Tasting the SU(5) nature of supersymmetry at the LHC

We elaborate on a recently found SU(5) relation confined to the up-(s)quark flavour space, that remains immune to large quantum corrections up to the TeV scale. We investigate the possibilities opened by this new window on the GUT scale in order to find TeV-scale SU(5) tests realizable at the LHC. These SU(5) tests appear as relations among observables involving either flavour violation or chirality flip in the up-(s)quark sector. The power of these tests is systematically evaluated using a frequentist, p-value based criterion. SU(5) tests in the Heavy supersymmetry (SUSY), Natural supersymmetry and Top-charm supersymmetry spectra are investigated. The latter scenario features light stops and scharms and is well-motivated from various five-dimensional constructions. A variety of SU(5) tests is obtained, involving techniques of top polarimetry, charm-tagging, or Higgs detection from SUSY cascade decays. We find that O(10) to O(100) events are needed to obtain 50% of relative precision at 3σ significance for all of these tests. In addition, we propose a set of precision measurements in ultraperipheral collisions in order to search for the flavour-changing dipole operators of Heavy supersymmetry.


Introduction
The Standard Model (SM) of particle physics describes to a rather good accuracy all kinds of elementary matter observed up to now -apart maybe from the elusive dark matter present in our Universe. A fairly fascinating feature of the SM is that these matter fields fit into complete representations of the SU(5) gauge group [1]. More precisely, the fields {Q i , U i , E i } and {L i , D i } can be respectively embedded into three copies of the 10 and JHEP05(2015)091 the5 representations of SU (5). 1 This feature can be interpreted as a hint that the SM gauge group SU(3) × SU(2) × U(1) arises as the low-energy limit of a Grand Unified Theory (GUT) containing the SU (5) sub-group at some higher scale (see refs. [2][3][4][5] for reviews). 2 Unravelling whether or not Nature is SU(5)-symmetric at short distance constitutes a challenging open problem of particle physics. Many realizations of SU(5)-like GUTs are possible, with various low-energy consequences. Moreover, the quantum corrections between GUT and low-energy scales potentially hide a lot of information. A SU (5) test has thus to face both challenges of avoiding a large model-dependence and being doable in the real world.
Besides the equality of the gauge couplings, a consequence of SU(5)-like unification of the matter fields is the relation between the Yukawa matrices of the down-type quarks and charged leptons. The fact that we do not observe equal masses for leptons and down-type quarks at low energy may be explained by the renormalization group (RG) flow, which does not respect SU(5) symmetry below the breaking scale. Moreover, eq. (1.1) is also modified by finite corrections at the GUT scale, coming in when integrating out the heavy degrees of freedom of the theory. It is thus necessary to study to which extent this relation can be fulfilled at both high and low energy, and a quantity of work has been done on the subject . This is a typical example of SU(5) test where observables are simple but model-dependence is large.
In the hypothesis that supersymmetry (SUSY) is present at a somewhat low scale, new forms of matter exist, that potentially offer new insights on SU(5) symmetry. We will work within this hypothesis throughout this paper. Supersymmetry is a natural companion of SU(5) unification as the simplest supersymmetric models (such as MSSM, NMSSM, . . . ) lead to gauge coupling unification with a good precision. This happens when no new degree of freedom shows up between the electroweak (EW) and the GUT scales, but also if new spacetime structure appears at some intermediate scale, or if additional fields in SU(5)-like representations are present. Apart from that, SUSY also naturally stabilizes the EW scale and provides candidates for cold dark matter in our Universe.
In the SU(5) framework, the two Higgs supermultiplets of the MSSM, denoted H d and H u , need to be embedded in a 5 and a5 representation, denoted H d and H u , respectively. Interactions between matter and Higgs fields are then given by the superpotential After SU (5) breaking, assuming that unwanted Higgs triplets are heavy due to some mechanism, the superpotential reads At the GUT scale, eq. (1.1) is still verified, with y u = y t = λ u . Furthermore, supersymmetry needs to be broken, such that the Lagrangian also contains the SUSY-breaking scalar JHEP05(2015)091 trilinear terms, L soft ⊃ a u h uqũ + a d h dqd + a h d˜ ẽ , (1.4) and SUSY-breaking scalar masses, whereq,ũ,d,˜ , andẽ are the supersymmetric partners of the SM fermions. With the assumption that the source of SUSY breaking is SU(5)-symmetric, SUSY-breaking terms can be parametrized by a singlet spurion and the aforementioned SU(5) relation is also verified for the corresponding trilinear terms, (1.6) It would be again tempting to use these relations in order to test the SU(5) GUT hypothesis. However, contrary to SUSY-preserving parameters, it turns out that the RG evolution of eq. (1.6) is much more model-dependent than the one of eq. (1.1). It depends on the other parameters of the model, on the hypothesis about SUSY breaking, on the hypothesis of having no new degrees of freedom, and so on. It may even be dominated by the SUSY breaking hidden sector dynamics. Therefore this relation can hardly constitute a generic test of the SU(5) GUT hypothesis. The scalar masses satisfy the SU(5) relations (1.7) The same arguments from model-dependence apply to these relations as for the one of eq. (1.6). Testing the SU(5) hypothesis in a way as model-independent as possible, even with lowenergy supersymmetry, seems therefore to be a rather arduous task. However it turns out that an additional SU(5) relation, first identified in ref. [36], opens intriguing possibilities. Building on this observation, the primary goal of this work is to find realistic tests of the SU(5) hypothesis, and to evaluate their potential for the Large Hadron Collider (LHC) era. Let us stress that our study opens a new possibility with respect to those relying on the quark-lepton complementarity of eqs. (1.6) and (1.7) (see, e.g., refs. [37][38][39]). Such correlations, although certainly interesting in specific models, can be hardly used as a generic test of the SU(5) hypothesis, as the RG corrections received by quark and leptons are fundamentally different.
The paper is organized as follows. In section 2, we review the sector of up-type squarks and discuss the SU (5) property which is to be exploited in the subsequent sections. The persistence of this property is closely examined and quantified in section 3. The relation is found to be exact up to O(1%) relative discrepancies. Section 4 comprises the strategy and tools needed to setup SU(5) tests at low energy. It includes in particular the up-squark effective Lagrangian, and a systematic frequentist method to evaluate the power of the proposed SU(5) tests.
We then investigate various SUSY scenarios. In section 5, we consider a scenario where all sparticles are too heavy to be produced at the LHC. A SU(5) test is possible through JHEP05(2015)091 the SUSY-induced flavour-changing dipole operators. Beyond standard LHC searches, we propose a precision search for dipole operators in ultraperipheral collisions. As an aside it is pointed out that EW fluxes in p-p, p-Pb, Pb-Pb collisions are expected to be coherent, and thus drastically enhanced.
In section 6 we consider the natural SUSY case. Depending on the stop/wino ordering, a test on stop flavour-violation involving charm-tagging and a test involving top polarimetry are proposed. The latter is favoured in presence of a light mainly right-handed stop. In section 7, both stops and scharms are assumed to be directly observable at the LHC. This top-charm SUSY scenario is found to be motivated in various 5d constructions. Tests relying on Higgs detection in cascade decays are presented. Summary and conclusions are given in section 8.

The sector of up-type squarks in SU(5) theories
In this section, we review the sector of up-type squarks in the light of the SU(5) hypothesis, and discuss a SU (5) relation which has, to our knowledge, only been reported in the earlier work of ref. [36]. Starting from the superpotential given in eq. (1.2), the 10 i 10 j term is symmetric, such that only the symmetric part of the λ ij u is selected. This enforces a symmetric top Yukawa coupling at the GUT scale, y u = y t u . (2.1) Contrary to the aforementioned SU(5) relation of eq. (1.1) which is potentially modified by model-dependent GUT threshold corrections or higher dimensional operators, eq. (2.1) remains exact whatever extra corrections are included. More precisely, for any operator of the form O ij 10 i 10 j , the antisymmetric part of O ij always vanishes in the contraction with 10 i 10 j . Let us emphasize that y u is symmetric, but generally not hermitian. For a non-SUSY theory, eq. (2.1) does not seem particularly exploitable. In this case the only physical parameters are mass eigenvalues and CKM angles, and this does not seem to be enough to find out whether or not y u is symmetric. This situation changes once we consider SUSY. More precisely, thanks to the required SUSY breaking, more degrees of freedom of the Yukawa matrices become physical. This is due to the presence of the scalar couplings, which in general are not flavour singlets. Just like the SU(5) relation eq. (1.6) for the SUSY-breaking terms comes along with eq. (1.1) in the sector of slepton and down squarks, we have a relation between the trilinear SUSY-breaking terms in the up-squark sector associated to eq. (2.1), a u = a t u . (2.2) As we will see in the following, this property provides a potential way to test the SU(5) GUT hypothesis. We remind that we work in the assumption that the SUSY breaking source does not break SU(5), i.e. the SUSY breaking is parametrized by a spurion F -term which is singlet under SU(5). Going beyond this simplest possibility, one observes that a spurion F -term in the 24 and 75 of SU(5) would lead respectively to a u = −a t u , and a u = a t u . This comes from the tensor product 10 × 10 =5 sym + 45 antisym + 50 sym . We do not consider these possibilities here.

