisfies JHEP09(2013)125 . T UV 4.4 (2.43) (2.44) term affects . 7 c UV ]. Starting with a = Λ 13 , 12 . PV-fermion UV Λ , giving rise to SUSY-preserving 2 / 3 UV for the purposes of loop-level calcu- Λ m , m

2 i / = 3 F , for example by appropriately coupling UV i m Λ K 2 PV UV − /

3 ], which is a gauge fixing of super-conformal m . All this means is that the regulator multiplet – 13 – 40 ± , b NL – ]) to construct a Wilsonian effective action. After 2 / X 2 UV 38 2 3 14 -term of the Weyl compensator. 14 m B C + Λ 2 = 2 / 2 3 PV m a 2 , coupling regulators in such a fashion is largely equivalent to making the Λ), with − 3 / NL We often say that anomaly mediation is “gauge mediation by the = as well as a X 2 / 12 2 3 (1 + m 2 C , we will show how the regulators must be included to get super-Weyl-invariant gaugino 2 PV-scalar − → m 3.4 . C so for simplicity we will assume regulators have no explicit coupling to SUSY 4 ], we described one-loop anomaly-mediated gaugino masses using the conformal 3 13 It is possible, however, to regulate SUGRA with a regulator multiplet whose scalar and In the language of section To avoid later confusion, we want to point out that there are two different types of ambiguities. The In section 13 14 12 replacement ambiguity discussed here isphysical effect whether that the can be regulatorshaving measured to do using do goldstino or with couplings. dousing how There supertraces to not is to write a experience define separate down the SUSY ambiguity a soft in mass SUGRA-invariant breaking, section spectrum, 1PI which up effective to action. is a a puzzling This ambiguity is in (partially) how the resolved same effect using a super-Weyl invariantonly and SUSY-preserving consider 1PI gaugino effective action. massesto We in will section this section, leaving our main resultmasses. on sfermion masses compensator formalism ofSUGRA. SUGRA Here, [ we will insteadwill use allow the us super-Weyl invariant to formulation connect ofreview directly of SUGRA, the which to super-Weyl the formalism, claims wethe will of logic follow de the of Alwis logic Kaplunovsky in ofdemonstrating and de refs. the Alwis Louis [ (which existence [ itself of follows anomaly mediation in the Wilsonian picture, we derive the breaking in the subsequent sections. 3 Anomaly mediation andIn super-Weyl ref. invariance [ However, we would insteadcourse get consider goldstino an couplings intermediate fromcouplings. case the with In regulator either a fields! event, combination oneis of One can phenomenologically mass show can equivalent that splittings of to modifying and changing lations, regulator goldstino couplings in this fashion fermionic components have athe common regulators mass to Λ themust SUSY-breaking have corresponding goldstino couplings by conservation of the AdS supercurrent: Since there is no mass splitting among the regulators, no mass splittings are generated. regulators”, in the senseacts that analogously the (SUSY-preserving) to mass theCrucially, splitting (SUSY-breaking) we at messenger will the mass threshold seebreak Λ threshold AdS that of SUSY. the gauge mass mediation. splittings generated by anomaly mediation do not mass splittings between the Pauli-Villars fermions and scalars: Any UV-divergent SUGRA calculation thataffected properly by includes this the regulator massanomaly modes splitting, mediation. will and be this fact is one way to understand the necessity of mass-squared JHEP09(2013)125 2 R E ]: − (3.3) (3.4) (3.5) (3.6) (3.1) (3.2) 41 , 15 ... 23 . , c . + 14 corresponds 2 + h Θ Σ to chiral U(1) 2 as | α | Σ M Σ W and superpotential | F , the corresponding α . Crucially, the only 1 9 E with Weyl weight and vector superfields K C W − w C E . Q M has Weyl weight 4 and 2 ]. The components of the ) † 1 6 Θ 2 of supergravity [ Σ 2 E component of 41 d + , M Σ Σ ⊃ − ( 23 Z F w R 1 4 e )[ † , w . + , so neither super-Weyl transformations nor transforms under . } , Σ Σ Σ : V Σ 2 C C F F F R W Θ 6 F 2 → 3 ,F ∗ , C − w C − ]) is not invariant under super-Weyl trans- C ∗ eM V , χ E C of chiral superfields 23 1 F M { w – 14 – Θ 2 corresponds to dilatations, Im ⊃ − C 2 → → | , d E ∗ = C Σ Σ 2 F Z w M C e , w 3. In the gravity multplet, 4 ) + − Q 3 is the scalar auxiliary field Θ / 2 | Σ K → F − M ] w | e 3 1 9 Q 23 − ( to conformal supersymmetry. The C | † correspond to different types of transformations which may be familiar Σ ). In that case, the tree-level lagrangian , and chiral curvature superfield α Σ + h.c. + E EC C D has Weyl weight 2 Σ Θ Θ α 2 ∗ 4 − (and its conjugate anti-chiral superfield d W e M Σ 1 3 Z → = C which transform as [ The usual SUGRA action (e.g. in ref. [ The complete super-Weyl transformations are given in appendix also have Weyl weight 0. For a vector superfield of weight 0, the gauge-covariant ⊃ − L Super-Weyl transformations do not include special conformal transformations, and superconformal w 15 E transformations do not includesuperconformal the symmetry transformations generated are by a subset of the other. has Weyl weight 0 as desired. The components of the super-Weyl compensator are and due to the non-vanishing Weyl weight of formations, so one needs(i.e. to introduce a super-Weyl compensator Ordinary matter fields have WeylW weight 0, so the potential superfield Kähler has Weyl weight 6. We will often talk about theV Weyl weights This auxiliary field appears inchiral the density determinant 2 of the SUSY vielbein from the superconformal algebra:rotations, Re and to a new symmetry which will play a keyfield role that in transforms understanding under anomaly mediation. The SUGRA lagrangianSUGRA. Super-Weyl can transformations are be thetorsion derived most constraints general of from transformations SUGRA that unchanged, aperfield leave and the gauge they may fixing bechiral parameterized superfield of by a super-Weyl-invariant chiral su- 3.1 Super-Weyl formalism for SUGRA JHEP09(2013)125 and (3.9) (3.7) (3.8) C ) is not a component + h.c. -invariance C 3.8 . The choice Σ F F C by imposing that M transformations 1 3 C ]). Σ − F , since these are the . We can investigate 14 ∗ ∗ C C F eliminates troublesome F M C i χ F and i yields canonically-normalized K C F C + is not a superfield itself. Instead, , ), one must gauge fix ∗ H H | | , ∗ M in some convenient fashion, leaving a ). In the super-Weyl case, the 3.4 K K 1 3 ) simply for calculational convenience, 1 3 M A ] C 1 3 It must be stressed that eq. ( − , since that degree of freedom is contained in the = 3.4 14 ∗ C − † 17 . F M C C ... transformations. F K – 15 – Σ ≡ ) that depends on undetermined, though one can fix arg F M + log ] (and implicitly missed in ref. [ C 1 3 3.4 SW C F + h.c. + 13 − , ∗ C log W 12 F ∗ 3 M − 1 3 − 3 C K/ F − e 16 3 C C ∗ being the harmonic (i.e. chiral plus anti-chiral) part of the Kählerpotential. + 3 C H = | ) is particularly convenient since this choice for Re ) should be thought of merely as a prescription for setting each component of L K 1 3.8 3.8 However, it is not so clear what is accomplished by gauge-fixing It should be stressed that this super-Weyl invariance (and the corresponding super- − The superconformal formalism does not contain This gauge choice leaves still leaves arg component of the conformal compensator (see appendix . Of course, other gauge-fixing prescriptions will give physically equivalent results, but = 1 yields the lagrangian in “SUGRA frame” (i.e. without performing any super-Weyl e † 16 17 Φ F is a pure gauge degree of freedom. the gravitino mass parameter has no phase. only two fields that are not inert under eq. ( Einstein-Hilbert and Rarita-Schwinger terms and thismatter-gravitino choice mixings. for this by examining the portion of eq. ( super-Weyl transformations). Effectively, thisWess-Zumino gauge gauge for choice the is realsupersymmetric superfield the relation equivalent amongst of superfields, since eq. going ( to C with This yields the lagrangian in “Einstein frame” (i.e. after having performed appropriate 3.2 Choice of gaugeTo fixing recover the familiar SUGRA lagrangian fromC eq. ( transformations). A more convenient choice is [ the combination regardless of what gaugeis choice the is key ultimately point made. missed in As refs. we [ will argue, this Weyl compensator) were introducedand into physical eq. results ( willuse the not super-Weyl actually transformations exhibittheory to super-Weyl without gauge-fix symmetry. spurious symmetriesare After or a all, degrees gauge of one redundancy freedom. can of the Because theory, though, physical observables will only depend on JHEP09(2013)125 (3.12) (3.11) (3.13) (3.10) ] using ]. This 14 43 , 42 ], this rescaling ) augmented with ), de Alwis found . After one solves 45 ∗ , 3.4 3.8 . 44 M . such that the rescaled a α -invariant combination a part of the super-Weyl α Σ , and hence no anomaly C W R 2 F W / α / i a α a . 3 Q C m F as in eq. ( ) → CW . ) contains a gaugino mass that CW i R i

