In hep-ph/0011311 the one Higgs doublet standard model was obtained by an
orbifold projection of a 5D supersymmetric theory in an essentially unique way,
resulting in a prediction for the Higgs mass m_H = 127 \pm 8 GeV and for the
compactification scale 1/R = 370 \pm 70 GeV. The dominant one loop contribution
to the Higgs potential was found to be finite, while the above uncertainties
arose from quadratically divergent brane Z factors and from other higher loop
contributions. In hep-th/0108184, a quadratically divergent Fayet-Iliopoulos
term was found at one loop in this theory. We show that the resulting
uncertainties in the predictions for the Higgs boson mass and the
compactification scale are small, about 25% of the uncertainties quoted above,
and hence do not affect the original predictions. However, a tree level brane
Fayet-Iliopoulos term could, if large enough, modify these predictions,
especially for 1/R.Comment: 7 pages, LaTeX, Contribution to "Summer Institute 2001",
  Fuji-Yoshida, Japan, August 200

Barbieri, Riccardo

Hall, Lawrence J.

Nomura, Yasunori

English

arXiv

In [1]the one Higgs doublet standard model was obtained by an orbifold projection of a 5D supersymmetric theory in an essentially unique way, resulting in a prediction for the Higgs mass m_H = 127 +- 8 GeV and for the compactification scale 1/R = 370 +- 70 GeV. The dominant one loop contribution to the Higgs potential was found to be finite, while the above uncertainties arose from quadratically divergent brane Z factors and from other higher loop contributions. In [3], a quadratically divergent Fayet-Iliopoulos term was found at one loop in this theory. We show that the resulting uncertainties in the predictions for the Higgs boson mass and the compactification scale are small, about 25percent of the uncertainties quoted above, and hence do not affect the original predictions. However, a tree level brane Fayet-Iliopoulos term could, if large enough, modify these predictions, especially for 1/R