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A crucial point is that eq. (2.2) survives when rotating the supermultiplets into the Yukawa matrices eigenvalue basis (generally referred to as the super-CKM basis). Generally, the singular-value decomposition of the complex Yukawa matrices is y f = U † fŷ f W f , where U f and W f are unitary matrices andŷ f is the diagonal matrix containing the (positive) eigenvalues. The rotated trilinear terms are thereforeâ f = W † f a f U f . The fact that the Yukawa coupling y u is symmetric implies U * u = W u , such that we can write y u = W t uŷ u W u . Further, we obtainâ u = W † u a u W * u , which is a symmetric matrix if a u is. One can remark that both properties eqs. (2.1) and (2.2) are necessary to keep a u symmetric in the super-CKM basis. 3 Let us emphasize that the relations (2.1), (2.2) of our interest are confined to the flavour space of up-(s)quarks, and are therefore expected to be more stable against quantum corrections than the quark-lepton relations (1.6) and (1.7). Moreover, let us note that there is no connection between the SU(5) relation a u = a t u and potentially dangerous FCNCs in the down-sector. In our later analyses of SUSY scenarios, we assume some up squarks to be observed, which means that constraints from down-sector FCNCs are assumed to be fulfilled by construction.
What we have found up to now is summarized by the implication The next step is to see how this translates into physical properties, which can be tested at the TeV scale. Since the property given in eq. (2.3) is confined to up-type squarks, we only need to scrutinize this sector. The object comprising all necessary information for our purpose is the up-squark mass matrix contained in the Lagrangian whereũ = (ũ L ,c L ,t L ,ũ R ,c R ,t R ) t contains the six up-type squarks. The 6 × 6 mass matrix M 2 u is hermitian by construction.
In the broken electroweak phase and within the super-CKM basis, the up-squark mass matrix M 2 u takes the form (2.5) Here, the parametersM 2 Q andM 2 U are the scalar masses of the left-and right-handed upsquarks in the super-CKM basis, and M SUSY is the typical SUSY scale. The H u Higgs field of the MSSM is decomposed into its vacuum expectation value v u and its light component h (the SM-like Higgs boson) with fraction c α . The heavy neutral component H is not shown. More detailed expressions can be found in any SUSY review (see, e.g., ref. [40][41][42]) and are not needed for the rest of this work.

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We can see that the propertyâ u =â t u implies that the off-diagonal block of M 2 u is symmetric. This property will be at the center of our attention in this work. Finding evidence for a symmetric off-diagonal block of M 2 u would constitute a rather striking hint in favour of the SU(5) hypothesis. Finding evidence against a symmetric off-diagonal block in M 2 u would prove the SU(5) hypothesis to be wrong. From this matrix we see that the physics needed to test this SU(5) relation will have potentially to do with up-squark flavour violation, chiral (LR) symmetry, and will possibly involve the Higgs boson.
While the propertyâ u =â t u is exactly valid at the GUT scale, it could be lost in the renormalization group (RG) evolution and not observable at O(TeV) scales, just as it is the case for the SU(5) relations eqs. (1.6) and (1.7). In the following, we are going to quantify how much information about the propertyâ u =â t u disappears through the RG flow between the GUT and the TeV scales.
Let us first consider the relevant RG equations [40][41][42]. We have to study the beta functions of the Yukawa and trilinear matrices, y u and a u , since these beta functions conserve the propertyâ u =â t u at any scale if they are symmetric. The one-loop MSSM beta function of the up-type Yukawa matrix is given by In this expression, the gauge and the trace terms are flavour singlets, and y u y † u y u is symmetric by hypothesis. The only non-symmetric contribution is the term y u y † d y d . In SUSY, the relative magnitude of the up-and down-type Yukawa couplings depends on tan β = vu v d , such that the contribution of this term to the asymmetry grows with tan β. Given the observed low-energy quark masses, one expects y † d y d to compete with y † u y u only at very high tan β. Moreover, the y u y † d y d contribution is suppressed by the elements of the CKM matrix, as y d and y u would commute if they were simultaneously diagonalizable. One can thus expect that y u stays symmetric to a very good approximation at low energy.
The one-loop MSSM beta-function of a u reads Again, the gauge and trace terms are flavour singlets. But the other terms are generally not symmetric, because a u generally does not commute with y † u y u , nor does a d with y u y † d . Thus, a u stays symmetric to a good precision at low energy only if the RG flow is dominated by gauge contributions. Therefore having a symmetric a u at low energy seems not to be such a generic feature. In practice, however, the beta-function is often dominated by the large gluino mass contribution. Moreover, as the M 2 3 contribution is positive, it decreases a u with the energy, such that the non-symmetric terms become smaller and smaller in the beta function. Therefore, one can expect that although β au is generally non-symmetric, its asymmetry at the TeV scale remains rather small in many concrete cases (see section 3).

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Overall, we expect y u and a u to remain symmetric to a good approximation at scales that can be tested, e.g., at the Large Hadron Collider (LHC). Somehow, this SU(5)-type relation persists because it occurs inside the flavour space of the up-(s)quark sector, while the other SU(5)-type relations eqs. (1.1) and (1.6) are relating different sectors, and are thus more sensitive to quantum corrections. The theoretical, model-dependent deviations fromâ u =â t u induced by the RG flow must be considered as an irreducible uncertainty. Quantifying this uncertainty by irr , the implication eq. (2.3) becomes therefore We numerically estimate irr in section 3 for typical scenarios within the MSSM.
Beyond the MSSM, one can also expect that y u and a u be symmetric at low-energy to a good precision. The condition is that the non-symmetric terms in eqs. (2.6) and (2.7) do not dominate, and that the possible hidden sector contributions do not contribute either to irr . For example, a hidden SUSY breaking sector would need to be flavour singlet to not spoil the symmetry. From now on, let us simply keep in mind that having irr 1 is not absolutely generic, although it is present in many classic cases. In the rest of this paper we work in the super-CKM basis unless stated otherwise, and will not write the hat on a u anymore. The low-energy SU(5) relation will be commonly denoted by a u ≈ a t u . Before coming to low-energy observables and to concrete numerical examples, let us set the notations of the up-squark parameters. The complete low-energy up-squark mass matrix in the super-CKM basis is denoted (2.9) We will denote the masses of the up squarks by mũ i (i = 1, ... , 6) with the convention mũ 1 < mũ 2 < · · · < mũ 6 . At the TeV scale, the squared mass matrix M 2 u defined in eq. (2.5) is diagonalized according to where Rũ is the 6 × 6 rotation matrix containing the flavour decomposition of each squark mass eigenstateũ i .

Persistence of the SU(5) relation at low-energy
In the previous section, we saw that the SU(5) relation a u ≈ a t u remains true at low scale up to a relative discrepancy measured by the parameter irr . This discrepancy is induced by the RG flow, and a qualitative study of the MSSM RG equations suggests that irr JHEP05(2015)091 Table 1. Supersymmetry-breaking parameters at Q = M GUT of the MSSM reference scenario used in our numerical analysis. Masses and trilinear couplings are given in GeV. Two values for tan β are considered, tan β = 10 and 40.
is parametrically small in most of the parameter space. In this section we numerically evaluate the magnitude of irr . It is mandatory to know this magnitude as it constitutes an irreducible bound on the precision at which one can potentially test a u ≈ a t u . A too large irr would imply that this relation is spoiled by large model-dependent quantum corrections.
Here we present the results for a reference scenario of the MSSM. The scenario is defined at the GUT scale by the parameter values given in , and a d = a t at Q = M GUT . The only CP -violating phase is the one of the CKM matrix.
We compute the SUSY spectrum in the DR scheme at the scale Q = 1 TeV according to the SPA convention [43] using two-loop RGEs. We use the spectrum calculator SPhenoMSSM obtained from SARAH [44,45] and SPheno [46,47]. To quantify the discrepancy between a u and a t u , it is convenient to use a quantity normalized with respect to the SUSY scale. Therefore instead of using irr , we introduce the asymmetry normalized to tr(M 2 u ), which is arguably representative of the SUSY scale. More precisely, we define which is related to irr by A ij = irr |(a u ) ij |/Tr{M 2 u } 1/2 . Note that the asymmetry A ij being dimensionless, it is independent of the overall scale chosen for our reference scenario.
There are in principle three asymmetries to be investigated: A 12 , A 13 , and A 23 . However, the 1−2 and 1−3 mixing terms appear to be too constrained by the requirements that no tachyons should be present in the low-scale spectrum and that loop corrections to the SM Yukawas should not be too large. Moreover, the largest asymmetry is expected to come from the 2 − 3 sector, so that A 12 , A 13 < A 23 . We focus therefore on the asymmetry A 23 .
As discussed in section 2, the main dependence of A 23 is with respect to the parameters (a u ) 23,32 and (a d ) 23,32 . We vary these parameters assuming that the SU(5) hypothesis is true, i.e. (a u ) 23 = (a u ) 32 at Q = M GUT . We also take (a d ) 23 = (a d ) 32 for simplicity.

