aa r X i v : . [ h e p - t h ] J a n Scalar-Vector Eο¬ective Field Theories from Soft Limits
Filip PΕeuΔil π, β , β π Institute of Particle and Nuclear Physics, Charles University,V HoleΕ‘oviΔkΓ‘ch 2, 180 00 Prague 8, Czech Republic
E-mail: [email protected]
We give an overview of the implementation of the soft-bootstrap method applied to the landscapeof theories where the Special Galileon couples to a massless vector particle. We also describethe corresponding traditional Lagrangian approach for this model, which takes into account theformal geometrical interpretation of the Special Galileon as ο¬uctuations of a π· -dimensional braneembedded in a 2 π· -dimensional ο¬at space. β In collaboration with Karol Kampf, JiΕΓ NovotnΓ½ and Jaroslav Trnka β Speaker Β© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ calar-Vector EFTs from Soft Limits
Filip PΕeuΔil
1. Introduction
In the last few decades, the on-shell amplitude methods have undergone a considerable devel-opment, and many new and surprising results have been obtained (for a recent review and a completelist of references, see [1]). These developments concern not only the originally considered (SUSY)gauge theories and gravity, but also the class of non-renormalizable eο¬ective ο¬eld theories (EFTs).Lagrangians of EFTs are organized as inο¬nite towers of vertices with increasing mass dimension,and their form is ο¬xed by the relevant symmetries. In some cases, the symmetry requirements are sostrong that the Lagrangian of the corresponding EFT is unique up to a ο¬nite number of free couplingconstants. This is typically the case of spontaneously broken symmetries, when the eο¬ective theorydescribes the dynamics of the Goldstone bosons. In such a case, the π -matrix of the theory has apeculiar IR behavior, which is expressed in terms of soft theorems for the scattering amplitudes. Itoften appears that these soft theorems can be taken as an alternative on-shell deο¬nition of the theoryitself [2] and can be used for the recursive reconstruction of the tree-level π -matrix [3]. Therefore,the on-shell amplitude methods in conjunction with soft theorems provide us with a powerful toolfor exploration of the landscape of EFTs and for ο¬nding new EFTs with interesting properties.This program, dubbed as the soft bootstrap, has been completed for the single ο¬avor scalar EFTsin [4], where the the classiο¬cation of the exceptional theories with enhanced soft limits has beenperformed. The case of SUSY EFTs has been explored by these methods in [5], and the systematicstudies of multi-ο¬avor scalar theories have been initiated in [6].In this contribution, we present our preliminary results illustrating the power and limitationsof the method in the case of the coupling a massless vector particle to a massless scalar withGalileon power counting . Physically, such an EFT might correspond to the coupling of the photonto the modiο¬ed gravity in the decoupling limit, where the only interacting degree of freedom is theGalileon, or to the interaction of the vector and scalar degrees of freedom of the massive gravitynear the decoupling limit.
2. Bootstrap method
In this section, we describe brieο¬y the application of the bootstrap method to the explorationof the landscape of EFTs containing a massless scalar (sGal) and a massless vector (we call it the BIphoton in what follows). The sGal nature of the scalar means that tree amplitudes A should possesan enhanced single scalar soft limit, i.e. A = O (cid:0) π (cid:1) , where π β The idea of the bootstrap method is to start with some lowest-order seed amplitudes (which arecontact by construction), glue them together to ensure the right factorization, and then add somecontact terms with free constants. These are to be determined using some additional information, inour case a certain set of soft limits. The whole process is then iterated as far as we can computativelygo to construct higher-order amplitudes. For an illustration of the ο¬rst iteration, see Figure 1. This hypothetical theory was ο¬rst mentioned in [5, 7]. calar-Vector EFTs from Soft Limits Filip PΕeuΔil A = β + Figure 1:
First iteration of the bootstrap method, construction of a 6-point amplitude from seed ones.