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    LBNL-48987         A Constrained Standard Model: Effects of Fayet-Iliopoulos Terms  Riccardo Barbieria, Lawrence J. Hallb,c, Yasunori Nomurab,c  a Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy b Department of Physics, University of California, Berkeley, CA 94720, USA c Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA                     This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.  DISCLAIMER  This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California. arXiv:hep-ph/0110102v1  7 Oct 2001UCB-PTH-01/39LBNL-48987A Constrained Standard Model:Effects of Fayet-Iliopoulos TermsRiccardo Barbieria, Lawrence J. Hallb,c, Yasunori Nomurab,ca Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italyb Department of Physics, University of California, Berkeley, CA 94720, USAc Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAAbstractIn [1] the one Higgs doublet standard model was obtained by an orbifold projection ofa 5D supersymmetric theory in an essentially unique way, resulting in a prediction for theHiggs mass mH = 127 ± 8 GeV and for the compactification scale 1/R = 370 ± 70 GeV.The dominant one loop contribution to the Higgs potential was found to be finite, whilethe above uncertainties arose from quadratically divergent brane Z factors and from otherhigher loop contributions. In [3], a quadratically divergent Fayet-Iliopoulos term was foundat one loop in this theory. We show that the resulting uncertainties in the predictions for theHiggs boson mass and the compactification scale are small, about 25% of the uncertaintiesquoted above, and hence do not affect the original predictions. However, a tree level braneFayet-Iliopoulos term could, if large enough, modify these predictions, especially for 1/R.In ref. [1] we obtained the 1 Higgs doublet standard model from a 5D supersymmetric theorywith both the standard model gauge particles and the Higgs boson in the bulk. The Scherk-Schwarz (SS) mechanism is employed to remove unwanted particles and symmetries: the entiresuperpartner spectrum, as well as other particles implied by 5D Lorentz invariance, are raisedto the compactification scale, 1/R. In ref. [2] we demonstrated that this is the unique suchconstruction in 5D, up to a small deformation in the orbifold boundary condition.A relevant property of the model is that the Higgs potential is calculable in terms of thecompactification scale 1/R, up to small effects from supersymmetric counterterms. The Fermiconstant determines 1/R ≈ 370 GeV, so that the Higgs mass is predicted. The leading 1 loopdiagrams for the Higgs potential are exponentially insensitive to physics at energies above thecompactification scale, but UV sensitivities can arise through the supersymmetric counterterms,which must therefore be studied. In ref. [1] the leading counterterms were found to be brane Zfactors for the top quark superfields. These are quadratically divergent, and affect the masses ofthe Kaluza-Klein (KK) modes of the top quarks and squarks, which enter the radiative diagramsfor the Higgs potential. The Higgs mass and compactification scale therefore have a sensitivity tounknown UV physics via a quadratic divergence at the two loop level, introducing uncertaintiesin the predictions for the Higgs mass and 1/R of 1% and 20%, respectively. Further uncertaintiesin the Higgs boson mass of about 6% arise from other higher loop contributions. In ref. [3] aquadratically divergent Fayet-Iliopoulos (FI) term was noticed, associated with standard hyper-charge, which escaped our attention. This introduces a quadratic sensitivity to unknown UVphysics at the one loop level, and its consequences are studied in this note.It is indeed immediate to see that the diagram of fig. 1, properly calculated for the differentKK components of the hypercharge D-term, gives rise to an effective Lagrangian termLeff =ξ√2(δ(y) + δ(y − πR/2))DY , (1)whereξ ≃ g′2∫d4p(2π)41p2≃ g′2Λ216π2, (2)andDY =D0√2+∞∑n=1Dn cos2nyR, (3)with D0 and Dn canonically normalized in 4D. Λ is an ultraviolet cutoff and g′ is the U(1)hypercharge coupling. In the loop of fig. 1, only the Higgs zero-mode contributes to the zero-mode of DY , whereas the 2n-th KK modes of DY receive contributions from the n-th KK modesof the Higgs field and of its charge conjugate. The matter hypermultiplets do not contribute toa FI term on the brane due to TrY = 0 over the standard matter multiplets.In the appendix, we comment on the theoretical issues about the generation of the FI term.1DYh, hcFigure 1: One-loop diagram generating the FI D-term.It is important to notice that Leff is perfectly compatible with the residual supersymmetriesof the full Lagrangian after the orbifold projection. This Lagrangian, other than the fully super-symmetric term in 5D, L5, must include the most general 4D Lagrangians at y = 0 and y = πR/2compatible with the (different) N = 1 supersymmetries at each of the fixed points. The FI termsin eq. (1) can indeed be there, as can be supersymmetric kinetic terms for the different fields orany other N = 1 supersymmetric operator of higher dimension. If not inserted from the start,one has to expect them from the loop expansion. In turn, their effect on the calculation of theHiggs potential has to be discussed along the lines of ref. [1].The model, being based on a non-renormalizable Lagrangian, is defined in terms of a cutoffscale M , which we take to be the scale at which the perturbative expansion in the top Yukawacoupling ceases to