i T F log Q i log F 3.12 i − E K E K 1 3 G

T ) cannot be the complete answer, since Θ 2 Θ 2 1 3 ] claimed this was the complete formula = 2 (3 2 d d 2 + C 3.13 13 2 π 2 , Z F Z g / only appear in the ) 3 16 ) 12 – 16 – R ∗ R m − T T M = = − only serve to shift the vev of − i ] is that to preserve super-Weyl invariance in a UV- G G C T and T ], refs. [ SW 2. Due to the Konishi anomaly [ 14 F ( (3 are not in dispute. F − 2 14 C h 2 i transformations. This is incompatible with the assertion π F 3 π F 1 ], it is possible to regulate all UV-divergences in SUGRA in ambiguous λ i 16 Σ 16 K ), m F 13 , = = 3.7 ], de Alwis was only claiming the absence of gravitino mediation. The Kähler- proportional to the gravitino mass L 12 3 L λ ∆ ∆ : m which has the vacuum expectation value C ) to become ∗ F M 3.11 1 3 by performing the (anomalous) rescaling . i − 18 C Q C F equation of motion, physical observables do not (and cannot) depend on the gauge F ∗ ≡ However, we see immediately that eq. ( The key observation of ref. [ M In the language of ref. [ 18 SW it is notthat invariant the under physical predictions of this theorymediated should terms proportional be to invariant under such super-Weyl Following the analysis offor ref. the [ gaugino mass,no and contribution by to gauge-fixing mediation. Immediately this presentsdepends a only conundrum, on since eq. ( This term can betransformations remains deduced non-anomalous. from It the ister convenient requirement to fields canonically that normalize the the mat- U(1) matter field have Weylmodifies weight eq. ( regulated theory, the Wilsonianthe effective counterterm action must consist of eq. ( higher-derivative regulators in aimplies that version the super-Weyl of symmetry discussed Warr’sany above regularization is physical not results scheme anomalous, we and [ see consequently, derive that must be anomaly completely mediationsuper-Weyl super-Weyl invariance. (despite invariant. its Indeed, name) we is will necessary to preserve both SUSY and fixing of 3.3 Counterterms in theAs Wilsonian emphasized effective in action ref.a [ way that preserves SUSY and super-Weyl invariance. This was shown in ref. [ F Thus, different gauge-fixings for the As expected from eq. ( JHEP09(2013)125 ), ≡ 3.15 (3.15) (3.16) (3.17) (3.14) , with SW , they instead F S ] showed PV PV . 14 and ) manifestly i L . F i c 3.12 . K 3 1 + h + . Expanding eq. ( 2 / LS E 3 m PV Λ ) E R T in the conjugate representation: E Θ 2 − 2 ), we have the super-Weyl invariant S d ]. The kinetic terms suggest that the G LS, log 2 T Z ∗ component of 2 46 1 3 3.13 , (3 2 + M ]). In this case, the tree-level expression in 2 + 8 i 2 37 π , UV g S C 16 Λ 36 V 1 3 − − – 17 – in some representation of a gauge group, one can log e † = Q → S ⊃ − in a diffeomorphically-invariant action. Indeed, there ). One could try to make the replacement − e SW C 2 2, but since the Pauli-Villars mass term is Λ / PV F 3 L − ) L manifest, but as emphasized emphatically (and correctly) log V m R e T † ), the physics should depend on the combination SW L − F 3.7 − G h that one can write down a non-local 1PI effective action that depends only T -dependent gaugino mass at one loop. Adding this loop-level con- E (3 ∗ 3.5 19 2 Θ 2 is a chiral density and not a chiral superfield, and one cannot include π 4 M g d 16 E Z to violate super-Weyl invariance in order for physical results to be super- − ], 2 = ) is precisely the term needed to restore super-Weyl invariance of the action. = -term that is not super-Weyl invariant! Doing calculations with these regu- . Given a chiral superfield (which does contain 13 B , PV ∗ 3.12 2.5 needs , the Pauli-Villars fields break super-Weyl invariance. However, ref. [ L ) will combine with loops of the regulators to yield a super-Weyl invariant result. 12 M ]) . PV physical λ 3 1 14 Λ 3.13 Now, because the Pauli-Villars regulators have a SUSY-preserving mass Λ To understand how this effect arises, consider a Pauli-Villars regulator, as anticipated m SW We will see in section − C F in the same representation of the gauge group and 19 C gaugino mass on we find which is a lators will yield an tribution to the tree-level contribution from eq. ( regulator fields have Weyl weight of that eq. ( exhibit boson/fermion mass splitting due to the Θ Gauge fields can beadjoint representation. similarly By regulated using by many such introducingall regulators divergences and chiral of including superfield appropriate SUGRA couplings, regulators can be in removed the [ eq. ( in section regulate its contributions to loopL diagrams by introducing two superfields, The resolution to theref. [ above puzzle isWeyl that invariant. the This Wilsoniancutoff, is effective where familiar the action Wilsonian from action (as must Yang-Millsnon-gauge be defined invariance gauge non-gauge of in invariant the theories in cutoff order (see with compensate also for a ref. the [ hard Wilsonian arbitrary extra factors of ainclude chiral arbitrary density extra factors in of a det SUGRA-invariantis action, no just local as one termsuper-Weyl cannot that invariant. one can add to the3.4 Wilsonian action Effect to of make eq. the regulators ( F to make the dependence on in refs. [ transformations. By eq. ( JHEP09(2013)125 20 . i F i (3.18) K 1 3 + 2 / 3 m , which is allowed to . To avoid SUSY- = W . C 3.4 F 4 , we can study a 1PI effective , α ). 2.5 W ) . In general, the 1PI effective action 3.17 ) is absent but the regulators have e ( NL S 3.12 X α W ] (since the superconformal framework auto- E 3 , if the regulators couple to SUSY breaking in – 18 – Θ 2 2 2.5 d Z 1 4 here is really the vev of the superpotential ] — one cannot neglect contributions to the gaugino mass dependence in the gaugino mass, this effect would show up 2 / L ⊃ 3 2 , again reproducing eq. ( 14 / – m 3 . SW C m 12 , an appropriately SUGRA-covariant, super-Weyl-covariant, and F F e ) then yields the correct gaugino mass with on , but this will just give extra soft masses and goldstino couplings in = 0. In that gauge (and only for that gauge), there are no regulator S = 0), and is effectively what was done in the original anomaly-mediated is a chiral superfield with the gauge coupling as its lowest component NL i 3.13 ∗ ∗ X S M M ] (though not in this language). For any other gauge — including the choice h 2 , ]). The running of the coupling with the momentum scale is encapsulated by dependence in the associated goldstino couplings. 1 ) used by refs. [ 2 17 / 3 3.8 m At one-loop, the 1PI effective action for the gauge multiplet is One disadvantage of the 1PI action is that it will inevitably be non-local, since it One can avoid this subtlety of regulator contributions by making a gauge choice such It is worth noting here that -terms proportional to -terms, so eq. ( 20 the dependence of chiral version of thecoupling, d’Alembertian. which is This sufficient 1PI if action we depends are only on interested theappear in in holomorphic one-loop the gauge gauge expressions. fixing To of describe The superfield (see ref. [ breaking terms in theaction regulators that as does discussed not have inwill explicit section depend dependence on on agreement with flat spacemediated intuition, effects. whereas we are interested in isolating the anomaly- one loop, and extend the logic to sfermions atinvolves integrating two out loops light in degrees section ofus to freedom. extract On all the anomaly-mediated effects otherabout from hand, the the the action contributions 1PI directly, of without action having allows regulators to explicitly worry as we did in section nian effective action. However, thelevel, super-Weyl since formalism there is exists entirelysymmetry valid a (i.e. at variety it the of quantum is regularizationeffective not action schemes that anomalous). that exhibits preserve allWeyl Therefore, the of invariance). we the Here, super-Weyl relevant should we symmetries will be of write the able down theory the to (including relevant write 1PI super- action down to a describe 1PI gauginos at invariant Pauli-Villars fields, inB which case eq. ( 3.5 1PI effectiveWe action argued and above goldstino that there couplings is no way to make super-Weyl invariance manifest in a Wilso- B This is essentially thematically strategy sets used in ref.literature [ [ of eq. ( due to the UV regulators. Alternatively, one can regulate the theory with super-Weyl- mediated result. As discussedsuch in a section way to removeas the an that the vev This expression is manifestly super-Weyl invariant, and reproduces the familiar anomaly- JHEP09(2013)125 ) Φ in 2 3.8 = 0 3. / (they 3 − (3.22) (3.23) (3.19) (3.21) (3.20) ∗ m 23 C M ) yields the ) in 2 / 3.18 2 3 (they would only m , ( 2 ) gaugino mass and C 2 O D gauge freedom to set / 21 3 attains a vev: Σ i † m F ( . K C , ) into eq. ( 2 V . O † a ± µν ) is not as manifestly chiral . e D . This slightly complicated form is F β α 3.22 L C i is in fact ambiguous. All choices e 3.22 W † G W 1 F β 16 e i C µν . D K − σ L , is chiral and has Weyl weight a 1 3 eff β e α α λ G F D in eq. ( + D µ

eff W ) acting on 2 i ∂ λ e / F R e c 3 F 2 ˙ i αβ 8 µ √ K , m − , like equals the ordinary flat space d’Alembertian σ i can be fixed using the gauge choice in eq. (

