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Muon magnetic moment and the pseudo-Goldstone Higgs boson
Oleg Antipin,* Stefania De Curtis,† Michele Redi,‡ and Carlotta Sacco§
INFN, Sezione di Firenze, Via G. Sansone, 1; I-50019 Sesto Fiorentino, Italy(Received 21 July 2014; published 12 September 2014)
We compute the correction to the muon magnetic moment in theories where the Higgs is a pseudo-Goldstone boson and leptons are partially composite. Using a general effective Lagrangian we show that insome regions of parameters a sizable new physics contribution to the magnetic moment can be obtainedfrom composite fermions that could explain the 3.5σ experimental discrepancy from the standard modelprediction. This effect depends on the derivative interactions of the Higgs that do not modify the couplingof the Higgs to leptons and it does not require extremely light fermions, allowing us to easily avoid LHCbounds. Our derivations can be in general applied to dipole operators in theories with Goldstone bosonHiggs.
DOI: 10.1103/PhysRevD.90.065016 PACS numbers: 13.40.Gp, 12.60.Fr, 12.39.Fe
I. INTRODUCTION
In this paper we study new physics contributions to themuon anomalous magnetic moment in theories where theHiggs is a Goldstone boson (GB) and leptons are partiallycomposite, see [1] for a review. These models are stronglymotivated by the hierarchy problem because the Higgsboson, being a composite state, is not sensitive to scalesmuch shorter than its size. This points to a scale ofcompositeness around TeV that is being tested at the LHC.Our main motivation here is the long standing muon
magnetic moment anomaly
Δaμ ¼ aexpμ − aSMμ ¼ ð2.8� 0.8Þ × 10−9 ð1Þ
[aμ ¼ ðgμ − 2Þ=2] whose size suggests a new physicscontribution of the order of the standard model (SM)electro-weak correction. In renormalizable theories whereSM fields mix with heavy leptons the contribution scales as
Δaμ ∼g2ψ
ð4πÞ2m2
μ
Λ2ð2Þ
where Λ is a new physics scale associated with the heavyfermions and gψ their coupling to the Higgs. At face valuethe effect is typically too small unless the fermions are aslight as 200–300 GeV, in agreement with explicit models[2–5]. In theories with GB Higgs new diagrams arise fromthe nonlinearities of the theory demanded by the sym-metries and also UV contributions from the compositesector dynamics are expected. We wish to show that thesize of these effects could account in certain regions ofparameters for the anomaly (1), compatibly with bounds
from flavor physics, LHC searches and electro-weakprecision tests.
II. PARTIALLY COMPOSITE MUON
We work within the framework of composite Higgsmodels with partial compositeness. The Higgs is a GB ofsome strongly coupled theory with global symmetry Gspontaneously broken to a subgroupH at a scale f > v. Forconcreteness we will focus on the minimal models based onSOð5Þ=SOð4Þ but our results can be extended to otherpatterns of symmetry breaking and different representa-tions, see [6]. SM fermions are partially composite, mixingwith states of equal quantum numbers under the SM gaugesymmetries.The Lagrangian for the composite states can be described
in the most general fashion using the Callan-Coleman-Wess-Zumino (CCWZ) formalism [7]. We focus here onnew composite fermions and do not include vector reso-nances for simplicity and because they are typicallyheavier. Composite states are classified according to theirrepresentation under the unbroken group. The most generalLagrangian compatible with the symmetries can be con-structed with the aid of the connections eμ and dμ bywriting down all possible invariants under the unbrokengroup. The connections are explicitly reported inAppendix A for the coset SOð5Þ=SOð4Þ. Elementary fieldscan be introduced assigning them to a representation ofSOð5Þ and writing the most general couplings to thecomposite states using the GB matrix U.For concreteness we study in detail the scenario where
the left and right chirality of the muon couple to compositefermions in the 5 of SOð5Þ but we provide the tools forcomputing in general dipole moments in models with GBHiggs. For the top quark, a model with the same structurecan be found in [8], see also [9]. We refer to [8] andAppendix A for details on the notation. We focus on asingle generation and comment on the flavor structure in
*[email protected]†[email protected]‡[email protected]§[email protected]
PHYSICAL REVIEW D 90, 065016 (2014)
1550-7998=2014=90(6)=065016(9) 065016-1 © 2014 American Physical Society
Sec. IV. The composite states decompose into a quadrupletand a singlet under SOð4Þ,
5 ¼ 4þ 1∶ ψ4 ¼1ffiffiffi2
p
0BBB@
iðE−2 − NÞE−2 þ N
iðE−1 þ EÞE − E−1
1CCCA; ψ1 ¼ ~E ð3Þ
with Lagrangian
Lcomp ¼ ψ4ðiD −m4Þψ4 þ ψ1ðiD −m1Þψ1
þ idaμ½cLψ a4Lγ
μψ1L þ cRψ a4Rγ
μψ1R þ H:c:� ð4Þ
where
Dμψ1 ¼ ½∂μ þ ig0Bμ�ψ1
Dμψ4 ¼ ½∂μ − ieμ þ ig0Bμ�ψ4
daμ ¼ffiffiffi2
p
fDμπ
a þ…; ð5Þ
g0 and Bμ are the SM hypercharge coupling and field and πa
are the four components of the Higgs doublet. The secondline in (4) contains the leading derivative interactions of theHiggs with the fermions (controlled by the symmetrybreaking scale f) that are characteristic of GB theories.These will play a crucial role in the computation of Δaμ.Based on general power counting arguments we assumecL;R to be of order one [10]. The derivative couplings are ingeneral complex unless the composite sector respects CP.Moreover cL ¼ cR if parity is preserved.