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∆M Bs (17.76 ± 3.37) ps −1 Table 2. Low-energy flavour data taken into account in the numerical analysis. For a complete analysis of the reference scenario, we take into account constraints from flavour physics at low-energy, among which the B s mesons oscillation frequency ∆M Bs and the ratio of the K-meson mixing parameter normalized to the Standard Model value The list of constraints is given in table 2. For each constraint, experimental and theoretical errors are combined in quadrature. The central values and experimental errors can be found in refs. [48], while the theoretical errors are taken from refs. [49][50][51]. In our reference scenario, the SM Higgs mass is found to be about m h 0 = 124 GeV, which is consistent with the latest ATLAS result, m h 0 = 125.36 ± 0.41 (sys+stat) GeV [52], combined in quadrature with a theoretical error estimate of ∼ 3 GeV.
We evaluate the asymmetry A 23 at the TeV scale for a scan in the parameters (a u ) 23 = (a u ) 32   indicate the asymmetry A 23 at the scale Q = 1 TeV in percent. Note that the bounds |(a u ) 23 | < 600 GeV and |(a d ) 23 | < 400 GeV are taken into account in order to avoid tachyons in the resulting mass spectrum. The leading constraints are BR(B 0 s → µµ), ∆M Bs (see figure 1), and BR(b → sγ) (at high tan β, see figure 2). We find K / SM K ∼ 1 due to the fact that we have chosen real SUSY-breaking parameters. We have numerically verified that this observable constitutes a stringent constraint on CP -violating phases. Finally, let us note that for both low and high tan β, BR(τ → µγ) never exceeds the upper experimental value. Moreover, the value of the Higgs mass also does not strongly depend on the variation of (a u ) 23,32 or (a d ) 23,32 .
Coming back to our main center of interest, we can see in figures 1 and 2 that the asymmetry A 23 does not exceed a few percent. As expected from RGE considerations, we can conclude that none of the asymmetries A ij exceeds this value on a large part of the MSSM parameter space. In the following, we find that, during the LHC era, such a level of precision will be most probably difficult to reach. We will therefore not mention irr or A ij for the rest of this paper.

Strategies and tools for testing the SU(5) hypothesis at the TeV scale
Any strategy that can be set up to test the SU(5) relation a u ≈ a t u necessarily relies on a comparison involving at least two up-squarks. Apart from this relation, the squark matrix is in general arbitrary, so that each of the six up-type squarks can take any mass. Some of the squarks can be light enough to be produced on-shell at the LHC, while others may be too heavy such that they appear only virtually in intermediate processes.

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As a result, a panel of possibilities for SU(5) tests will appear, depending on the exact features of the up-squark spectrum. It can be convenient to split the possibilities of SU(5) tests into three categories, depending on whether the tests involve only virtual, both virtual and real, or only real up-type squarks. In the following, we outline the tools that are relevant for each of these cases. Note that the SU(5) tests on virtual up-type squarks will necessarily involve one-loop processes, because the interaction terms probed in the tests are R-parity conserving. In contrast, the SU(5) tests on real squarks can in principle rely on tree-level processes only.
Dealing with the eigenvalues and rotation matrices of the full 6 × 6 mass matrix is in general rather technical, and may constitute an obstacle on the quest for simple SU(5) relations among observables. However, depending on the pattern of M 2 u , two expansions can be used in order to simplify the problem: the mass insertion approximation (MIA) and the effective field theory (EFT) expansion.
The MIA is an expansion with respect to small eigenvalues-splittings, and is in particular valid in presence of small off-diagonal elements. In contrast, the EFT expansion can be applied if a large hierarchy is present in M 2 u . The heavy up-type squarks with mass matrix M 2 are integrated out, the expansion parameter is E 2M −2 (see section 4.1), and M 2 can have arbitrary off-diagonal entries. The two expansions, which we detail in the following, are therefore complementary.
It is clear that the feasibility of the SU(5) tests we will setup depends crucially on the amount of data available -whatever they involve real or virtual up-squarks. This feasibility needs to be quantified using appropriate statistical tools. Whenever a SU(5) test can be obtained through a definite relation among observables, we will use a frequentist p-value in order to evaluate to which precision this relation can be tested, for a given significance and amount of data. We will systematically report the expected precision associated with this test.

Tree-level up-squark effective theory
The pattern of the squark masses can in principle exhibit some hierarchy, and this actually occurs in many classes of new physics models. In such a situation, the physics of the light squarks can be conveniently captured into a low-energy effective Lagrangian, where heavy squarks are integrated out. This has, e.g., be discussed in ref. [36] for the situation where only two squarks are light and the remaining four can be integrated out. The higherdimensional operators present in this tree-level Lagrangian will serve as a basis for the SU(5) tests based on both virtual and real squarks.
Following ref. [36], the up-squark mass term of eq. (2.4) can be reorganized such that whereφ contains the heavy states and φ the light ones. The relevant piece of the corresponding Lagrangian has the general form

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where the operators O andÔ represent the interactions with other fields, that are potentially exploited to probe the up-squark sector.
Assuming that the matrix norm ofM 2 is large with respect to the energy E 2 at which the theory is probed, i.e. ||M 2 || = i,j |M 2 i,j | 2 1/2 E 2 , the heavy squarksφ can be integrated out. We are thus left with the low-energy effective Lagrangian of light squarks, We keep only the leading and the subleading terms of the E 2M −2 expansion relevant to build SU(5) tests. This effective Lagrangian contains in principle higher dimensional couplings and derivative terms, which are either subleading or irrelevant for the observables we are going to consider, and are thus neglected. Eq. (4.3) has been obtained using the field redefinition Φ → 1 − 1 2M 2 †M −4M 2 Φ in order to canonically normalize the light squarks kinetic terms. The { , } denotes the anti-commutator.
From eq. (4.3) we see that flavour-violating couplings of the light squarks enter at first order and are controled byM −2M 2 . The flavour-conserving couplings will instead be modified at the second order. The mass matrix M 2 , associated to the light squarks, receives a correction independent of M 2 at first order, and corrections proportional to M 2 at second order. The imprint of the heavy up-squarks in the Lagrangian of eq. (4.3) appears as corrections to the masses and couplings of the light up-squark states. Physically, these corrections have to be understood both as tree-level exchange of heavy up-squarks, and as the first terms of the expansion with respect to the small parameters that describe mixing of heavy and light squarks. While in ref. [36] it has been applied to the special case of two light squarks, this framework is more general and can be used for any number of states that are accessible at the LHC, while the others are integrated out. Finally, we remind that, although this effective theory approach might bear some resemblance with the mass insertion approximation, the two approaches are fundamentally different.

One-loop effective operators
When all the up-squarks are heavier than the typical LHC scale, they can only appear off-shell. Potential tests of the SU(5) hypothesis are thus based on virtual squarks only. All squarks can be integrated out, and the resulting low-energy Lagrangian then contains the SM plus possible other light SUSY particles. Using other light SUSY particles to test a u ≈ a t u does not seem to provide attractive possibilities. In particular gauginos are flavor-blind, and bottom squarks depend on their own flavour structure.
Here, we focus instead on the case where the whole SUSY spectrum is heavy. The resulting low-energy Lagrangian then has the form

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The effective operators with dimension larger than 6 are not relevant for our present study. The dimension-six Lagrangian takes the form where the α i are dimensionless constants accompanying the operators O i . This Lagrangian contains 59 operators, which have been first fully classified in ref. [54].
In SUSY models with conserved R-parity, all CP -even operators are generated at loop level. In general, they receive contributions from a large number of loop diagrams, and we shall identify which among these operators can be potentially relevant for a SU(5) test. We are thus looking for one-loop diagrams involving • at least one (a u ) ij h uqiũj vertex, as it encloses the SU(5) relation eq. (2.2) that we want to test, • fermions on the outgoing legs in order to access information on the flavour structure of the (a u ) ij coupling.
These requirements naturally lead to the sector of up-type flavour-changing dipole operators of the form The key point to obtain a test of the SU(5) hypothesis, is to be able to distinguish between O V u,ij and O V u,ji . The generation of these operators and the possibilities they offer to build a SU(5) test will be discussed in section 5.