) α – 19 – + 2 contains a goldstino if χ † , the choice of β 1 3 † ) and derive super-Weyl-invariant gaugino masses i a C 4 W C D λ K C W ˙ = ( † α a e β 1 3 1 8 3.18 λ D C , ) because it only contributes at D λ ( 1 χ i α m 1 2 D ≡ − 1 2 3.19 2 † = + α D is equal to the gauge choice for the conformal compensator C W 1 8 is only gauge-invariant for an abelian gauge symmetry. It can be easily )) and terms with spacetime derivatives on Note that e 2 C C − L ⊃ − / e † 1 ), though, and we will choose to work with . 3 22 in eq. ( 2 C / m 3 D ( R m → O ( e has a spinor index. O α ), but it can be verified to be chiral (up to terms that we have dropped at this order. So for the purposes of getting the ], and yields identical results. Plugging eq. ( W 3 α 3.19 ] never explicitly wrote down the form for W 14 ] for how the Kählerand Sigma-Model anomalies contribute to the 1PI effective action. 6 , 3 This gauge choice for In practice, though, it is much more convenient to use the As we will discuss further in section = 0. The remaining components of Ref. [ As written, this form of Strictly speaking, this is only the piece of anomaly mediation related to the super-Weyl anomaly. See ∗ 21 22 23 needed because modified for non-abelian gauge symmetries by appropriate insertionsrefs. of [ used in ref. [ expected soft masses and goldstino couplings from traditional anomaly mediation: never contribute at yield terms with derivativesthe on goldstino goldstinos, equivalence which limit). canas be in The ignored form eq. at of ( thisat this order order). in acting on goldstino couplings, we can simply make the replacement where we have dropped terms with superspace derivatives on multiple copies of In this gauge,gravitino-goldstino the kinetic graviton mixing andcurvature terms gravitino superfield to are worry canonicallygauge about. normalized (and in and fact We gives theretuned no can are contribution to also in zero). no this drop Similarly, gauge if the the cosmological chiral constant has been Note that the fermionic component of It is then possible toand expand out goldstino eq. couplings. ( M such that (to linear order in fields) described in appendix are equivalent at the canonical gauge coupling (including two-loop effects), one needs an alternative action JHEP09(2013)125 . ) 2 λ ). / 2 3 R (as m / 4.1 2 3 and 2 from (4.2) (4.1) m / (3.24) 3 m 2 ( / 3.5 3 m O 3 m − = ∗ would arise from M 2 / is not an order pa- , 3 i 2 m F / i 3 K m 1 3 g g β . − , Q ) = R , e λ 2 ) below), we can extract ( / 3 Z 4.6 † m 2 1 , c = CQ † i ∗ F i M is the superfield associated with wave function – 20 – EC 6 1 K 1 3 − Θ Z 4 d + = | 2 / Z ), we will need to construct appropriate supertraces to 3 R 2 / = m 2 3 will preserve (AdS) SUSY. L m | ( g g R β O ]), then the gaugino mass proportional to − is a super-Weyl invariant version of the d’Alembertian acting on 13 , ]. In this section, we want to show that this effect can be understood = e necessarily appear in the regulated SUGRA action. Crucially, 1 do not have associated goldstino couplings. does not come with a goldstino coupling, which tells us that it is not [ λ 12 2 2 2 2 / m / / / 3 3 3 2 3 to help identify all places where the goldstino field can appear. Because m m m m that depend on the lowest component of the chiral curvature superfield R F e for SUSY breaking in the SUGRA multiplet, which is valid at order R is a chiral matter multiplet, F is the beta function for the relevant gauge group. Note that the piece of Q g β The obvious choice for the 1PI effective action is Thus, we have seen how anomaly mediation is a necessary consequence of SUSY in- is in fact ambiguous at e space. With an appropriatethe gauge lowest choice component (see of eq. the chiral ( curvature superfield and effects proportional to mass-squareds, as well as unfamiliar one-loop goldstino4.1 couplings. The orderAs parameter already for emphasized SUSY breaking arameter number for of SUSY times, breaking the but gravitino is mass simply a measure of the curvature of unbroken AdS rameter We then use extract unambiguous “soft mass-squareds” andhand, “goldstino we couplings”. can With then these use tools the in 1PI effective action to derive the familiar two-loop scalar soft Here, renormalization, and chiral superfields. Ourpreserve key SUSY task and in which this pieces section break is SUSY. to To do figure this, out we which first pieces identify of the eq. order ( pa- proportional to as being a consequence ofderive the AdS sfermion SUSY. spectrum To by do constructing1PI so, a effective super-Weyl-invariant we and action will SUSY-preserving for follow chiral the multiplets. logic of section proportional to 4 All-orders sfermion spectrumIt from is anomaly well-known that mediation anomaly mediation yields sfermion soft mass-squareds at two loops the parts of variance and super-Weyl invariance. Becauseproportional of to the underlying AdSis SUSY not algebra, terms an order parameter for (AdS) SUSY breaking, so anomaly-mediated soft masses and proportional to an (AdS) SUSY breaking effect.was Had the we instead case worked in in refs. a [ gauge where where JHEP09(2013)125 R ). in 2 F / R 2 3 (4.5) (4.3) (4.4) F m ( O that do break = 0). So while 2 / 2 3 R m is an order parameter (i.e. eff 2 ). When SUSY is broken ), which has the goldstino / F 2 2 3 / 2 3 for simplicity. ..., m ). Thus, gravitino couplings m 2.36 ..., − + ..., NL , ): ..., = 4.4 + L = + X 2 Pl e G L + 2 R eff L R µ 2.22 e M G 1 F e 12 F σ G 3 λ M ∗ σ R R eff eff − from eq. ( F F F F R M eff = F 1 9 2 2 NL F i 3 2 3 2 / √ √ i X , since applying its equation of motion would : 2 3 3 µ √ − − ) into eq. ( R . In an arbitrary gauge, we will define m R − ψ µ = = = – 21 – Pl 1 12 σ τ ν 2.25 λ ) contain terms not relevant to our discussion. In M R − ψ ψ ψ = ). can yield effects proportional to ρ µ µ | terms since we only care about effects up to 4.5 | 1 12 D D to be independent of D R µ R is an order parameter for SUSY-breaking, it is helpful R ν µ 2 F ψ 2 µν ≡ σ σ Z 2 R D i σ / D F R 1 4 3 4 1 Pl Pl F for finite 1 − µνρτ 1 m − M M R Pl F − 1 is tuned to yield flat space or not. Since M eff F = 0 for unbroken AdS SUSY (i.e. , it is crucial to identify all places where the goldstino field can appear. R 1 controls the amount of gravitino-goldstino mixing in the super-Higgs F ) (instead of R F 4.4 is the Ricci scalar. Upon using the Einstein equation, this takes on the value ), namely the highest component of R 2 / itself does not break SUSY, 3 2 The most straightforward case is when there are direct couplings between the visible As expected, To better understand why The SUGRA multiplet does contain a SUSY-breaking order parameter at order 2 / m 3 ( sector fields and the SUSY-breaking superfield as its fermionic component.soft This case masses is and not goldstino interestingtake couplings for the our in wavefunction purposes superfield agreement since with it flat generates space intuition. We therefore only serve to reintroduce derivatives acting either on4.2 gravitinos or goldstinos. Goldstinos inSince the SUGRA our multiplet ultimate goaladvertised in is table to compute the sfermion soft masses and goldstino couplings where we have alsowhich used look the innocuous Einstein can equationis secretly from contain non-zero. eq. (SUSY-breaking) ( goldstino Thisnext couplings will subsection. when be The of ellipsesparticular, great of we can eq. importance ignore ( when any We we can also track ignore goldstino terms proportional couplings to in the to note that mechanism. This can bemotion seen one by can examining obtain the by various plugging forms eq. of ( the gravitino equation of and the cosmological constantm is tuned to zero,SUSY. then This distinction lies at the heart of the confusion surrounding anomaly mediation. regardless of whether for SUSY breaking, soterms is of eq. ( O where JHEP09(2013)125 ∗ M and (4.8) (4.9) (4.6) (4.7) R ) in the 2 / is always 3 3 . We find m or multiple C ( C F O 25 , R ), which are nec- F µ ∂ 4.5 , ), we find that ..., ., M appears in a soft SUSY- c + µ . 4.5 )). Upon picking out the ∂

! i . , ) then tells us any goldstino + h 4.2 F µ 2 eﬀ i b . † L 4.5 F = 0 does not break AdS SUSY, ! Ω , K e ) 2 , thus giving us an easy way to G i 6 µ

i ∗ R eﬀ R ..., √ itself can also contain a goldstino, ψ 1 3 L 8 F F F µ e M i 2 + G + − σ i C h = (see eq. ( 2 2 † √ , † i K 2 µ ! h Θ C D / ψ ( 3 1 3

F eﬀ 2 1 4 h , / L Θ + µ m F i 3 − ) carried out to linear order in ﬁelds, which e σ 2 G

χ m = i

√ E ! i i 3.8 ∗ – 22 – F K eﬀ Θ 2 i h L M 2 F 1 3 Θ + e d K 1 3 2 G

− √ Z 1 1 3 , but these are most easily tracked by making the replacement R 1 2 E F = by super-Weyl invariance, but eﬀects proportional to = Θ + + Ω 2

/ C = 1 + 3 E M R 1 6 m Θ F C 4 only occurs (without derivatives acting on it) within the SUGRA 1 2 − d 24 = (which has vanishing vev) may have associated goldstino couplings, but they R ]. Z = = ∗ F R , and the components of these superﬁelds can be extracted by the µ F µ 48 ]. In this gauge M R µ , G 1 3 ∂ G 47 23 − 6= 0 does.