make sense. It is M ≃ 5/R, as can be seen from an actual perturbativecalculation or from naive dimensional analysis. We assume that perturbativity is maintainedup to M , even after the inclusion of all possible other terms in the Lagrangian. This limits theeffects of the various counterterms mentioned above on the Higgs potential. The closeness of Mto 1/R should not be viewed as an obstacle. In the chiral Lagrangian the relation of mρ to fπ isnot very different and yet the usefulness of the chiral Lagrangian itself is not disputable.Suppose that we useΛ ≃ M ≃ 5R, (4)to estimate the size of the ξ term in eq. (2). We get ξ ≃ 0.03/R2, which gives effects well withinthe uncertainties already discussed in ref. [1] both on the Higgs potential and, a fortiori, on thesuperpartner spectrum. Radiatively generated brane kinetic terms, and higher loop corrections,give larger effects, as quoted above.The symmetries of the theory allow a tree level D-term as in eq. (1), although with differingmagnitudes on the two branes. This introduces a correction to the Higgs squared mass parameterδm2φH (ξ) =g′2ξ, (5)2120125130135140145150155160-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3mH (GeV)ξR2Figure 2: The physical Higgs boson mass mH as a function of ξR2.where ξ is now the average value of the D-term on the two branes. The sign of such a term isunknown. It should be compared with the finite top loop contribution to the Higgs potential,which gives a mass squaredδm2φH (top) = −63ζ(3)8π4y2tR2≃ −0.08R2. (6)If δm2φH (ξ) were positive and bigger than |δm2φH (top)| its presence would prevent symmetrybreaking. For other values of ξ, we have minimized the Higgs potential and predict the Higgsmass and 1/R as a function of the dimensionless parameter ξR2, as shown in figs. 2 and 3.Vertical dashed lines at ξR2 = ±0.03 show the effects to be expected from the radiative FI term.The experimental limit on the top squark mass requires ξR2 to be larger than about −0.1, andelectroweak symmetry is broken only if ξR2 is less than about 0.5. The figures are not extendedto values of ξR2 above 0.3 because corrections to the top KK masses from ξ have not beenincluded in the calculation of the Higgs potential, and in this region they become important.We note that a partial cancellation can take place between δm2φH (top) and δm2φH(ξ), with acorresponding increase in 1/R and m2H itself. As ξR2 approaches the maximum value consistentwith electroweak symmetry being broken, larger values of 1/R result, but at the price of anincreasingly precise cancellation among the two contributions to the Higgs mass. Very largevalues of 1/R are therefore disfavored, although even in this case there is a strict upper limit onthe Higgs mass in the region of 180 GeV. The hypercharge gauge interaction is weakly coupledat the cutoff scale M , so that ξ cannot be estimated using strong coupling arguments. A totallynaive guess, ξ ≃ g′M2, would be clearly excluded. The theory above M must lead to suppressedtree-level values for certain brane interactions, including the FI term and flavor-dependent kineticterms for the light generations.3300400500600700800900100011001200-0.05 0 0.05 0.1 0.15 0.2 0.25 0.31/R (GeV)ξR2Figure 3: The compactification scale 1/R as a function of ξR2.A value of ξR2 ≃ 0.3 raises the compactification scale by about a factor of 3, thereby removingthe fine tuning needed to satisfy the experimental constraint on the ρ parameter found in ref. [1].Such a value of ξR2 is an order of magnitude larger than the radiative contribution, but overan order of magnitude smaller than a naive guess in 5D, and does not require fine tuning in theHiggs mass squared parameter.For values of ξ which are not too large we find analytic approximations:m2H(ξ) ≃ m2H(0) + M2Z(cos[πR(0)mt] − cos[πR(ξ)mt])+ g′ξ(1 − cos[πR(ξ)mt]), (7)and1R(ξ)≃ 1R(0)(1 +g′ξM2Z)14. (8)Furthermore, for ξ sufficiently small, as in the one loop calculation, the predictions made inref. [1] still hold:mH = 127 ± 8 GeV, (9)1/R = 370 ± 70 GeV. (10)The role of the FI term in the Higgs potential is to provide an additional contribution tothe Higgs mass squared parameter. In the case that this contribution is positive, the effect isequivalent to a deformation in the translation orbifold boundary condition discussed in ref. [2].Indeed, one can ask what measurement will provide a distinction between these theories. Theanswer lies in the details of the scalar superpartner spectrum with masses near 1/R. The bound-ary condition deformation leads to a universal shift in the scalar masses, while the FI term leadsto shifts that depend on the hypercharge of the scalar.4The DY field has a kinetic coupling to the real scalar field σ of the hypercharge chiral adjointfield: DY ∂yσ. On performing a KK expansion one discovers that the effects of the FI termcoupled to Dn, for n 6= 0, are canceled by vacuum expectation values (VEVs) of σn. Thephysical effects of the FI term discussed above all result from the coupling to the zero mode D0.The VEV of σn leads to mass mixing amongst KK modes from the gauge interaction g′[XΣXc]θ2for any hypermultiplet (X, Xc). This leads to a violation of momentum in the fifth dimensionallowing single production of excited KK modes, such as g0g0 → q̄0q2, and new decay modes ofthe excitations, such as q2 → Z0q0, where subscripts label the KK modes.In the constrained standard model introduced in ref. [1], electroweak symmetry is brokenradiatively via a finite 1 loop contribution involving the top quark and its superpartners and