Mass-insertion approximation
If the relative difference between the eigenvalues of the mass matrix M 2 u is small, an expansion can be made by taking this difference as the expansion parameter (see, e.g., refs. [55][56][57][58]). This is the so-called mass-insertion approximation (MIA). 4 In the super-CKM basis, the mass matrix is written in terms of the mass insertions δ αβ u (α, β = L, R) as follows, where m 2 q is an average squark mass, that can, e.g., conveniently be chosen to be m 2 q = Tr{M 2 u }. Note that, here, the δ αβ u are 3 × 3 matrices in flavour space, and we have translates on the chirality-flipping, flavour-violating mass insertion matrices as

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or equivalently, in terms of the generation-mixing entries, The MIA also applies when only a subset of the M 2 u eigenstates is nearly-degenerate. The MIA is commonly used in the broken electroweak phase with H ≡ v/ √ 2 = 0. However, the effect of electroweak breaking in the mass matrix is small by assumption, as it participates in splitting the eigenvalues. This is in particular verified whenever M SUSY v/ √ 2. Therefore electroweak breaking does not have influence on the mean squark mass mq by assumption. As a result, the MIA can also be used in the unbroken electroweak phase. In this case, the VEVs have to be replaced by the original Higgs fields. Electroweak symmetry breaking appears in the squark mass matrices exclusively through the chiralityflippling mass insertions δ LR , δ RL . More precisely, the latter are proportional to the VEV according to δ LR,RL ∝ v/M SUSY , while the chirality-conserving ones are not. Thus, the use of the parameters δ LL and δ RR remains unchanged. In contrast, for the chirality-flipping, flavour-violating mass insertions, which we are interested in, we have to replace for i = j. This is also a way to recover the physical Higgs boson in the complete mass matrix eq. (2.5).

Expected precision
When the SU(5) test consists of a simple relation, for example among event rates at the LHC, one can use a frequentist p-value to evaluate the potential of the test. Assume that there is a relation R(O i ) among observables quantities O i , that satisfies whenever the SU(5) hypothesis is verified. Here, E denotes the expectation operator. We use a p-value test to quantify whether or not the observed value of R, denotedr, is compatible with zero. This test is not trivial to perform in general, but simplifies whenever the probability density function (PDF) ofr can be approximated by a half-normal laŵ r ∼ 2N (r, σ)θ(r) (see e.g. ref. [59]), where θ is the unit step function. In this case, the compatibility ofr with the r = 0 hypothesis can be expressed in terms of a statistical significance as Here, Z = Φ −1 (1 − p) translates the p-value into a significance given in terms of standard Gaussian deviations. Φ is the standard Gaussian repartition function. For a given relation satisfying eq. (4.11) and an hypothesized amount of data, one can evaluate the expected value of σ. For a given significance Z, the quantity

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represents the expected precision to which the relation eq. (4.11) can be tested. Throughout this paper we will systematically report the value of P Z when a relation like eq. (4.11) is available.
As an example, consider the relation aX 1 = bX 2 . The X i are assumed to be normal variables, centered on µ i and with variance σ 2 i . Defining R = |aX 1 − bX 2 |, the PDF of R takes the form f R ∼ 2N (r, σ 2 )θ(r) where r = aµ 1 − bµ 2 , and the variance is given by In order to compute the expected significance defined in eq. (4.13), we use this value of σ assuming that the SU(5) hypothesis is true. Note that this implies r = 0 by construction.
In practice it is useful to normalize R such that E[R] ∈ [0, 1]. The expected precision can then be expressed in percent. Concretely, having P Z = 20% means that a violation of the relation E[R] = 0 by 20% or more (i.e. E[R] > 20%) can be assessed with a significance of Zσ. The use of a normalized quantity is more convenient to interface the SU(5) test with other studies -either experimental or theoretical. Normalizing R implies that one frequently encounters the random variable |A − B|/(A + B), where A and B are potentially correlated. We will use the notation E The following relations are useful in order to derive the expected precision, (4.14) These approximations are in particular valid in the Gaussian limit, where a ratio of random variables is described by the Fieller-Hinkley distribution [60]. For a large number of events N A ≡ µ A ≈ σ 2 A and similarly for C, one has for example (4.17)

Heavy supersymmetry
We can now apply the strategy and tools presented above in various SUSY scenarios. In this section we assume that the SUSY scale is large with respect to the scales probed by the LHC. This of course increases the fine-tuning of the electroweak scale which is a primary motivation for low-energy SUSY. 5 In such a situation it is appropriate to integrate the whole SUSY sector including the charged Higgses. As discussed in section 4, the dimension-six

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operators that one should scrutinize to test the SU(5) hypothesis are the up-sector dipoles Our ability to build a SU(5)-test relies crucially on distinguishing the chiralities (i.e. between L and R) at some level. The only quark for which this can be done is the top quark, as the lighter quarks hadronize too fast. Our test has therefore to rely on top polarimetry, which itself relies on the V − A structure of the leading top decay t → bW (see e.g. ref. [62][63][64][65]).
We therefore focus only on the dipole operators involving a top quark, The operators given in eq. (5.1) are generated at one-loop by penguin diagrams. They receive their main contributions from the chargino-down-squark, charged Higgs-down-quark and gluino-up-squark amplitudes. The latter contribution is expected to dominate because it is enhanced by the QCD coupling. Moreover, contrary to the well-known case of down quarks, higher-loop corrections in the up-sector from flavour-changing self-energies [66,67] are tan β-suppressed, so that they cannot become large. We therefore end up with dipole operators mainly generated by the gluino loop.
From now on we assume that the up-squark mass matrix is degenerate enough so that the MIA applies. The expression of the electroweak and chromo-dipoles operators are then given by and the α i3 coefficients are obtained by replacing δ LR Here, we have defined x g = mg/mq, the form factors satisfy F (1) = 1, F (x) → 0 for x → ∞, and are given in appendix A. Moreover, mq denotes the average squark mass introduced in section 4.3. The above expressions have been deduced using loop functions from refs. [68,69] and by making use of section 4. The coefficients of the EW operators in the broken phase, Here s w and c w are the sine and cosine of the weak angle.
As already pointed out above, for the purpose of obtaining a SU(5) test, the contributions that we are interested in are the chirality-flipping ones, δ LR . The chirality-conserving contributions from δ LL and δ RR are suppressed by a factor y t v u /mg = m t s β /mg) with respect to the contributions from δ LR . This factor is compensated if one considers that δ LR ∝ va u /m 2 q , as it is usually the case in SUSY models with soft breaking. However, the magnitude of the a u -term can be larger than mq (for example from naive dimensional analysis [70]). One should therefore let δ LR ∼ O(1).
Considering usual models, one should also notice that at low-energy to a very good aproximation. We can now identify the consequences of our SU(5) relation a u ≈ a t u at the LHC. This relation implies the following equalities between the coefficients of the dipoles in eq. (5.2), Concretely, the dipole operators induce both new top decays and top production modes at the LHC. These processes will be discussed in the subsections 5.2 and 5.3 of the present paper.

SU(5) test through single top polarimetry
We have seen above that measuring the top spin is a necessary ingredient to build a SU (5) test that distinguishes between the O V 3i and O V i3 dipole operators. The expected precision associated with such polarization-based tests can be evaluated in a generic way. The top spin has to be measured with some polarization-sensitive observable z with distribution f Z . For the t → bW decay, z can be for example the angle between the top and the lepton from the W -decay, or the b-quark energy in case of a boosted top [62][63][64][65]. The usual way to proceed to get information on the top spin is to split the phase space into two subsets D = D + + D − . The total sample of N events is then split into two subsamples N ± = N | z∈D ± satisfying The choice of D ± is in general a freedom of the analysis, although it is preferable to have N + ∼ N − to minimize the statistical error. In what follows, we will systematically choose the subsets of phase space D ± such that E[N + ] = E[N − ] if the SU(5) hypothesis is satisfied. The power of the SU(5) test depends on the power of the spin-analyzing variable, which is parametrized by κ ∈ [0, 1]. Here we assume that the z-dependent distribution used to analyze the spin has the form f Z ∝ (1 + κ P t z), where z ∈ [−1, 1] and P t = ±1 is the top polarization [62][63][64][65]. A similar approach can be done for any other distributions. The lepton angle in t → bW has a maximal spin-analyzing power κ = 1 because of the In order to quantify the precision at which the SU(5) relation can be tested, we introduce the normalized asymmetry which satisfies Assuming B background events for the total signal, the expected precision defined in section 4 reads Note that P Z → ∞ for κ → 0, as no spin information is available in this limit. The expected precision is shown in figure 3 as a function of the background and total event numbers B and N . The SU(5) hypothesis starts to be testable when P Z is lower than 100%.
For κ = 1 and B = 0, we see that the hypothesis starts to be testable for N 16 at 2σ and for N 36 at 3σ significance. About 144, 900, and 3600 events are needed to test the relation at respectively 50%, 20%, 10% precision with 3σ significance. Clearly, and contrary to a mere signal discovery, a substantial amount of signal events is necessary in order to test the SU(5) hypothesis.