and i F i R K

does not have hidden goldstinos. E Extensively using the gravitino equations of motion of eq. ( The most subtle case is to identify goldstino fields hiding in the SUGRA multiplet. Somewhat less obviously, the Weyl compensator can be written as: Terms containing There are also goldstinos lurking in µ 25 24 since 2 will always featuregoldstino a equivalence derivative regime, acting and can on be the ignored goldstino. here. Such terms will always be of where the ellipses includegravitinos terms or containing goldstinos. Fortional simplicity, to we the have assumed metric, as that it the is Ricci in tensor any is homogeneous propor- space. track such goldstinos. superfields methods of refs. [ G whereas These arise through theessarily gravitino SUSY equations invariant. of The motion SUSYarising transformation shown in of in such eq. eq. a ( ( fashion must be accompanied by an This gauge choice clearlybreaking shows that term, wherever itaccompanied will by have ando associated not have goldstino associated goldstino coupling. couplings. After Of all, course, is the minimumSchwinger necessary terms to [ havegoldstino canonically-normalized direction, neglecting Einstein-Hilbert other fermions, and and dropping Rarita- terms with multiple goldstinos, and different (super-Weyl) gauge fixingsit give different convenient goldstino to dependence work in in the gauge where This is effectively the gauge choice of eq. ( JHEP09(2013)125 , † 27 X F )). (4.12) (4.10) (4.11) ). This ), not Θ 2.31 † 4.11 (see eq. ( ) for Θ (Θ χ, with infinitesimal 1 − eff 26 4.10 /F µ 2 D / ) can be thought as the 2 3 , a product of SUSY-breaking µ i σ m F † i 4.12 χ K f 2 = 2 C / ). This action accounts for the 3 i eff m + 4.1 /F . F R 1 . eff F − L 2 F eff e 2 G L − ∗ F e √ 2 G F by first finding the vevs of these fields, and . Finding the Einstein frame is more difficult beyond a √ µ appearing in eq. ( − – 23 – 2.1 Θ + G − C = φ → ∗ ). In the context of anomaly mediation, these con- 2 φ ) will have associated goldstino couplings. These can Θ s 2 m , and / ( 3 is some soft mass), corresponding to corrections to the R −C O m , ), so we can neglect any further SUGRA corrections. In m = 2 C / 2 3 , not m . The tachyonic scalar masses are removed by a SUSY-breaking NL are all ( ), one is really only making the replacement of eq. ( ı i ¯ 2 O breaking X (where ∗ C − F 2 ı ¯ K /p ( 2 when uplifting from AdS to flat space. This indeed has a corresponding SUSY i m φ L F i ∗ φ (but crucially K R R F F , or i The simplest application of this method for finding goldstino couplings is the tree- Note that with this particularly convenient gauge choice, we can identify all of the In practice, the use of gravitino equations of motion is less than transparent, which is the reason why we The situation is more subtle for terms with coefficients like F i 27 26 and SUSY-preserving effects. Inarises such since cases, for one(Θ), only recalling makes that half we of have the a transformation hermitian of action. eq.relied ( on the Einsteintree-level, frame however, lagrangian which in is section whyequations we of choose motion. to work in SUGRA frame in this section and exploit gravitino tributions are already particular, at thisd’Alembertian order in the flat operator space. terms at order field self-energies: where the coefficients sociated with goldstino couplings,terms we now can need arise to fromquantum consider corrections the what coming possible 1PI from loop SUSY-breaking effectivemust diagrams be action of careful massless in to particles. include eq. bothin For ( this local a reason, and 1PI non-local one effective terms action. when For considering a SUSY-breaking chiral multiplet at quadratic order in fields, there are three coupling 2 goldstino coupling in flat space proportional to 4.3 Supertraces and theNow 1PI that effective we action have identified our SUSY-breaking order parameters and how they are as- This will allowwithout us having to to identify wrestle with goldstino complicated couplings component manipulations. directlylevel from analysis the of sfermion section spectrum, K be found bySUSY making parameter a global SUSY transformation of those terms goldstino couplings in then making the replacement At the component level, this implies that any terms in the lagrangian with coefficient JHEP09(2013)125 = T a that C (4.16) (4.17) (4.13) (4.14) (4.15) T = s vanishes. C or S S . 2 / ) under SUSY, with 3 . F. 1 m Q − ) on the full lagrangian, 4.12 µ which is invariant under ) SUSY-breaking contri- is not a good measure of 2 i 2 / D † i 3 C 1 C χ m − ( m µ ). A non-vanishing value of O D † µ T Q σ L e NL G , i f X . This will not be the case, however, C T eff Q . α 2 T G f F )), D C ) and is an unambigous measure of SUSY- − 2 . In the following subsections, we will find = ˙ + 2 αα to exclude such contributions. a − C / µ ∗ 4.11 C + s 2 3 a from anomaly mediation is unambiguous and T 4.13 σ C C G m + Q χφ S – 24 – † NL s in our explicit calculations. L yields vanishing = C e → G + 2 X 2 S T ˙ and † α = T S Q ) does not break SUSY in the limiting case (see eq. ( D . In the context of anomaly mediation, non-vanishing . Thus, a single coefficient S S i 2 = 2 eff C − G eff Λ F θ = S L 4.12 F / S = 4 e G 2 G d 2 i S G X √ C G F ), there will be non-local goldstino couplings. In the case of Z = − but = 2 1 = Λ 4.12 T T = ]) T goldstino L ⊃ 14 G L ] considered a similar supertrace over the also does not break SUSY if it arises in conjunction with a fermion mass term after an = 0 but 17 s . For example, at tree level in AdS SUSY, one would use the appropriate S C frequently arise but they in general depend on how the theory is regulated. In 2 / 3 T ), which we take to be m The simple field redefinition (or the appropriately super-Weyl- and SUGRA-covariant = 4.13 Analogously to eq. ( Of course, there is another independent combination of the The non-local action in eq. ( Obviously, 28 1 . 28 − AdS f AdS SUSY transformations (which haswhich terms would proportional yield to these relations to be modified, but always by terms proportionalauxiliary field to redefinition. We will therefore define For global flat-space SUSY, for AdS SUSY ordo for not SUGRA, break where SUSY.λ there Such can effects be will non-vanishing always values be of proportional to the inverse AdS radius global flat-space SUSY, one can simplyinfinitesimal transform parameter the terms in eq. ( which yields values for contrast, we will find thatirreducible, so the we supertrace will mainly focus on This is the unique independentscalar choice mass which in vanishes for AdS tree-level discussedis SUGRA in still (the section a tachyonic SUSY-breakingThis effect, can and arise can most be notably present from even terms like when the supertrace butions to the self-energy for the components of vectoreq. superfields. ( SUSY-breaking by itself. On the other hand, the supertrace is invariant under the transformation ofbreaking. eq. ( ref. [ C equivalent, see ref. [ eliminates all three terms for JHEP09(2013)125 = 3 and c (4.18) S = to find 1 ] (which c − 14 which would , 4.2 i α c W = ] to extract the ) U 0 i ˙ α c † U 23 U ) 30 . † D P ) 1 V P − ± P (( ( . For example, requiring e C i However, this procedure 0 2 P c α c U 0 6 α D c + 32 1 D U ) ) †− U ) + in whether the regulators feel SUSY † R P C , respectively, and are defined in 8 U P ˙ ˙ α α ) ) )( 2.5 In order to actually determine the α † † α − P e 2 G by recursion and find the component P P G † 31 (( 0 7 ( ( † c 2 4 D U c c ( P n − α 6 = 1 limit (corresponding to + + c e (and their hermitian conjugates) are super- , and D U 1 = 0. C ˙ 29 + U U α α R e C ) α possesses a sensible analogue of integration by † P † 1 D U ) e 1 – 25 – = 0. This is the choice made in ref. [ G ) . † C P ), 2 e 2 i †− 1 8 in the U P P ) has not been yet defined. Its definition is the c R , we choose not to impose any constraint on them. as a Taylor expansion. ˙ ( ( P i α 8 a C ( 0 1 0 3 are not. c − ) vanish in the limit of global flat-space SUSY, so α ˙ 0 5 α c c 4.1 D † Q , and c e − a S G ) U + + ˙ 2 D that are charged under a gauge group, the operators of D α G P † e + in the limit of global flat-space SUSY. Given a generic † 4.18 1 α , ( U U = − D and that = 0), though it is not chiral. e Q ) U † G P ( Z PP 8 ) e C are left completely undetermined. We could impose certain 1 4 c P 2 P ˙ U C α † ) i † + † − 1 α = c P P P e 5 C U ( ( G ( , which would lead to constraints on the c 5 7 8 1 3 e 1 and all other P c c c c c = † is generically ambiguous, our final results for the supertrace − 2 , there are a limited number of options (neglecting fractional powers + + + + + P c is a super-Weyl-invariant version of the d’Alembertian acting on scalar e . We could then determine = U = 2, e appearing in eq. ( 6 / U c 1 ) by treating U e e − ), though it does not satisfy e are independent of the 4.1 = 4 S . For matter fields 7 c G is chiral for chiral C ) would need to be modified by appropriate insertions of = 6 U c and e 4.18 At this point, we could use the full machinery developed in ref. [ Many of the terms in eq. ( The operator S There could also be additional possible operatorsAnother proportional obvious to candidate the is field strength And we have. This ambiguity is a reflection of an ambiguity in how to write down a SUGRA-invariant 1PI effective 2 = 30 31 32 29 / , one would have to explicitly take into account virtual effects to all orders in a specific i 4 action, which is inbreaking. addition to the ambiguity discussed in section not give any contributions to self-energy corrections or goldstinoc couplings at the desired order. components of form of eq. ( is overkill for our purposes, since we will ultimately use the trick in section parts would set they denote c regularization scheme, which isfor beyond the scope of this paper. Because our final results eq. ( the associated coefficients desirable properties for that The operators and superfields Weyl covariant versions of appendix superfields, which reduces to spinless superfield of derivatives): The operator last ingredient we needwill to see computing that sfermion while corresponding soft goldstino masses coupling and goldstino couplings. We 4.4 The super-Weyl-invariant d’Alembertian JHEP09(2013)125 ). T C so NL . By 33 (4.21) (4.19) (4.20) (4.22) R X ]. This F . 0 5 25 dependence c 7 c and + i Therefore, for 5 . c (which have no = 0 [ F = 0. Because a i 34 † 2 + in place of i ), which is clearly / i K ). ] (though they are 4 2 3 C )). This , can be linear, 1 αβγ c , C K 2 m h ), and contributions to . / 4.20 C + R 2 3 C.9 vanishes. 4.14 W R F 0 3 7 F m c c 4.15 . Starting with unbroken = 2 d + + R ˙ ) (with α 2 3 ) limit without missing any α + / of eq. ( c , we can use interpolation to 2 3 (which do), we need a limiting 2 G R . Here, one can use the gauge † 4.16 S + m R i F i D 0 1 F 2, 2 → ∞ d term (see eq. ( / K / h 2 3 7 are well-known [ / + c R ≡ Pl defined in eq. ( 3 m 1 S 1 to four: 1 4 M d T m 12 0 1 e d = ] is dramatically simpler than the SUGRA 2 + − = 0, one can use the global flat-space SUSY 52 coefficients via R 2 – and ≡ ) or by the i R D c 2 – 26 – 1 4 49 i , 4.18 do not appear in the 1PI effective action. 1 , keeping careful track of all of the factors of F = 1, d i 24 R , d U 0 6 ), the effects of all such terms can be eliminated by K − C c e 2 or ), it can be shown explicitly that they have no effect 1 3 a / 2 2 2 3 + / / D + 3 3 0 2 a m 2 c ( m m / D for two different values of ( 3 O + ) or they take the form of eq. ( = 0, and since O = S m 0 1 c terms with supercovariant derivatives on ∗ = 0 in rigid AdS, so the term associated with ≡ we need only consider the first term in eq. ( ˙ 4.13 α ≡ M depends on exactly how one regulates the theory. In unbroken rigid AdS, this α + S 0 1 is dimension two, its only dependence on ). At G SW . Thus, it is sufficient to discuss two limiting cases where the behavior T . Furthermore, this is exactly the term which is already considered in 2 F rigid AdS R i / S c 2 3 e C F , d † to set 1 6 m should still be considered SUSY-breaking. It can be readily shown that non-zero values ( c C R O SW + F = coefficients are related to the F for all 2 i c d S = flat + e C 1 simplifies. F c U ≡ The second limiting case is unbroken SUSY in rigid AdS where The first limiting case is flat space but arbitrary will only be induced by the first line of eq. ( Fractional or negative powers of Terms of the latter form do contribute to the parameter e 1 T 33 34 d proportional to of implies that the valueambiguity of does not arise; where the case which captures terms proportionalSUSY to in the scalar AdS, curvature wephysics. can The rigid luckily AdS consider SUSYalgebra, algebra the corresponding [ rigid to ( thereduces the limit number of independent operators in the anomaly mediation literature, sousually the stated results as for being the soft mass-squared and notflat the space supertrace). analysis cannot distinguishassociated between goldstino effects couplings) and proportional those to proportional to physics up to (up to boundary terms).transformations like At eq. ( In either case, they yieldthe no purposes contribution of to finding the supertrace independent of the contained therein. In fact, one does not even need to be all that careful, byThere noting that is a limited set of the possible terms in the parentheses that can contribute to determine of choice algebra to find the components of super-Weyl invariance, we know our results can only depend on two parameters: Moreover, because if we know the behavior of goldstino couplings once we know the dependence of the supertrace on JHEP09(2013)125 , µ 36 at b F, φ, 1 1 Q ) χ − − (4.25) (4.26) (4.27) (4.23) (4.24) χ 2 e 2 in AdS, ( √ = 0. This √ , ) Z U ) i 7 ) except for c 7 e c c + Θ + Θ − F 4.18 F 1 (1 φ 1 φ (1 + 1 − γ 1 − ) γ − is (nearly) chiral: − ) R ) , ) R F ) γ Q ) γ F γ ) γ + F e 1 + ). ), so one will find additional + 2 10 2 ( + 2 2, and all other 10 / − 2 γ / 2 3 / − Z γ 3 2 3 − 4.13 2.24 ) m + ˙ γ − 7 + ˙ ( γ . m 2 c ( 1 2 2 + ˙ = γ + ˙ 1 2 35 γ ( 2 4 ( 2 2 − c γ / dγ − γ ( ) log 2 3 (1 + ( 2 2 ) = SW γ 2 d / ) such that m γ F 2 3 3 2 7 ) 2 SW i 1 8 c 8 1 c 7 , we first find the behavior of c m F c + Θ ≡ 2, 1 8 + 1 8 + / ˙ φ γ 7 4.4 1 + c + , we can now find an appropriate super-Weyl (6+4 γφ − γφ − 2 F − / F 1 3 1 ) = γ SW 4.4 – 27 – , − 7 + (6 + 4 − m F 1 c + ˙ Z c γ 2 1 2 1 2 . This approach makes it clear that the results in AdS space γ − γ 2 γ 2 2, + ˙ / ( by Taylor expansion. − − )) will depend on no parameter in eq. ( / log log / 3 , 2 (1 3 7 F, , up to the transformation eq. ( SW d d c m i γ F F 2 2 SW Q c F m ( 2 e ) γ F 4.13 F 1 2 1 2 2 e 2 2 1 + 1 8 ≡ 2 SW ( 2 + Θ SW − − − γ F F Z χ φ φ 8 1 + Θ = + Θ + Θ + 1 2 2 ) ). χ 6 F √ c 2 1 ) ) + ( 4.1 − √ Z + Θ = 0. ( ( γφ γ f + Z φ Z + Θ C SW )). As outlined in section SW e = φ 2 F F ) = ) = / 2 1 2 3 Q 1 2 ) flat m − − ), so we neglect such terms in the following. This choice also has the appealing feature of ( ( e ) is first non-zero at one-loop (two-loop) order, our results will hold to any loop 2 by recursion, and ( / Z φ F γ O 3 3 ( ˙ ) does not commute with SUSY transformations due to eq. ( rigid AdS Z ) in the flat space and rigid AdS limits. Since we have argued that the final result U m ≡ ≡ ( γ 2 e n ( ) = / e ( e φ O Z e F 2 3 Applying the procedure outlined in section e This choice corresponds Alternatively, one could simply work with the component form of the AdS SUSY lagrangian. In that Z ( m , we will only present the answer for a choice of ( 36 35 Z 7 case, terms proportional to positivemust powers be of completely independent of the is not chiral outsideor of at AdS space, butautomatically deviations setting from chirality only appear in terms with gravitinos, order (at invariant interpolation valid for any spacetime curvature, where the anomalous dimensions are defined as While O (up to the transformation of eq.c ( find 4.5 Soft masses andWe now goldstino have couplings all forwhich of chiral follow the multiplets from ingredients eq. to ( determine the soft masses and goldstino couplings One can then use the AdS SUSY algebra to easily extract the components of JHEP09(2013)125 k F k with 2 K / (4.32) (4.34) (4.30) (4.31) (4.33) (4.35) (4.28) (4.29) i -terms 3 F A Q m 1 defined in − . ij ∗ b C F F - and ) B γ ) φ 7 ) (dropping factors ∗ c φ to extract goldstino 4.1 ) , 2 , / transformations. The F. 2 3 ), not from 4.2 1 Σ k ` − m F 4.6 , F F + (2 + 4 vanishes in flat space but is 2 k ` + , ∗ / γ . k 3 R K ., K R k c F 1 3 Q 1 3 + ˙ γF F m . F j ) ) is super-Weyl invariant, we are ) 2 k + j R + + φ, γ Q γ i ( K 2 + h F 2 2 + 2( / / / 4.32 ) k SW 2 3 Q 3 3 + j 2 + 2 37 SW φ