KKresonances. Corrections to this picture arise from supersymmetric brane interactions. There isa quadratically divergence brane FI term, as pointed out in ref. [3], but this leads to only a 2%correction in the Higgs mass, which is smaller than other corrections. It is perhaps surprising thata 1 loop quadratic divergence is so mild relative to the finite 1 loop top quark contribution. Thisresults from several factors: the top Yukawa coupling is larger than the hypercharge coupling,there is a color factor of 3, the Yukawa couplings of the KK towers are√2 times larger thanthat of the zero mode, and finally the cutoff of the theory is only about a factor of 5 abovethe compactification scale. A tree level brane FI term could be present. Since the hyperchargecoupling is highly perturbative even at the cutoff, it seems likely to us that this tree contributionfrom physics at the cutoff is comparable to the quadratically divergent radiative correction andtherefore also negligible. However, if it is larger by an order of magnitude it could lead tosignificant changes in the predictions of the theory as shown in figs. 2 and 3.AcknowledgementsY.N. thanks the Miller Institute for Basic Research in Science for financial support. This workwas supported by the ESF under the RTN contract HPRN-CT-2000-00148, the Department ofEnergy under contract DE-AC03-76SF00098 and the National Science Foundation under contractPHY-95-14797.AppendixIn this appendix we comment on theoretical issues about the generation of the FI term. Wehave seen that brane-localized operators, eq. (1), are radiatively generated in the model of ref. [1].The coefficients of the two FI terms on y = 0 and πR/2 branes are the same, so that in the 4Dpicture the Lagrangian is given byL4D =√2 ξ(1√2D0 + D2 + D4 + · · ·), (11)where Dn is the n-th KK mode of DY with “mass” 2n/R. The question is what happens for the5FI term if we take the supersymmetric limit of the theory. Does the FI term remain non-zero?To answer this question, let us describe our model using Z : y → −y and T : y → y + πRrather than Z : y → −y and Z ′ : (y − πR/2) → −(y − πR/2). Then, our model correspondsto taking Z as the “standard” Z2 parity reducing 5D N = 1 supersymmetry to 4D N = 1supersymmetry, and T as a direct product of the SS rotation with the SS parameter α = 1/2 andthe overall negative sign for the Higgs hypermultiplet. In the notation of ref. [2], it is written asZ = Σ3 ⊗ 1, (12)T = e2πiασ2 ⊗−1, (13)where α = 1/2 corresponds to the model of ref. [1].Suppose we take SS parameter α to be equal to zero. Then, 4D N = 1 supersymmetry remainsunbroken, and both Higgs and Higgsino KK towers have mass (2n+1)/R. In this case, one mightconclude that no FI term is generated since the matter content is completely vector-like; we havefull hypermultiplet states, h, hc, h̃, h̃c, at each KK level. However, the situation is not so simple.Although the matter content is vector-like, the interactions are not; Dn (n:odd) interactions donot have a charge conjugation symmetry. As a consequence, non-vanishing brane-localized FIterms are generated even in this supersymmetric case, α = 0. Indeed, a simple calculation showsthat the terms of the formLeff =ξ√2(δ(y) − δ(y − πR/2))DY , (14)are generated radiatively. In the 4D picture, this isL4D =√2 ξ (D1 + D3 + D5 + · · ·) . (15)It is important to realize that this special form of the FI terms is guaranteed by a symmetry;in the 5D picture there is a charge conjugation symmetry which is accompanied by a spacetimereflection with respect to y = πR/4, and it allows only brane-localized FI terms with oppositecoefficients at y = 0 and πR/2. Incidentally, if α takes some arbitrary values, the situation isbetween the two extreme cases α = 0 and 1/2; the size of the FI terms on the two branes aredifferent in general.Thus, we conclude that, if we have only one hypermultiplet in the bulk, brane-localized FIterms are always generated (even if supersymmetry is not broken). However, there needs to be aslight care for this statement. In the supersymmetric case of α = 0, the generated terms do notcontain the FI term for the unbroken (zero-mode) U(1) hypercharge, in contrast with the case ofα 6= 0 (see eqs. (11, 15)). This is a reasonable result since the appearance of a FI term is deeplyrelated to the U(1)-(grav.)2 anomaly in the usual 4D supersymmetric theories. In 4D theorieswith U(1)-gravitational anomaly canceled, the FI term is never generated unless we break either6supersymmetry or the U(1). In the present case with α = 0, the “FI term” appeared in thesupersymmetric limit. However, it is a brane-localized term and not a true FI term in the 5Dsense. In other words, in the 4D picture the generated FI terms are only for higher KK modesand not for the zero mode. Since the U(1) symmetries corresponding to the higher KK towersare non-linearly realized (spontaneously broken), the generation of these FI terms do not conflictwith the above general theorem in the 4D supersymmetric theories, nor break supersymmetrysince the FI terms for Dn (n > 0) are completely absorbed by the expectation values for thephysical scalar field σn coming from the gauge multiplet in 5D.References[1] R. Barbieri, L. J. Hall and Y. Nomura, Phys. Rev. D 63, 105007 (2001) [arXiv:hep-ph/0011311].[2] R. Barbieri, L. J. Hall and Y. Nomura, arXiv:hep-th/0107004.[3] D. M. Ghilencea, S. Groot Nibbelink and H. P. Nilles, arXiv:hep-th/0108184.7

A Constrained Standard Model: Effects of Fayet-Iliopoulos Terms

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