Existing LHC searches
Let us now discuss the various LHC processes that we expect to be relevant to perform the SU(5) test described previously. The dipole-induced anomalous top couplings induce the two-body decays Three and four-body decays also exist, they are obtained from the ones above by adding an extra Higgs in the final state and replacing Z by W + W − . 6 Among the two-body decays, t → qZ and t → qγ carry information about the dipoles, that can be accessed through polarization of the outgoing gauge bosons. The t → qγ process has been searched for at the Tevatron and LHC [71,72], the leading CMS bound is provided in table 3.
We now turn to dipole-induced top production. All the LHC processes we can consider have the particularity of featuring a single top. For a proton-proton collider, the main partonic processes one can think about are g q → t, g q → tZ, g q → tγ, g g → t q. Apart from the latter, various LHC searches have already been performed in these channels, see [73][74][75][76][77][78]. The sensitivities translated on dipole coefficients are given in table 3. The leading bounds on α γ 3i,i3 , α Z 3i,i3 , α G 3i,i3 come from CMS and ATLAS searches for rare top decays, top-γ production and single top production.
We observe that all these bounds seem to be far from the sensitivity required to observe the dipoles from SUSY, which are loop-generated. The typical order of magnitude for the SUSY dipole operators (eq. (5.5)) α G 3i,i3 /M 2 SUSY (α W 3i,i3 /M 2 SUSY ) is indicated in the last column of table 3. It might be in fact possible that none of the searches above will ever be able to reach the SUSY dipoles at the LHC, even with 3000 fb −1 of integrated luminosity.
For illustration, let us consider the ATLAS gq → t search discussed in ref. [75,76]. Knowing the expected background B ≈ 10 5 and the 95% confidence level bound on 6 As the dipole operators of eq. (5.2) involve a Higgs field, the effective vertices that they produce can be either taken with H = v/

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|α G 3i | 2 + |α G i3 | 2 for L = 14.2 fb −1 one can readily estimate the event rate by interpreting the significance as Z = S/ √ B + S. 7 We obtain σ ≈ 52 |α G 3i | 2 + |α G i3 | 2 /M 4 SUSY pb −1 . Using the typical SUSY-induced values of α G 3i shown above and extrapolating to L = 3000 fb −1 , we find that only ∼ 28 signal events would be collected in this analysis. The background would need to be drastically reduced in order to obtain a O(1) significance. A mere discovery of the SUSY dipoles using these analyses being apparently difficult, applying top polarimetry is even more compromised, as a substantial amount of events is necessary, as shown in figure 3.

Proposal for ultrapheripheral searches
Given that the backgrounds are rather large in the LHC analysis described above, we now propose an alternative type of measurement relying on ultraperipheral collisions (UPCs). These are the quasi-diffractive processes that occur when the incoming protons or heavyions interact with an impact parameter larger than the sum of their radius (see ref. [79] for a review). In the ultraperipheral regime the remnant of these nuclei are only slightly deflected and proceed in forward direction. The protons have a good chance to remain intact and can be tagged in the forward detectors planned for both ATLAS [80] and CMS [81]. Tagging both protons in fully exclusive processes provides a powerful background rejection, because the complete kinematics of the events are known (see e.g. refs. [82][83][84][85]). Heavy-ion collisions have no pile-up, and do not need forward tagging. For both proton and heavy ions, UPCs are therefore an attractive tool for precision physics.
As an accelerated charged object, the proton radiates short pulses of electromagnetic field. The associated photon flux f γ emitted by the proton is proportional to the squared electric charge, f γ ∝ e 2 ∼ 0.1. Recent prospects for photon fusion in UPCs can be found in ref. [86]. In the case of heavy ions, a striking feature is that all charges inside the nucleus radiate coherently, such that the photon flux scales as f γ ∝ Z 2 e 2 . This phenomenon is known and exploited since a long time (see, e.g., ref. [87,88] and references therein). The enhancement factor is, e.g., 82 4 ∼ 4.5·10 7 for Pb-Pb collisions. This enhancement has to be contrasted with the low expected heavy-ion run luminosity, typically 1 nb −1 per year. This type of experiment is useful if the large photon flux can compensate the small luminosity.
We now point out that the same reasoning can qualitatively apply to electroweak charges. The W -and Z-boson fluxes can be treated similarly to the photon [89]. In the UPC regime, contrary to a seemingly widespread belief, electroweak charges radiate coherently just like the electric ones. Although the W and Z boson masses are responsible for the "short range" of the EW interaction in a non-relativistic view, in relativistic QFT these masses have to be put in balance with the centre-of-momentum energy of the diboson collision. The maximum energy from a flux is estimated to be roughly γ/R. For pp collisions, the maximum energy of the vector collision system is √ ω max ∼ 4.5 TeV [86].
This is large with respect to m W or m Z , so that the flux of transverse W or Z is essentially identical to the one of the photon. Also, by the equivalence theorem, the flux of longitudinal

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W -and Z-bosons can be replaced by the flux of corresponding Goldstone bosons. For heavy-ion collisions, we have √ ω max ∼ 260 GeV and ∼ 160 GeV respectively for p-Pb and Pb-Pb, such that √ ω max = O(m W,Z ). In this case the EW fluxes are thus expected to be suppressed in comparison with the photon flux case. Understanding more precisely the features of these EW fluxes deserves a dedicated analysis. Note that with lead ions in initial states, the emission of W + , W − transforms P b into Bi and T l ions respectively. Such isotopes are unstable, but are long-lived with respect to the LHC scale.
Here we simply focus on the effect of the total EW charge, i.e. the overall prefactor of the W -and Z-fluxes. For coherent fluxes f W ± and f Z , we have f W + ∝ 2g 2 ∼ 0.9, f W − ∝ g 2 /2 ∼ 0.2, and f Z ∝ g 2 /c 2 w 1/4 − s 2 w + 2s 4 w ∼ 0.07. For the proton, the EW fluxes are therefore slightly enhanced for W -bosons because of the magnitudes of the involved couplings. For the heavy-ion EW fluxes, in contrast to the electromagnetic case, the neutrons inside the nucleus are charged under the weak interaction. We obtain For Pb-Pb collisions, we have therefore an enhancement of 4.1 · 10 6 for f 2 Z and 9.4 · 10 9 for f W + f W − with respect to the p-p case. For W W the enhancement factor is 200 times larger than the enhancement for γγ. Taking into account the gauge couplings, the factor from f W + f W − is 10 3 times larger than the factor from f 2 γ . Provided that the W fluxes are not strongly suppressed, processes involving intermediate W -bosons in heavy-ion collisions at the LHC are thus rather attractive.
The proton cannot emit a single gluon and stay intact because of color conservation. For UPCs however, an intriguing possibility is that the gluon comes from a Pomeron emitted by the proton [90]. In that case the proton remains intact and can be tagged. However, the Pomeron itself cannot remain intact, such that the process is not fully exclusive. Taking all features into account, we now propose a selection of processes that deserves particular attention in order to probe the dipole operators O G,γ,W with high sensitivity. The processes are pictured in figure 4: • γ γ → t q in p-p collisions (Central exclusive production). The two protons can be tagged and the reaction is exclusive. However, because of the e 4 suppression for photon fluxes, the cross-section might be too small to detect the O γ dipoles.
• g g → t q through Pomeron fusion in p-p collisions (Double Pomeron exchange).
The two intact protons can be tagged. However, this process is not fully exclusive as the Pomerons break up. A detailed study using diffractive PDFs would be necessary in order to evaluate the potential of this channel. This process is sensitive to the O G,γ dipoles.  • γ g → t q in p-p collisions (Diffractive photoproduction). The proton radiating the photon can be tagged. One may also require a second intact proton. The remnant of the subsequent Pomeron would then be observed in one hemisphere only. This process is sensitive to O G , and may be an interesting trade-off between the two above processes.
• W W → t q in Pb-Pb collisions (Central exclusive production). This configuration fully relies on the huge enhancement factor from the coherent action of the EW charges of the nucleons. A computation of the W -boson fluxes is required. This process is sensitive to the O W dipoles.