F i γ m 2 j m m ) ijk φ F γ / φ γ λ 2 3 + 1 8 i i ) ) 1 6 . These are the familiar one-loop anomaly- SW φ ]. m − j i γ k + 2 + ( F γ γ 2 + ) F 7 ijk , φ − (2 i c γ j 1 ( ∗ + A + 1 2 2 + ), we ﬁnd 2 – 28 – K i φ 1 6 Q j ij − ) 1 3 i γ − γ − µ ( γ 1 + + ) Q 1 6 (2 + 4.10 + F ij 7 j 1 2 ij − 2 c 2 + i µ φ = / ) and use the trick in section µ γ i φ 3 1 6 → − + ( 1 2 ∗ ( φ ij m 4.6 b F ij = F in ﬂat space, and ij ijk =

γφ (2 + 4 λ B µ -term has a corresponding goldstino coupling ) = − 2 ij ) is still invariant under the super-Weyl -term and the goldstino coupling is proportional to / 2 1 2 1 2 1 + R b B 3 − ij W B F χ SW B = ⊃ = m γ 4.34 µ and a superpotential F + . Because the result in eq. ( ij D + ˙ i V = 2 2 ijk µ γ / / 2 B equation of motion trivial, but induces non-zero 2 3 3 A σ γ † -terms will have corresponding goldstino couplings proportional only SW m m − F ( A F ( iχ 7 c − 2 SW 2 − φ F 1 8 1 - and ∗ ), this goldstino coupling is independent of tuning the cosmological constant. B ) if there are superpotential terms. Generalizing to multiple fields + − φ 2 2 / / but not to 3 3 = ). Performing the shift in eq. ( i ) for clarity): m m L for unbroken SUSY in AdS. F ( ( i 2.1 ( These It is now straightforward to expand the superspace action of eq. ( 2 Note that the result in eq. ( O O / K Z 3 2 37 emphasizing the role of AdS SUSY. factor arises by isolating the goldstino direction out of the fermion in eq. ( eq. ( At The difference between the mediated results that can be found in ref. [ to free to choose thecouplings. gauge For of example, eq. the ( where we have expanded the associated scalar potential terms are which renders the at anomalous dimensions To extract the sfermion spectrum, is it helpful to perform the shift m of remembering that JHEP09(2013)125 ]. 1 ): . For (4.36) (4.37) (4.38) 2 / defined R 2 3 ). After F S m , modified G ( , this asso- i 4.14 2 O S (arising here from ]. The difference R . F 54 ) i . . Like , ) γ ] have no correspond- 2 R expected from ref. [ R . / 1 As discussed further in − 2 3 F F M ) S i ∗ m is a cross term between a (2 39 + γ 2 M defined in eq. ( 2 / / − 2 3 SW 2 3 S ) is the true scaling dimension F m (2 m γ arises purely from the structure 2 ) for related reasons. / 2 transformations. )( − 3 − − i / 2 3 Σ γ m 4.29 , this second term vanishes in flat F k ` m − terms to be partially cancelled off by AdS R F F ` k i 1 12 (2 γ K K − 3 1 1 3 = 2 ), we can read off the value at that the tree-level result is RG-stable once and ˙

+ + to extract the goldstino coupling k R i B 2 ) could have been anticipated, since anomaly 2 γ F F / / 4.29 γ . As anticipated, this diﬀerence vanishes with 3 – 29 – k 3 4.2 + R m − K m 2 1 3 / )) does not implicitly contain ) is invariant under 2 3 k 2 + / m F 4.5 3 2 k 4.37 / m 3 i K ˙ i γ m ˙ γ terms are known to be RG-stable from the general arguments