Let us consider a tree amplitude A with mass dimension π and π external legs, composed ofvertices π π with mass dimensions π π and with π π external legs. Then it holds π β = Γ π ( π π β ) , π β = Γ π ( π π β ) . (1)We deο¬ne the power counting parameters π A and π π for the amplitude A and for the vertices π π as π A β‘ π β π β , π π β‘ π π β π π β . (2)It is self-evident that all the tree amplitudes of an EFT with vertices that all have the same parameter π π = π also have π A = π . We call such EFT single- π theory in what follows . The question we aretrying to answer is whether there is a unique theory with the Galileon-like power counting π = Since we consider only the massless theories, we can employ the massless spinor-helicityformalism. The principles of Lorentz invariance and locality imply that contact amplitudes arepolynomials in square and angle spinor brackets, and they are constrained by the power countingand the little group scaling. In a single- π theory, the mass dimension π of any contact π -pointamplitude A should be π = π ( π β ) +
2. Also, the amplitude should scale as
A β π§ β π A whenever the π -th leg (with helicity β π ) is scaled using | π ] β π§ | π ] and | π i β π§ β | π i .Let us now classify the seed amplitudes. Assuming the helicity conservation and the parityconservation, there are three 4-point seed amplitudes possible A ( , , , ) = π h i [ ] h i [ ] h i [ ]A ( + , β , , ) = π h i [ ] [ | | i [ | | iA ( + , + , β , β ) = π h i [ ] h i [ ] . (3)The indices π πΎ , π π in A π πΎ π π (or in the free couplings π π πΎ π π ) denote the numbers of BI photons andsGals, respectively. For instance, the ο¬rst amplitude A corresponds to the scattering of zero BIphotons and four sGals (it is thus a pure sGal amplitude). Among single- π EFTs, there are distinguished theories such as the NLSM, DBI, BI, or Galileons. This means that the number of helicity-plus and helicity-minus BI photons is the same when we assume all theparticles as outgoing. Note that these amplitudes obey the sGal limit and also the multichiral soft limits in the sense of ref. [8]. calar-Vector EFTs from Soft Limits Filip PΕeuΔil
The ο¬rst iteration means to construct the 6-point amplitudes by gluing the 4-point seed verticesand adding independent 6-point contact terms with unknown constants. The latter are to bedetermined using soft limits. The amplitude A is a pure sGal amplitude, and it is well-known.For the 6-point amplitude A , we symbolically get A = A β A + A β A + π Γ π = π ,π A CT24 ,π , (4)where β β β β represents the gluing corresponding to an exchanged particle with the helicity β andthe sum is over the π =
29 independent 6-point contact counterterms. We have found that justdemanding the O (cid:0) π (cid:1) sGal soft limit for any scalar leg is enough to ο¬x all the constants π ,π andalso one of the two 4-point constants π and π . The remaining unο¬xed 4-point constant representsthe overall normalization of the amplitude. Similarly, for the amplitude A , we get A = A β A + A β A + π Γ π = π ,π A CT42 ,π , (5)where π =
42. In this case, all π ,π s and one of the two 4-point constants π and π can be ο¬xedvia imposing the multichiral soft limit [8] for the BI photons in addition to the O (cid:0) π (cid:1) sGal limit.Finally, for the amplitude A , we get π = A = A β A + π Γ π = π ,π A CT60 ,π . (6)Since this amplitudes has no external scalars, we cannot use the sGal O (cid:0) π (cid:1) limit. We have foundthat requiring A = O (cid:0) π‘ (cid:1) behavior in the limit where two BI photons with the same helicitybecome soft (i.e. a stronger requirement than in the previous case) is suο¬cient to ο¬x all the π ,π s. The π -point amplitudes for π > π -point contact termsand then ο¬xing as many as possible of the free constants by soft theorems. The resulting constructionis schematically summarized in Figure 2. Using this approach, we have proven numerically that A , A , and A are uniquely ο¬xed just by the sGal soft limit, and A is not ο¬xed by the sGalsoft limit alone. Also, it appears that the amplitude A cannot be uniquely ο¬xed by any type ofmultichiral soft limit. This results seem to be in accord with the soft BCFW recursion. Indeed,assuming that the theory exists, the amplitudes A π πΎ π π satisfying π πΎ < π π + π = π πΎ + π π are already ο¬xed, and the amplitudes A π π + ,π π could bereconstructed adding just one extra soft condition (e.g. some multichiral soft limit). In our symbolic formulas, adding helicity-conjugated graphs (i.e. those with opposite helicity assignments to theexternal BI photon legs), which are present due to the helicity conservation, is implicit. That means requiring A = O ( π‘ ) when all the same helicity BI photons become soft, π Β± π = O ( π‘ ) when π‘ β Up to an overall normalization. Cf. also a similar discussion in [7]. calar-Vector EFTs from Soft Limits Filip PΕeuΔil A
04 1 A
22 1 A A
06 29 A
24 42 A
42 5 A A
08 696 A
26 2152 A
44 1280 A
62 94 A A ,
10 ? A
28 ? A
46 ? A
64 ? A
82 ? A , S p ec i a l G a lil e on G a lil e on - li k e B I Figure 2:
The web of the amplitudes and the contact terms. We use the notation π ππΎππ A π πΎ π π , where A π πΎ π π is eitherthe set of contact term, or the amplitude, and π π πΎ π π is the number of independent contact terms contributingto the amplitude A π πΎ π π . Whenever two nodes of the web can be connected by an oriented path, the contactterms corresponding to the starting point of the path contribute to the amplitude attached to the endpoint ofthe path. Provided the theory exists, the green-colored amplitudes can be uniquely reconstructed recursivelyfrom the sGal soft limit alone, the red-colored ones should be ο¬xed by some additional requirement. Theorange-colored amplitudes A π π + ,π π could be reconstructed using just one such extra soft limit.