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• gγ → t q, gW → t q in p-Pb collisions (Single diffraction dissociation). No proton tagging is necessary in this case, so that ones lets the gluon be emitted by the proton in order to take profit of the large gluon PDF. The luminosity is higher than in the Pb-Pb case, about 200 nb −1 per year. One takes profit of the γ-or W -flux enhancement from the heavy ion. This process is sensitive to the O G dipoles. Therefore this configuration might be a good compromise between p-p and Pb-Pb collisions.
All these processes feature a top-quark plus a jet in the final state. Would the expert reader obtain an estimate for one of these processes, the potential to test the SU(5) hypothesis through top polarimetry could be directly drawn from figure 3. Identifying the jet charge -if possible -would be useful to reject fake QCD events [91]. Finally, we remark that all these processes are relevant for generic dipole operators searches beyond SUSY. Also, the enhancement of W -boson fluxes from the total EW charge has implications for the searches of other anomalous couplings like triple and quartic gauge interactions.

Natural supersymmetry
We now investigate SU(5) tests in the framework of "natural" (or effective) supersymmetry. The mass spectrum of this scenario features a first and second generation of up-squarks that are considerably heavier than the squarks of the third generation. This pattern is motivated by both naturalness considerations [92] and by ultraviolet (UV) constructions like partiallysupersymmetric composite models [93] or supergravity contributions generically present in five-dimensional models [94].
Clearly, in natural SUSY, the up-squarks effective theory (see section 4) will consist of two mostly stop-like squarks. Their mixing is not constrained and can potentially be large, which is often needed to satisfy the constraint coming from the observed Higgs-boson mass. The effective operators that appear when integrating out the heavy up-squarks can potentially induce flavour-changing stop decays, and violation of unitarity relations in the stop chiral decays [36].
All the information about the SU(5) relation a u ≈ a t u is enclosed in the higherdimensional operators of the effective Lagrangian for the stops. Potential SU(5) tests have to rely on the effect of these effective operators in the stop sector. Therefore these tests involve both real squarks (the stops) and virtual squarks. We assume that both stops (t 1 andt 2 ) are produced at the Large Hadron Collider (LHC). The total production crosssections of stop pairs at next-to-next-to-leading order (NNLO) are σtt = 89 fb −1 ± 10%, 8.6 fb −1 ± 15%, 0.71 fb −1 ± 20%, and 0.08 ± 25% fb −1 for stop masses of mt = 700, 1000, 1400, and 1800 GeV, respectively [95].
In the following, we develop SU(5) tests that already have been sketched in ref. [36]. We refer to this paper for certain details of the derivation. We evaluate the potential of these tests using the frequentist treatment described in section 4. We consider two cases concerning the mass ordering between the stops (mt 1,2 ), the bino (mB), and the wino (mW ). We assume that the lightest neutralino is mostly bino-like,χ 0 1 ∼B, and the second-lightest neutralino is mostly wino-like,χ 0 2 ∼W , as it is typically the case in scenarios with gaugino mass unification at the GUT scale.

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6.1 The mt 1,2 > mW > mB This mass ordering allows the stops to decay either into the lightest neutralinoχ 0 1 ≈B or into the second-lightest neutralinoχ 0 2 ≈W . In order to build a SU(5) test, we are interested in the flavour-changing decays We assume that the stop masses are unknown, and that only the event rates of the stop decays into binos and winos, respectively denoted by N Y and N L , are experimentally accessible. Moreover, we assume that a certain fraction, denoted by N c Y and N c L , of these events can be charm-tagged. The remaining events N c then contain both up-quark events and miss-tagged charm jets.
Assuming the same charm-tagging efficiency c for N Y and N L , 8 we have where γ Y,L are the actual fractions of charm events. As shown in ref. [36], the SU(5) Let us remark that the large QCD error on the underlying cross-sections roughly cancels out in the above ratios of event rates. Moreover, no information on the stop mixing angle nor the stop masses is necessary to perform the test. Following section 4.4, we define the normalized test quantity as such that The expected precision (see section 4.4) associated with this test is found to be Note that P Z → ∞ in the limits γ → 0 (no charm jets expected), γ → 1 (no up jets expected), or c → 0 (no charm-tagging). The expected precision is shown in figure 5, where the actual charm fraction is set to γ = 0.5, and the charm-tagging efficiency is set to c = 0.5. We see that N Y = N L 27 events are sufficient to start to probe the relation at 3σ significance, i.e. to have P 3 < 100%. Testing the relation with 50%, 20%, or 10% precision at 3σ requires respectively N Y ∼ N L ∼ 110, 675, or 2700 events. For comparison, assuming stop flavour-violating branching ratios of 0.05 and 300 fb −1 of integrated luminosity, we expect about 1340, 130, and 11 events for stop masses of mt = 700, 1000, and 1400 GeV, respectively. 6.2 The mW > mt 1,2 > mB For this mass ordering, the stops decay only into the bino according tot 1,2 → t L,RB . Performing top polarimetry on a decaying stop potentially gives access to the stop mixing angle [96][97][98]. Here we point out that the same kind of procedure also provides a SU(5)-test.
In the language of the up-squark effective Lagrangian of section 4, the operator coupling to (t L ,t R ) is O ∝B(t L , −4t R ) [40][41][42]. The factor −4 comes from the different hypercharges of the left and right top components. In the stop effective Lagrangian, the effective mass term mixest L andt R . The mass eigenstatest a ,t b are obtained by where s θ and c θ are sine and cosine of the stop mixing angle θ t . No implicit assumption is made about the mass ordering oft a ,t b , i.e. we do not know in principle whethert a =t 1 , t b =t 2 or vice versa. 9 The stop angle can be large and is important for phenomenology. Using the effective Lagrangian of the stop, it can be shown [36] that the top-stop-bino coupling is distorted with a structurẽ if the SU(5) hypothesis is true. The parameter x encloses the effect of effective operators and therefore x 1. The matrix K is instead non-symmetric if the SU(5) hypothesis is 9 The stop rotation matrix is real as we consider real SUSY breaking terms. The stop angle is obtained up to O(x) corrections using the event rate expressions with x = 0 (see appendix B), i.e. at leading order in the EFT expansion. Note that only one stop is necessary for that purpose. The stop mixing angle is for example obtained from thẽ t a forward-backward asymmetry, that satisfies The x parameter is obtained using the full event rates expressions of appendix B, i.e. at next-to-leading order in the EFT expansion. The measurement of N a,b± provides two inequivalent ways of knowing x. The event rates N a,b± satisfy therefore one nontrivial relation if the SU(5) hypothesis is verified. This SU(5) relation can be put in the normalized form where we have introduced the usual asymmetries One has E[R] = 0 if the SU(5) hypothesis is satisfied. The information about x carried by the expected event rates is conveniently measured by I[N a,b± ] = |∂ log N 1,2± /∂x|. It turns out that this information depends crucially on the stop mixing angle. This can be seen in figure 6, where we show I[N a,b± ] as a function of the latter. The information becomes small for θ t ∼ 0 (no stop mixing) and vanishes exactly for θ t = π/4 (maximal stop mixing). In between these two limit cases one has I[N a± ] = O(1), I[N b± ] = O(0.1), and the reverse for the interval [π/4, π/2].
The associated expected precision is found to be approximatively in the θ t ∈ [0, π/4] case. We have P Z → ∞ for κ → 0 as the test relies on top polarimetry. For θ t ∈ [0, π/4], the event rates N b± are less sensitive to x than the event rates N a± . The expected precision depends thus mainly on the amount oft b produced. For this reason we drop the 1/N a term is the equation above. The exact formula is given in appendix B. As for θ t ∈ [0, π/4] moret b thant a are needed, scenarios where thet b is the lightest are more interesting. For these values of the mixing angle, this lightest stop is mainly righthanded. The same reasoning can be applied for θ t ∈ [π/4, π/2]. This time, a larger amount oft a is necessary. A spectrum wheret a is the lightest stop is therefore more favorable for the SU(5) test. Again, this lightest stop would be mainly right-handed. We conclude that scenarios with a light mainly right-handed stop are always more appropriate to carry out this SU(5) test, for any value of the stop angle. The expected precision is shown in figure 7 for θ t = 0.4, as well as for N a = 20 and 1000. We see that with a spin-analyzer of efficiency κ = 0.5 and N a = 20, N b 137 events are needed to probe the relation at 3σ significance, i.e. to have P 3 < 1. Testing the relation JHEP05(2015)091 with 50% or 20% precision at 3σ requires N b ∼ 589 or 7560. For comparison, for 300 fb −1 of integrated luminosity, one expects about 26700, 2580, 213, and 24 events for respectively mt = 700, 1000, 1400, and 1800 GeV.