1 4 i i 1 ˙ γ 12 − γ 4 1 Because ]. While we have not computed them explicitly, such boundary terms are − = 9 − and S = i 38 i = . S i proportional to (2 γ i ] and will be suppressed for almost no-scale models [ is G Q − G S † S i 53 , the result in eq. ( S S Q G 37 , as it is intimately related to SUSY-preserving anomaly mediation effects in ], while we argue in appendix 2 / 40 = 0) [ 3 20 and , this whole expression is RG-stable, as it must be since it comes from a 1PI ): – i m B S SW 17 F 4.17 = 0, the two-loop anomaly-mediated masses familiar from ref. [ We can again use the trick in section The key result of this paper is the sfermion supertrace i The same factor appearedFor in any the auxiliary negative field curvature, shift one of eq. expects ( the As in footnote i 38 39 40 K boundary effects, asnecessary in for ref. the [ structure of the AdS SUSYthe algebra gravitino to equations be of maintained motion in of the eq. unbroken ( limit. which is independent ofvanishing the curvature AdS SUSY. Whereas the second termof proportional unbroken to AdS SUSY, theSUSY-preserving first and term a proportional SUSY-breaking to effect and vanishes in the no-scale limit. unbroken. Curiously, such two-loop(where goldstino couplings alsobetween vanish in the no-scale limit ciated goldstino coupling isarise RG-stable. because there Thethen are tree-level lifted and SUSY-preserving by one-loop scalar anh goldstino masses amount couplings proportional in to theing the goldstino bulk SUSY-breaking coupling, of as order AdS, such parameter masses which are are also present in the bulk of AdS when SUSY is in eq. ( As advertised, there are no goldstino couplings proportional to appendix effective action. The ˙ in refs. [ one accounts for goldstino-gravitino mixing. The second term isstarting the at tree-level one-loop mass ordera splitting to contribution in to include AdS the discussedmediation effectively anomalous in tracks scale-breaking dimension. section effects,of and The (2 the fact operator thatspace, we which have is why it does not appear in the original literature. performing the auxiliary field shift of eq. ( The first term is the usual two-loop anomaly-mediated result for JHEP09(2013)125 R F has : 2 / (4.40) (4.42) (4.41) T 2 3 m . ) and the ) . − R R i γ F = F i 2 4.15 / is independent γ R + 2 3 7 F 2 T c / m 2 3 2 7 c = 0) (4.39) and SUSY-breaking m dependence vanishes 1 + ( 2 i 2 i i 7 / γ T − G c + 2 3 1 + , arising as it does from K 7 2 h c -violating terms (gaugino / m + 3 k 2 ). This indicates that the R i m F 1 + defined in eq. ( γ k ) 4.18 7 K + T ] to understand anomaly media- c = 0), all 1 3 = 0 then 2 3

i has important effects at all orders. i i k i + (flat space, S i K F 2 K h / k h 3 K = − G + (2 + 4 m 1 3 i for various values of the curvature. The i R S γ 1 + F k 2 + ˙ F / . k defined in eq. ( 2 i 3 – 30 – γ = 0 in the flat space limit with K 7 m i i c

˙ i γ i i γ 1 4 K k ) γ h is always proportional to 7 ) F + c 7 k . i c T : γ K , D 2 T 1 3 / − , 2 3 G 2 2 + / m T − G 3 2 are shown in table 2 ) + (2 + 4 , but these two effects are difficult to disentangle because / i 2 m + (2 + 4 i 3 S γ R -terms), anomaly mediation generates soft masses proportional i after tuning for flat space. Having successfully isolated these two / i i γ 2 3 ˙ γ G F m 2 B γ − / 4 1 m + ˙ 2 3 For the sfermion soft mass-squareds, the situation is far more inter- + ˙ 2 2 i − − / m 2 i γ and 3 γ 41 = (2 = = − i m i S S S 1 i i 1 8 24 1 8 S G − − − − G depends on exactly how one regulates the theory (i.e. on the correct choice of i = = = = 0 in rigid AdS SUSY. -terms, and S i ˙ α T T T A i i without corresponding goldstino couplings, making it clear that these are SUSY- T α G 2 / G 3 − G . Furthermore, in unbroken AdS SUSY ( For completeness, we give results for the parameter Results for 7 i Strictly speaking, we have not carried out the calculation of gaugino masses beyond one-loop order. m c for a given regularization scheme). Note that if T 41 e effects proportional to happens to equal effects, we see that the familiar two-loop anomaly-mediatedWe sfermion sketch how soft to mass-squareds do this in appendix tion as being amasses, SUSY-preserving effect in AdSto space. For the preserving effects. esting, since there are SUSY-preserving effects proportional to of since 5 Conclusions This paper completes the task originally started in ref. [ As expected, the difference the structure of AdSresidual SUSY. dependence However, on these thevalue results of parameter are harder to interpret, since the goldstino couplings. Indeed, the difference associated goldstino coupling While anomaly-mediated sfermion softloop mass-squareds effect, are this colloquially expression described makesconstant as it to clear a zero, that two- since this anomaly is mediation an has artifact important of tree-level tuning and the one-loop cosmological effects on answer is particularly striking when JHEP09(2013)125 ) ]) 2 / 2 3 14 m ( O can have T ) is indeed an is unambiguous, S 2.31 . 2 / 3 ] (and implicit in ref. [ m 13 , 12 , we identified the independent trace S ) offers strong evidence that AdS SUSY (and – 31 – 2.31 , though the physical significance of that dependence is not SW F This paper has focused on formal aspects of anomaly mediation, and therefore has not Along the way, we have learned a number of lessons about AdS SUSY and SUGRA. which is perhaps known to SUSY aficionados but is unfamiliar to us. Even in global sunori Nomura, Riccardo Rattazzi, andvigorous Raman discussions Sundrum. on We thank these Senarath issues.of de Energy J.T. Alwis (DOE) and for under cooperative Z.T. research aresupported agreement supported DE-FG02-05ER-41360. by by J.T. the the is DOE also U.S.F.D. under Department is the supported Early by Career the Miller research Institute program for DE-FG02-11ER-41741. Basic Research in Science. mediation will yield irreducible physical effects proportional to Acknowledgments We benefited from conversations with Allan Adams, Daniel Freedman, Markus Luty, Ya- T flat space SUSY, it wouldon be phenomenology. helpful to Finally, know(weak the what scale) big effects SUSY a question is non-zerobroader facing value in question, particle of but fact we physics realized can in say in that 2013 nature. if is (AdS) We SUSY whether and of SUGRA course do have exist, no then anomaly insight into this effects break SUSY. Ideally, onegoldstino would couplings want could to be findthe measured, an underlying since experimental AdS context this curvature.even where would probe these give Measuring the an such value experimentalclear of a handle to coupling on us. to Third, two-loop in precision addition would to the supertrace was motivated in part by the possibilitythe of sequestered sequestering, and limit one would is likebe to physically know useful obtainable whether to without fine-tuning. knowattractive IR whether fixed To point, that the as end, irreduciblewe one it would goldstino have expect would used coupling in conformally in goldstino sequestered eq. couplings theories. ( Second, as a probe of which effect preserve SUSY and which Weyl-invariant 1PI effective action in hand,in there how is to residual write ambiguity down starting a at yielding SUGRA-invariant theory. the Luckily, the same supertrace soft mass-squareds known in the literature. addressed a number of important phenomenological questions. First, anomaly mediation Second, to incorporate quantum effects,Unfortunately, one it has is to work impossibleeffects with of to a anomaly write regulated mediation at SUGRA downfields tree-level, action. a since at there loop-level. Wilsonian are Instead, important actionmanifest, we effects countering that of used the the captures a (gauge-dependent) regulator claims 1PI the inabout effective refs. full the action [ non-existence to of anomaly make mediation. super-Weyl Third, invariance even with a SUSY-preserving, super- needed to preserve the underlying AdS SUSY structure. First, the peculiar behavior ofirreducible anomaly goldstino mediation coupling is already innot evident flat eq. at space ( tree-level, SUSY) and is the the correct underlying symmetry structure for SUGRA theories. are accompanied by tree-level and one-loop goldstino couplings, and all three terms are JHEP09(2013)125 ) ), = i i A.1 2.3 (A.2) (A.3) (A.4) (A.1) W h , and thus Pl . M is not a gauge 3 , a chiral field / Φ Φ K F − e 3 ≡ − ]). Ω equation of motion to find , ]. This formalism is reviewed in ] allows us to use the “global ..., 40 47 Φ , 59 – Φ F 38 39 ,F i ..., χ ..., i + h.c. + This gauge is + j K j 1 3 43 Q ], but it is only valid in flat space. Given this , i Q i 1 47 Q Q i + h.c. + i ij W ij Ω to build a superconformally invariant action at W – 32 – h ] via a conformal compensator W 3 h 1 2 Φ 3 Arg 58 Φ 1 2 , we can solve the – i/ θ + 2 − Φ i + 55 6 d / i 2 Q / ı ¯ K/ Z 3 † Q e ) to find the pertinent features of supergravity, including Q m + = → = = proposed by Kugo and Uehara [ A.1 Φ i ΦΩ Ω Φ Q † W Φ Here, the extra gauge redundancies of conformal SUGRA are gauge θ 4 ) represents the action for the gravity multiplet as well as couplings of d is an auxiliary complex degree of freedom, corresponding to the complex 42 of supergravity. Unlike in the super-Weyl formalism, ]. ... Φ Z . The extra terms are suppressed by at least two powers of 3 F M = ... L 44 + 2 / 3 m ] using two-component fermion notation. = 0) is The most general Kählerand superpotential for unbroken SUGRA in AdS (i.e. The gauge choice for 47 i = For details on the conformal compensator formalismAn see alternative refs. gauge [ fixing was proposedFor in simplicity, we ref. assume none [ of the visible-sector fields are singlets. The physics does not appreciably i 42 43 44 Φ K ref. [ limitation, it would obfuscate the derivation of thechange sfermion if spectrum there in are curved singlets, space. as long as there is no SUSY breaking in the visible sector. where the ellipses representand higher-order rescaling terms. the InsertingF fields these expression into eq.irrelevant ( for our purposes. It is then simple to read off the cosmological constant, as well h where the field auxiliary field degree of freedom. scalar masses andsupergravity goldstino effects couplings from the in terms curved in space, the ellipsis. without having to worry about Here, we use globaland superspace the ellipsis variables ( tothe express matter only fields the to matter the gravity parts multiplet of (see, the e.g.,superspace” action, refs. [ terms of eq. ( with conformal weight 1.tree-level (dropping We Yang-Mills can terms use for convenience) In this appendix,working we in provide the conformal a compensatoranalysis third formalism in of derivation SUGRA ref. of to [ connection thefixed to to goldstino our previous recover couplings minimal in SUGRA eq. [ ( A Goldstino couplings from the conformal compensator JHEP09(2013)125 ) ] → NL 35 and – (A.5) (A.6) (A.7) (A.8) (A.9) NL X (A.12) (A.13) (A.14) (A.15) (A.16) (A.10) (A.11) NL 33 X , † NL X X 27 , ⊃ 26 . We also have . 2 K / ij 3 is never a singlet , ) for scalars in un- M m j -terms can be easily , 2 NL 3 / Q A 2.9 3 i √ X NL ): m Q = X † NL − k i A.1 X X Q = i j F i ¯ = 0, the Kählerpotential and , h ]. X Q ij , 2 i , X Ω