3. Lagrangian approach
On the Lagrangian level, the suο¬cient condition for the sGal soft limit is the invariance of theaction with respect to the generalized polynomial shift symmetry (here πΊ πΌπ½ = πΊ π½πΌ , and πΊ ππ = πΏπ = β πΊ πΌπ½ (cid:16) πΌ π₯ πΌ π₯ π½ + π πΌ ππ π½ π (cid:17) , (7)The building blocks [9] are the eο¬ective metric π ππ , the extrinsic curvature K πΌππ , and the scalar ππ ππ = π ππ + πΌ π π ππ Β· ππ π π, K ππ πΌ = β πΌ π π π π π πΌ π, π =
12i ln det (cid:16) π + i πΌ πππ (cid:17) det (cid:16) π β i πΌ πππ (cid:17) . (8)Any theory with even-point amplitudes only obeying the sGal soft limit and with the power counting π = L min = L sGal [ π ] β p | π | π ( π ) πΉ ππΌ πΉ ππ½ π ππ π πΌπ½ , (9)where L sGal [ π ] is the sGal Lagrangian, and π ( π ) = π (β π ) is an arbitrary real function . Theaction based on L min is invariant with respect to (7), provided the BI photon ο¬eld π΄ π transforms as πΏ π΄ π = β πΊ πΌπ½ (cid:0) π πΌ ππ π½ π΄ π + π΄ πΌ π π½ π π π (cid:1) . (10)However, L min alone gives a vanishing seed amplitude A =
0, and the amplitude A does notobey the multichiral soft limit. To reproduce the results of Sec. 2, it is thus necessary to addalso non-minimal invariants (starting from 4-photon terms) of the schematic form π πΉ K (12 Here πΌ is the free parameter of the sGal Lagrangian, see [9, 10] for details and for explicit formulas. See also [7], where the special case π = calar-Vector EFTs from Soft Limits Filip PΕeuΔilindependent terms), π (DK) πΉ (4 terms), π (D πΉ ) πΉ (9 terms), π (D πΉ ) πΉ K (10 terms), and π (D πΉ ) πΉ (5 terms), where D is the covariant derivative associated with the metric π ππ , and π π ( π ) = (β ) π + π π (β π ) are arbitrary functions. The resulting Lagrangian has inο¬nitely many freecouplings. Some combinations of them can be ο¬xed by the multichiral soft limit applied to A .
4. Conclusion
We have presented the preliminary results of the application of the bootstrap method to theconstruction of unique tree amplitudes for a parity-and-helicity-conserving theory where the sGalis coupled to the BI photon. We have successfully ο¬xed (up to one normalization constant) all the6-point amplitudes by soft limits. Continuing to higher amplitudes, though some amplitudes areο¬xed, we have not found any appropriate constraints to make all the 8-point amplitudes unique.Also, a compatibility check of the ο¬rst iteration with the second one has to be performed yet.We also give an overview of a possible Lagrangian description of such a theory, based oncovariant building blocks. Apart from the minimal (2-photon) part which must be always present, wehave also performed a full classiο¬cation of possible 4-photon non-minimal terms. The Lagrangianthen reproduces the bootstrap results up to 6-points. It also allows for generalizations with non-vanishing odd-point amplitudes and violation of the helicity conservation.
Acknowledgments
This work was supported by the Czech Science Foundation, project No. GA18-17224S, by theCzech Ministry of Education, Youth and Sports, project No. LTAUSA17069, and by the CharlesUniversity Grant Agency, project No. 1108120.
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