Top-charm supersymmetry
Let us now focus on classes of models that feature a heavy first generation of up-type squarks, i.e. with only the stop and scharm states potentially accessible at the LHC. In the super-CKM basis, this means assuming up-squark mass terms of the form where the λ ij form in general a hermitian matrix of O(1) parameters and Λ M SUSY . Such a framework is immune against D 0 −D 0 mixing induced from the up-squark sector [53] because the first up-squark generation is heavy. We assume that the down-type squarks are either aligned or heavy in order to avoid FCNC induced by the down sector. Let us recall that, in the SU(5)-like GUT context, slepton mass matrices are related to the squark mass matrices.
Phenomenologically, this top-charm SUSY framework constitutes an ideal playground to gain knowledge about how to test the SU(5)-like GUT hypothesis of our interest. It is also useful in order to carry out stop searches at the LHC taking into account flavour violation [99][100][101][102][103][104][105][106][107][108][109]. In particular, large top-charm mixing is found to both improve naturalness and to relax the constraints on stop masses [110].

Extradimensional realizations
Here we provide some elements of model-building to demonstrate how the top-charm SUSY structure can appear in presence of a compact extra-dimension. We focus on the features that make the mass matrix eq. (7.1) appear. The discussion below is not meant to be exhaustive, nor fully detailed. Although not all details are discussed, care has been taken that the elements proposed are compatible with the observed SM flavour structure and that a u -terms are potentially generated. The reader with no interest in model-building can safely skip this subsection.  [111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128] and the review [129]). The first generation up-squarks get a tree-level mass of 1/2R and come within a SUSY N = 2 hypermultiplet. The masses of the second and third generations are generated at one-loop [112], 2)

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where C is a model-dependent factor containing group theory invariants. The hierarchy of eq. (7.1) is generated due to this loop suppression.  [130][131][132]). Supersymmetry is broken by radion mediation parameterized by the radion F-term F T (i.e. a Scherk-Schwarz with small twist) and by a spurion F X on the y = 0 brane. The 10 1 is localized towards the y = 0 brane using a a 1 > 0 bulk mass, 10 2 is localized towards the y = πR brane using a 2 < 0, and the 10 3 has flat profile, a 3 = 0. A correct Yukawa hierarchy is obtained in that way. For the up-squark masses one obtains 10 where the second-generation term is exponentially suppressed with respect to (F T /2R) 2 . The five-dimensional cutoff M * is such that πRM * is of order of few units. It is natural to assume F X /M * ∼ F T /2R. In order to reproduce the up-squark Yukawa, one has a 1 ∼ 10, so that there is one order of magnitude between the first and third generation. The gaugino having a flat profile, its mass is of same order as the third generation of the soft masses M 2 1/2 ≈ m 2 Q,U 3 . Below the GUT scale, the theory reduces to the MSSM. The RG flow of the up-squark soft masses is dominated by the gluino, whose contribution is universal. The low-energy matrix is roughly given by m 2 Q,U 3 ≈ m 2 Q,U 3 + K 3 M 2 1/2 with K 3 = (4.5-6.5) [40][41][42]. One has therefore a low-energy matrix of the form eq. (7.1).  [70,133]. Assume that an IR spurion F X breaks the SM flavour symmetry, giving a soft mass O((F X /M * ) 2 ) only to the first generation of squarks and to nothing else. Radius stabilization and a vanishing cosmological constant [70,134,135] typically implies that radion mediation is suppressed by a warped factor with respect to the IR brane breaking F T /2R ∼ F X /M * . We therefore have The gaugino mass is generated by the radion, M 1/2 ∼ F T /2R. The low-energy mass matrix has therefore the structure given in eq. (7.1).

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•  ±). This setup has a natural holographic interpration has a N = 2 strongly interacting CFT coupled to a non-SUSY elementary sector [93,136,137]. Bulk masses of 10 i are set in order to reproduce the Yukawa hierarchy, with c 1 > c 2 > c 3 ∼ 1/2. Assuming left-handed zero modes for the quarks, one sets (∓, ±) BCs to the 1st generation and (±, ∓) to the two other generations. From the latter BCs a light mode emerges, with m ≈ (4c 2 − 1) 1/2 k c+1/2 for c > 1/2 and m ≈ √ 2/| log | 1/2 k for c = 1/2. This mode does not exist for the first generation, whose first mode is given by m KK = (c 1 /2 + 1)πk . The bulk masses being set in order to reproduce the up quark masses pattern, one ends up with 11 Notice one obtains an inverted hierarchy where the 2nd generation is the lightest. Extra source of SUSY breaking like the universal gravity contributions discovered in ref. [94] could be useful to lift the scharm masses and to generate a-terms. The phenomenology of this last model is rather intriguing as the light stops and scharms are accompanied by their complex conjugate, as a remnant of the broken N = 2 hypermultiplet. We stress that this model would deserve a dedicated study.

Effective Lagrangian
The

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However, unlike for the Natural SUSY case, looking at the higher-dimensional operators is not the only possibility available, because a SU(5) information from a u ≈ a t u also remains at leading order in the low-energy mass matrix M 2 . This is the relation of the topcharm sector m 2 26 = m 2 35 . We have therefore the possibility of testing the SU(5) hypothesis using only the real up-squarks. From now on we thus focus only on the top-charm sector.

SU(5) test through Higgs production
Let us first consider a case where all stop and scharm masses are nearly degenerate. This possibility happens in particular in low-energy GUTs, where no large stop mixing is needed in order to have the correct Higgs mass, see e.g. ref. [128]. Nearly-degenerate squarks imply that the mass insertion approximation (MIA) is valid for the sector of stops and scharms, and mt L ≈ mt R ≈ mc L ≈ mc R ≡ mq. These states should be produced in an equally abundant way at the LHC, as their production occurs mainly through gluon fusion.
The off-diagonal elements of the up-type trilinear matrix are identified with mass and δ LR 23 . The former relates to a truly bilinear term, i.e. the scalar masses, while the latter is induced by a trilinear term, the squark-Higgs scalar coupling. This fact is somehow hidden if one lets the Higgs be on its VEV. The physical Higgs exclusively couples to the LR components of the squark eigenstates. This can be seen from the complete low-energy mass matrix in eq. (2.5). In order to set up a SU(5) test, one may therefore use the Higgs as a probe of the squark eigenstates. Detecting a Higgs gives a direct access to the coupling δ LR u 23 , i.e. to (a u ) 23 . The LHC SUSY processes of interest are thus stop and scharm pair production, followed by a flavour-violating decay into a squark and a Higgs-boson in one of the decay chains. These processes are depicted in figure 8. We further assume that the squarks decay into the bino. These processes can be identified requiring a single top, a hard jet (from a charm quark), a Higgs, and large missing transverse energy (E T ). Higgs production through upsquark flavour-violating decays have been also studied in refs. [108,109]. Note that in the degenerate case, not all particles can be on-shell in the decay chain producing the Higgs.
As in previous cases, this test has to rely on a distinction between the chiralities, which is possible only for the top quark. Reconstructing the events is necessary in order to select the ones where the Higgs comes fromc L,R → ht R,L and reject the ones fromt L,R → hc R,L . There is in principle enough information from the decay chain to carry out this distinction. The former of these processes is shown in first row of figure 8. The latter is shown in second row. Other processes leading to the same final states are also possible but are suppressed by extra mass-insertions (see second row of figure 8).
Provided that the cascade decay withc L → ht R can be isolated, top polarimetry then readily provides a SU (5)  This situation is similar to the one of section 5 and will not be further discussed. Instead we focus on a somewhat different spectrum, where stop mixing is large while the scharms are nearly degenerate. The MIA applies to the scharm sector, but not inside the stop sector. Instead, we rotate exactly the stops into their mass eigenbasis. After rotating the stops (see section 6 for the definition of the stop angle), the scharm-stop mass matrix takes the form Following the MIA approach, the first matrix above corresponds to the squark mass eigenvalues, while the second matrix is treated as a mass insertion. By definition, m 2