are strongly constrained 2 NL

27 Q Ω h , L X h i 2 X X e / Ω G 26 NL . 2 3 F + [ 2 2 / X X terms by using our freedom to perform 2 / X m ijkX 3 , / 2 3 F † NL 2 Ω 1 3 2 m W

m NL h NL X m 3 √ 1 3 − i i − 1 6 X X ¯ i − . Upon rescaling the non-linear ﬁeld ij X + X + 2 +

Ω Ω θ † NL i Q 2 ]. The scalar masses and W h h ¯  – 33 – k / 2 . 3 X † X

, 3 [ 2 X , ¯  F Q / i + i Ω Q i X NL j − m 2 3 δ h Q ij 2 F NL ¯ 1 3 Q  m / X NL = = = i † Ω 2 3 i X = h i i i + − X j Q -terms proportional to the fermion mass matrix. Q m X 2 Φ , X i V i i /

2 can be made real by using our freedom to rotate B Q h F NL 3 2 F ij Pl W ¯ j X h X i h i h ijk − = 1 m Ω X W M n X ¯  ¯ h X i h + 2 W i k + / ¯ h 2 W + X 2 2 3 Ω / i h / i 1 6 M 3

X 3 3 + ij m ik Q Ω m 3 m ⊃ ⊃ h W and M − − h → ⊃ − ⊃ Ω i W i = = = = X Ω ¯  i Q 2 i W ij ij Ω h V B m h M is the goldstino. Because of the constraint and integrating out auxiliary fields, we find from eq. ( L e G Φ / The Kählerpotential and superpotential will also include direct couplings between SUSY breaking effects then lift AdS space up to flat space. We represent the source NL where we have eliminated anya possible transformation visible matter fields andof the goldstino SUSY couplings breakingunder for any sector. massless of visible the Forcan sector gauge simplicity, add fermions we symmetries are (e.g. start in the our theory). study In this simple case the operators we The amount of SUSY breakinga to canonically-normalized achieve goldstino flat with space mass is 2 thus enforces the condition X The coefficients perform Kählertransformations. A canonically-normalized goldstino (i.e. where superpotential terms involving the non-linear field Thus, we recover thebroken universal AdS tachyonic SUGRA, soft as mass-squared well in as eq. ( of SUSY breaking in the hidden sector by a non-linear goldstino multiplet [ as the fermion and scalar mass matrices: JHEP09(2013)125 † NL X j (A.21) (A.22) (A.23) (A.17) (A.18) (A.19) (A.20) Q i Q , The resulting , or equivalently i Φ and ) (multiplied by NL 45 † (i.e. the universal X Φ j L A.11 † NL Q e term, however, cannot be G i i X χ Q

i † NL ¯ X ), the fermionic component K X j , ) L Q ijX X ... i A.2 e G Ω Q i +

/F , . i χ 2 j /

i + χ , 2 3 ı ¯ ¯ φ ¯  X ∗ i 2 m / δ † NL φ ¯ X 3 2 ijkX ). The i 1 3 / X 2 3 m Ω W

2 h +

. Less obvious is that there are additional A.2 ). . m ¯ ¯  2 2 X 2 i L / / → ij NL 3 e 2 3 , we expanded this result to all loop orders, 2.3 G m + 2 ) Ω = 0. i m m ¯  X 4

† L 2 i ⊃ – 34 – 3 j 3 X e G ¯  2 NL + m i − √ Ω Q µ i = 0 limit. h a X ∂ = = = 1 3 NL Q µ ij ¯  ¯  i i 2 i could have been eliminated by a redefinition of . In the gauge from eq. ( iσ X M = a ijk )). The full goldstino coupling reads † term in the superpotential of eq. ( m i i − Φ ijX A Φ ( χ i j i W . In section ijX NL χ 2.31 Q h 2 ; namely j i K Ω / X 1 2 h φ 2 3 1 3 Q ) also yield goldstino couplings to visible sector fields from the 2 NL i ) in the / m + + 3 X X i -terms can be easily extracted from this lagrangian. Goldstino j 2.3 m W A.15 ij B h Q ): i Ω h Q h A.1 i j ij Q i W h Q , we found a universal tree-level goldstino coupling to matter scalars and , but in addition, the Kählerpotential cubic terms 1 2 1 2 2 NL ⊃ ⊃ X Ω W contains visible sector fermions (coupled to its conjugate scalar): Finally, we consider superpotential and Giudice-Masiero mass terms for the fermions. The terms in eq. ( after rescaling) gives an additional coupling (2 The problematic cubic term and Φ 2 45 In section fermions proportional in choosing a differenteliminated gauge by fixing any redefinition than that the preserves one in eq. ( these yield Yukawa interactions betweengoldstino matter couplings fields are and exactly those the of goldstino. eq. ( B Renormalization group invariance of irreducible goldstino couplings Φ contain derivative interactions with thethe goldstino. goldstino of After mass using 2 the equation of motion for Fermion masses and couplings are more difficult to read off. As already mentioned, the goldstino lives both in in agreement with eq. ( This introduces a plethora of new possible terms: This means that the Φ goldstino couplings from eq. ( goldstino couplings coming from of fermionic component of read off from eq. ( JHEP09(2013)125 ). 46 (B.2) (B.3) (B.4) (B.1) 4.33 is term by Z i γ . = 0 and a Wess-Zumino (flat space) i i ..., above are also RG stable. ) RG stable. K + h i γ B.1 2 terms in the form of eq. ( j F A and ˙ j .... i log Λ K γ 2 1 3 , + , it has long been known in the litera- ) 3 i 2 2 π γ + λ Q 2 (4 λ / 6 1 3 − log Λ and ˙ m )), just as a constant scalar mass would. This 2 i = ) . Using a Pauli-Villars regulator, the divergent χ γ 2 – 35 – π )) in flat space with y j λ vis B.1 3 L F (4 e G j W 1 2 x 2.30 K 2 i / = ˙ γ 2 3 eff in eq. ( Z 1 F m 12 γ 2 i − term given corresponding 2 from what would be needed to have the entire divergence in 2 = i / − γ 3 2 1 The presence of such a divergence would be fine if it could be m i M i γ 47 . The goldstino coupling seems to receive a correction from the log- − } 2 / the external wave function spinors for the goldstino and the visible-sector 3 2 χ m y φ, χ, F ] that mass terms of such a form are RG stable. This is true for the { = 2 and S i 20 = ) explained by wave function renormalization. Thus, one would seem to find that L – G e G 17 Q x B.3 For clarity, we give an example of how this occurs in one concrete model: a se- However, the tree-level term, proportional to a constant, is not so clearly RG stable. This differs by a factoreq. of ( fermion, respectively. completely absorbed byHowever, the we wave-function know that renormalizationthe it exact of cannot same the diagram beExplicitly, (up visible absorbed one to can sector in a see the soft fields. this scalar global by mass noting SUSY that that case, does the which not divergent one-loop features affect contribution its to divergent part). with with arithmically divergent diagram inpart figure of this diagram is quantum corrections tocouplings, gravitino making couplings the tree-level feed goldstino into coupling in quantum eq. corrections ( questered to theory goldstino (in the sensevisible of eq. sector: ( it clear that the goldstino couplings proportional to Naively, one would expect itfrom to the receive term quantum corrections proportional startingpuzzle to at is one resolved loop (separate by remembering that the goldstino and gravitino mix in SUGRA, so tions and are thusequations. completely RG For stable the terms —ture proportional that [ is, to their coefficientsitself, solve and their is own trueThe RG for same logic the for soft ˙ terms can be trivially extended to goldstino couplings, which makes effective action: Since these results follow from a 1PI action, they have incorporated all quantum correc- finding further couplings by carefully analyzing the SUGRA- and super-Weyl invariant 1PI JHEP09(2013)125 (B.5) (B.6) (B.7) ] 23 Effectively, ., c . 48 ), we are making + h . ... B.5 ). The two diagrams c . + B.2 2 ∗ + h , φ † µ † µ † ψ ψ χ µ µ µ σ σ σ i 2 µ / χ . Nevertheless, the logic still holds. 3 i ψ 2 χ / 2 φ χ G 3 / im 3 2 m F 3 4 i ). The diagram has the same logarithmic m √ crossterm, as the goldstino coupling correspond- + 3 2 2 Pl j χ B.2 G L M F r 2 e j G φ e 2 iλ K / 2 1 2 3 − / 3 – 36 – µ m + m 3 2 ψ i ∗ ν γ φ σ r ν µ L = σ G χ∂ i ν µ χ ¯  ψ σ ), the gravitino couples to visible-sector fields as [ ∗ ν µ ] for calculations here, but keep the sign and sigma matrix conventions φ σ D ν 2.5 µ 60 ∂ ¯ µν  ψ ) is not expected to depend on i 2 σ g / 3 F Pl 4.33 m M 1 3 2 2 . 2 / √ r 3 2 goldstino coupling runs at one-loop order, in conflict with the claims that − − m 2 / 2 defined in eq. ( = = 2 3 ⊃ m L G -terms of eq. ( 2 S i . One-loop diagram that renormalizes the goldstino coupling to visible-sector scalars A G ]. ⊃ ) as internal legs when computing such quantum corrections. 23 1 S i − Pl Using What we have not accounted for, however, is the mixing between the gravitino and the G arises from a valid 1PI effective action. We use the methods of ref. [ One can of course pick a gauge for the Rarita-Schwinger gravitino field which removes the quadratic This logic is less clearly applicable for the ˙ M 48 46 47 S i of ref. [ mixing and changes thisgoldstino equation system of after motion. computingnot quantum As pose corrections in a to the problem allby text, as orders we we in never will visible-sector have only couplings. to pick consider a This gravitinos does gauge or for goldstinos the (whose gravitino- couplings are suppressed where in the second linefeaturing we an have external specialized to gravitino the that theory can in give eq. contributions ( ing proportional to the to the left-hand so diagrams with an external gravitinousing may this yield corrections equation to of theby motion goldstino trading coupling (or after away making couplings ansure proportional appropriate that to field we the redefinition). are left-hand still side in of Einstein frame eq. at ( one-loop order. G goldstino in SUGRA. Recall that the equation of motion of the gravitino in flat space is and fermions in thedivergence in Wess-Zumino both theory global SUSY from andcoupling eq. SUGRA, and ( would seem to renormalize the tree-level goldstino the Figure 3 JHEP09(2013)125 ), 49 , ) B.3 (B.8) ˙ (C.1) αα a , ) σ † Σ , ˙ β Σ ˙ ˙ − α † α † L D ) ( Σ † a ( ˙ Σ α M φ ˙ ] β α χ ]. The super-Weyl † E D i 2 i D 41 . , is the corresponding 41 φ , 4 2( + − E ˙ 23 23 α α − , [ .... α ˙ † † α M 14 + G χ E D ..., Σ ) ) ) † 2 Σ Σ Σ ) + 2 − E + µ − † ψ (2 log Λ † Σ Σ ( 2 Σ Σ ) − are the Lorentz generators acting on 2 π λ = (2 = ( = (4 αβ ˙ ˙ ) = 6 † α α α 1 2 L α E D M G (2 χ y is RG stable. – 37 – L ) for the gravitino. When combined with the diagram 2 e , δ G / 2 3 x B.5 χ αβ 2 m φ χ / , δ L 2 . Each of these diagrams is logarithmically divergent, 2 3 † ) eff 4 ⊃ Σ F m Σ 2 2 β S i † i G χ D D = 1 4 2( φ ), we see this is precisely the logarithmic divergence that can be − , δE − a total , δ α and its conjugate anti-chiral superfield R B.4 , M ) D E M ) ) E Σ i Σ † † ) αβγ 2 † Σ Σ µ Σ 2 − W ψ + † + − Σ ) are shown in figure Σ Σ 3 Σ Σ − is the determinant of the supersymmetric vielbein, 2 B.5 = 2( = = ( = ( = 2( is a local Lorentz spacetime index, E . These two diagrams yield logarithmically divergent corrections to the goldstino coupling a α ), the goldstino coupling E a R 3 D M δ δ αβγ δ In fact, they are linearly divergent, but any ensuing subtleties will only affect the finite pieces, not the δE W 49 δ where spinors, logarithmically divergent ones. and chirality constraints ofby SUGRA a unchanged. chiral superfield Theytransformations may act be infinitesimally completely on parameterized the gravity multiplet as [ the one-loop level in thisrun, model, as we we confirm knew that had the to tree-level be goldstino coupling the does caseC not from our 1PI analysis Super-Weyl in transformations section Super-Weyl transformations are the most general transformations that leave the torsion we find Comparing this to eq. ( completely absorbed by the wave function renormalization of the visible-sector fields. At after using the equation ofin motion eq. in ( eq. ( side of eq. ( and they give equal corrections to the goldstino coupling. Combining these with eq. ( Figure 4 JHEP09(2013)125 (C.5) (C.6) (C.7) (C.8) (C.9) (C.2) (C.3) (C.4) (C.10) (C.11) (C.12) ) returns . † c . that trans- P , ( ) ˙ α † α + h P used to project C G α ˙ † α ) thus acts as an α . † D W D P , P α )( and α ( V C D ) W ] R P α 2 † E D 23 C Σ , ( 1 2 + Θ 2 is the real superfield having C 2 † = 0 ˙ 1 Σ α d ) = † ( C α . 0 ˙ α 4 Z P α w G U V α is too complicated to include here, 1 4 e + } α G = D α a † D + C ) D D C V α , transform as [ ( R 3. ˙ ]. The operator ˙ † α D α W 8 † ˙ − 3 † α V 14 D {D − D C 2 , as its lowest component. The ellipsis in the ˙ † αα a E C 1 U σ 4 µ D i b – 38 – ( Θ 2 1 4 + , 1 4 2 † , 2. The tree-level lagrangian then reads † − d − C C C ) ) = ˙ Z ≡ − , † α † . R R α D ˙ U ] can be made super-Weyl invariant by including a super- 8 8 α P α a ) + α ˙ † α W 3 23 − − D e D , δ / G D † 2 2 ) 2 † K Q † † 1 D C , − D ( Σ 4 Σ e ( C 2 , and vector superfields 3 w 2 † 1 2 + 1 − = 0 1 − C C ˙ = ( Q α ) can also be serve as an operator, which we denote by the non- † = 0 of Weyl weight Σ † 1 4 1 4 α ( ) = P C Q ˙ P P α δ G − † C δ ( ˙ † α α is the chiral curvature superfield, and are the Weyl weights of their respective superfield; for ordinary matter ≡ − ≡ − = ≡ = e D G 0 P ˙ ˙ † α α EC R ). When acting on a super-Weyl invariant spinless superfield, P P C w α α † P ( δ Θ e e P α G G 4 ( δ d D and P Z may always be expressed as some composition of the above objects. For example, w a = The SUGRA action of ref. [ Chiral superfields D L The superfield boldface a super-Weyl invariant (anti-)chiral superfield(anti-)chiral projector. [ These objects also obey appropriately-modified versions of the Bianchi identities: The super-Weyl compensator can alsoform be homogeneously used under to super-Weyl build transformations: versions of transforms as a chiral superfield of Weyl weight Weyl compensator where or gauge superfields, thesesuperfields still weights transform, vanish. due tothem Note the out. that non-trivial For transformation the a of vector higher the superfield components of of weight 0, matter the superfield the vector auxiliary fieldtransformation of of supergravity the chiral vielbeinsuper-Weyl are invariant action. omitted The terms transformation irrelevantbut for of the construction ofwhen a acting on a Lorentz scalar superfield chiral density, JHEP09(2013)125 , ) D.3 (D.3) (D.4) (D.1) (D.2) ). component α ) is the real 2 θ W R α β is chiral, eq. ( W U θ α P C 2 3 2 † d f W ., C . C ) ., R c c β . c . e . ( D , the appropriately super- e α e + h R + h , + h ) D α α 1 α β β ) to be SUGRA and super-Weyl W W U †− U f ). A non-vanishing W is the super-Weyl covariant chiral 1 ˙ ) α 1 1 α † C − D.2 − − P P 1 2 C D ( W P † 2 2 1 2 C † / α 2 3 C 1 2 C D when acting on chiral superfields, D f β C W 1 4 + ( ) 1 4 D β − e 1 β − ( D − = U e α α R 2 α 3 2 C C † – 39 – D α D C W C b † 2 W ) C D β † 1 2 ) ˙ † α ] D D ) effects such as gaugino masses, however, there are ( D + 2 ( α 1 50 e EC 1 / 16 R e α 3 17 for vector multipets, are beyond the scope of this work. R D , θ Θ †− only has a lowest component, it then follows trivially that 1 ]), one would need to consider additional terms. Such effects, the θ U 4 2 m 4 4 ( d C d 4.3 e †− 17 P R d 1 2 ) O † C Z Z − Z acting on a super-Weyl inert superfield with an undotted spinor P P 1 4 4 1 C 1 4 ( 2 1 ). It can be easily verified that when 1 4 a C of section + + has vanishing Weyl weight, and 2 1 C.6 L ⊃ L ⊃ L ⊃ T α = ), an integral of a local quantity over chiral superspace. (not to be confused with the chiral curvature superfield W α and 3 2 U e S 3.18 − R e ) effects, such as non-local contributions to the self-energies of the particles in the vec- C 2 / ), we used a 1PI effective action for the gauge multiplet built as an integral 2 3 = m α ( encapsulates the running of the coupling with the momentum scale (selected by ) is equivalent, after integrating over half of superspace, to the usual expression for 3.18 O f W yields a gaugino mass. If D.2 The only potentially ambiguous part of this equation is It is now a simple matter to generalize most of eq. ( e For R on 50 e tor multiplet (as consideredequivalents of in the ref. [ only two families of possible choices projector given in eq. ( reduces to eq. ( Weyl covariant version of index. If we only care about covariant where R which should be thought of asfor acting only on the first eq. ( the gauge kinetic lagrangian in chiral superspace (proportional to The superfield vector superfield with the 1PI gauge coupling as its lowest component. The dependence of or alternatively, remembering that In eq. ( over chiral superspace. Thisrenormalization is scheme, sufficient the for 1PIlocal extracting action quantity one-loop must over results, instead all but bewe of in may written superspace. write a as the general For an 1PI the integral action familiar as of case [ a of non- global SUSY in flat space, D 1PI gaugino masses JHEP09(2013)125 , 03 ]. , 1 is , , 03 (D.5) − , B 557 = 06 JHEP SPIRE JHEP , b Fortsch. IN , JHEP , Annals Phys. ][ , , JHEP would need to be = 0, , chosen, a (more (2013) 006 V a ]. b ± e Nucl. Phys. 01 , (2008) 105020 and SPIRE a IN JHEP ][ , D 77 hep-th/9402005 Gaugino mass in AdS space [ ]. ]. 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