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Contrary to the degenerate case, the stops and scharms have different production rates. Moreover the theoretical predictions suffer from a large QCD error. One way to proceed is to normalize N hj , N ht using observed data. In order to normalize N ht , we ask for the measurement of flavour-conserving decay chains ofc-pairs into two jets plus large missing E T . The corresponding event rate is noted N jj . Because of stop mixing, the same process cannot be used to normalize N hj . Instead we ask for one of the twot 2 to decay into Zt 1 . This event rate is noted N Zt . These processes are depicted in the second row of figure 9.
Normalizing N hj by N Zt and N ht by N jj cancels the cross-sections, leaving only the ratio of partial decay widths, We now consider the quantity which is obtained when neglecting the SM masses. 12 This factor may be evaluated using extra information from kinematic analyses, for example using the kinematic edges of the ht, hj, Zt invariant masses. The normalized SU(5) test then reads which satisfies (7.14) The associated expected precision reads Note that P Z → ∞ when θ t → π/4, i.e. the power of the test vanishes in the limit of maximal stop mixing as expected. We display only the leading statistical uncertainty that comes from the small flavour-changing event rates. The expected precision is shown in figure 10 for the intermediate value θ t = 0.4. Because of the mass ordering of this scenario, one expects N hj < N ht because thet 2 is heavier than the scharms and thus produced less abundantly. Assuming N hj N ht and θ t = 0.4, testing the relation with 50%, 20%, or 10% precision at 3σ requires respectively N hj 19, 116, or 464 events. Roughly twice less events are needed if θ t = 0. For comparison, assuming flavour-violating branching ratios of 0.05, and 300 fb −1 of integrated luminosity, one expects about 1340, 130, and 11 events for a squark with respectively mq = 700, 1000, and 1400 GeV.  Figure 10. Expected precision P Z for the SU(5) test based on Higgs detection in SUSY cascade decays eq. (7.13). N hj and N ht are the number of observed cascade decays fromt 2 andc L,R pair production, respectively. One fixed the stop angle to θ t = 0.4. Plain (dotted) lines denote 2σ (3σ) significance, respectively. Purple, orange and red lines respectively show P 2,3 = (50%, 20%, 10%) isolines of expected precision.
Other possibilities of normalization of the N hj , N ht rates are in principle possibleusing either observed or theoretical event rates. In any case the approach relies on evaluating the appropriate η parameter such that eq. (7.14) is satisfied. The expected precision eq. (7.15) is valid as long as the statistical uncertainty from N hj and N ht dominates.
Finally, the analysis of the specific mass orderings carried out above can be taken as a guideline to build more global SU(5) tests. In particular the flavour-changing squark decays with Higgs production are expected to always carry relevant information regarding a u ≈ a t u , for any mass ordering. Assuming an arbitrary mass ordering, a global SU(5) test can be setup by putting into a likelihood function the N hj , N ht event rates plus some information to normalize them (i.e. all the potentially useful information found above). This global test constitutes a natural extension to the cases analysed above and is subject to future work.

Summary and conclusion
A fairly intriguing feature of the Standard Model (SM) is that quarks and leptons fit naturally into complete representations of the SU(5) gauge group. This might be taken as a hint that the SM is the low-energy effective theory of a Grand-Unified Theory (GUT), either SU(5)-symmetric or symmetric under a group broken to SU(5) at an intermediate energy scale. Many realizations of SU(5)-like GUTs are possible, and imply a variety of low-energy features. Moreover, the large quantum corrections occurring when flowing down the SM effective theory from the GUT to the weak scale potentially wipe out a lot of precious information. Testing whether the world is microscopically SU(5)-like is JHEP05(2015)091 thus a rather challenging task. The large model-dependence coming from both the GUT realization and from quantum corrections constitute a irreducible theoretical uncertainty, from the point of view of testing the SU(5) hypothesis.
Low-energy supersymmetry is a natural ingredient of SU(5)-like GUTs, as it typically favours the high-energy unification of gauge couplings. As a happy coincidence, the superpartners potentially carry information about the possible GUT group. Low-energy SUSY is therefore both favoured by SU(5)-like GUTs and useful to reveal their nature. Still, testing the SU(5) hypothesis rises the double challenge of taming the model-dependence and finding observations that are doable in the real world. While such a program may seem at first view insurmountable, it happens that a new SU(5) relation between the up-squark trilinear couplings a u = a t u , first identified in ref. [36], opens a number of possibilities. This relation is exact at the GUT scale, and is immune to any GUT threshold corrections.
Moreover, the relation a u ≈ a t u is found to survive through the MSSM renormalization group flow, such that it is spoiled by typically O(1%) of relative error at the TeV scale. This relation can be thus taken as a window on the GUT physics, that can be exploited to test the SU(5) hypothesis. Because of the peculiar structure of a u ≈ a t u , all the SU(5)tests coming from this relation have to rely on either flavour-violation or chirality flip in the sector of up-squarks and quarks. The discussed SU(5)-tests are set up for the Large Hadron Collider (LHC).
Even though the SU(5) relation a u ≈ a t u is immune to many corrections, a modeldependence remains in the up-squark spectrum. Indeed, although certain patterns of the up-squark mass matrix can be justified in a number of ways, overall a lot of freedom remains. We consider therefore various typical scenarios for the up-squark spectrum. The scenarios that we have investigated are "Heavy supersymmetry" (i.e. M SUSY TeV, section 5), "Natural supersymmetry" (i.e. real stops are accessible at the LHC, section 6), and "Topcharm supersymmetry" (i.e. both scharm and stops are accessible at the LHC, section 7).
Depending on the up-squark spectrum, the SU(5) tests involve only virtual squarks, only real squarks, or both real and virtual squarks. Whenever the virtual squarks are heavier than the TeV scale, they can be integrated out to form a TeV-scale effective Lagrangian. The effect of virtual squarks is then enclosed into effective operators, either generated at tree-level or at one-loop. A complementary expansion is the mass insertion approximation, that we use both in the broken and unbroken EW phase.
In many cases, the SU(5) tests that we find consist in determining whether a relation among certain observables is satisfied or not. In order to quantify the feasibility of a test, we introduce a systematic procedure relying on the frequentist p-value. The associated expected precision tells, for a given amount of data, up to which magnitude a violation of the SU(5) relation can be assessed within a given statistical significance.
In the "Heavy SUSY" case, it appears that the SU(5) hypothesis can be tested using flavour-changing dipole operators in the up-sector. These dipoles from SUSY are loopgenerated. A top quark has to be present in the dipole in order to use top-polarimetry. The expected sensitivity as a function of signal and background event rates is evaluated. The common LHC searches from either anomalous single top production or anomalous flavour changing top decays seem however not sensitive enough to be able to discover SUSY dipoles.

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We propose thus an alternative kind of precision measurements relying on ultraperipheral collisions between proton and/or heavy ions. A set of exclusive processes is proposed in order to probe the SUSY dipoles operators. We also point out that, contrary to a seemingly widespread belief, the W /Z-boson fluxes from proton and heavy ions are coherent in the UPC regime. The vector boson fusion cross-sections can thus be expected to be rather large.
In the "Natural SUSY" scenario, already investigated in ref. [36], the two mass orderings mt 1,2 > mB ,W and mW > mt 1,2 > mB are considered. A test associated to the former relies on flavour-changing stop decays and on charm-tagging, while no knowledge of stop masses nor the stop angle is needed. We evaluate the expected precision as a function of charm-tagging efficiency and event rates.
For the second mass ordering, the SU(5) test relies on top polarimetry. Its potential greatly depends on the stop mixing angle, and vanishes in the limit of no stop mixing and maximal stop mixing. In contrast, the potential for arbitrary stop mixing is more promising. We evaluate the expected precision as a function of the stop mixing angle, event rates and spin-analyzing power. A few events from the mainly left-handed stop are sufficient, while a more copious production is needed for the mainly right-handed stop. This test is favoured by a SUSY spectrum with a mainly right-handed lightest stop.
We finally introduce the "Top-charm SUSY" scenario, featuring a heavy first generation of up-squarks, and demonstrate that it can arise in a variety of five-dimensional GUT realizations. SU(5) tests can be found using the SM Higgs boson to probe the LR components of the squarks. The tests therefore rely on cascade decays involving Higgs production and flavour-violation. For degenerate scharms and stops, a test using top polarimetry is presented. For the case of mt 2 > mc L,R > mt 1 , a test relying on the reconstruction of the decay chains involving the Higgs is discussed. The expected precision as a function of the stop angle and event rates is evaluated.
The SU (5) tests that appear in the various SUSY scenarios considered are summarized in table 4. The typical amount of events needed to reach an expected precision of 50% at 3σ is also shown for each of the tests. For these numbers the experimental conditions are chosen to be the most favorable, e.g κ = 1. The number of needed event ranges roughly from about 10 to about 100. For the required precision, the tests involving top polarimetry require typically O(100) events, because they rely on the shape of kinematic distributions. All these tests rely on the existence of a given relation among certain observables. In cases where no such relation is available, a more global hypothesis testing has to be done. This will be the subject of a further work [138].

A The dipole form factors
The form factors from penguin diagrams are [68,69]  N bL = N b (s θ + xc θ ) 2 (s θ + xc θ ) 2 + 16(c θ + xs θ ) 2 , (B.7) N bR = N b 16(c θ + xs θ ) 2 (s θ + xc θ ) 2 + 16(c θ + xs θ ) 2 . (B.8) Here N a,b is the production rate oft a,b . The expected precision associated with the test of eq. (6.10) reads Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.