The mixing with the elementary fermions is given by,
−Lmixing ¼ yL4fðl5LÞIUIaψ
a4 þ yL1
fðl5LÞIUI5ψ1
þ y�R4fðμ5RÞIUIaψ
a4 þ y�R1
fðμ5RÞIUI5ψ1 þ H:c:
ð6Þ
where
l5L ¼ 1ffiffiffi2
p
0BBBBBB@
−iνLνL
iμLμL
0
1CCCCCCA; μ5R ¼
0BBBBBB@
0
0
0
0
μR
1CCCCCCA: ð7Þ
Diagonalizing the mass matrix one finds the followingexpression for the muon mass
mμ ≈f2ffiffiffi2
p�yL4
yR4
m4
−yL1
yR1
m1
�shch ð8Þ
valid to leading order in the mixings. We recall that thetrigonometric dependence (sh ≡ sin h=f, ch ≡ cos h=f) isdetermined by the representations of the global symmetry.One can always choose the phases so thatmμ is real and wewill assume this choice in the rest of the paper.
A. Contributions to aμWe parametrize the dipole moment operator of the
muon as
Xμ
4mμμLσ
μνμReFμν þ H:c: ð9Þ
For mμ real, aμ ¼ Re½Xμ�, while the imaginary partcontributes to the electric dipole moment (EDM).
FIG. 1 (color online). Diagrams contributing to Δaμ. On the first line the diagrams with gauge and Yukawa interactions are shownwhile on the second line the ones with Higgs derivative interactions.
ANTIPIN et al. PHYSICAL REVIEW D 90, 065016 (2014)
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At 1-loop the new physics contribution to Xμ arises fromdiagrams with heavy fermions χ in the loop with charge-2, -1 or 0 and SM gauge fields or Higgs. To leading orderΔXμ is generated by diagrams with one left and one rightmixing corresponding to the function G in the expressionsreported in Appendix B.There are two classes of contributions drawn in Fig. 1.
The first corresponds to diagrams with heavy compositefermions in the loop and W, Z or Higgs with nonde-rivative interactions. These are analogous to the onesconsidered in renormalizable theories with vectorlikefermions [2–5,11,12] except that the couplings of thecomposite leptons have new contributions from the con-nections eμ and dμ. With the standard formulas collected inAppendix A, the contribution of heavy fermions comingfrom this first class of diagrams reads,
ΔXZμ ≃ mμmχ
4π2v2ðgZLÞðgZRÞ�
ΔXW−μ ≃ −
mμmχ
8π2v2ðgW−
L ÞðgW−
R Þ�
ΔXWþμ ≃ mμmχ
8π2v2ðgWþ
L ÞðgWþR Þ�
ΔXhμ ≃ 1
16π2mμ
mχðλLÞðλRÞ� ð10Þ
(v ¼ 246 GeV) valid to first order in the mixings and in thelimit mχ ≫ mZ;W;h. These contributions can be, in general,complex and generate both electric and magnetic dipolemoments. Within an explicit model the couplings in theequation above are obtained by rotating the matrices ofcouplings to the mass basis. For this purpose, given thesmallness of the muon mass, it is sufficient to use therotation matrices to first order in the mixings. In the CCWZparametrization this is particularly simple since the onlyoff-diagonal terms in the mass matrix are the elementary-composite mixings. The contribution from Higgs exchangeis not sub-leading contrary to the SMwhere it is suppressedbym2
μ=m2h compared to the gauge one. Note that in theories
with vector-like leptons without GB structure the Higgs canhave additional non-derivative interactions with the heavyfermions that dominate [11,12].The second type of contribution is strictly associated
to the GB nature of the Higgs and is analogous to the oneconsidered for dipole moment of baryons in QCD, see [13]and Refs. therein. The term in the Lagrangian (4)proportional to cL;R contains a derivative interaction ofthe Higgs with the composite fermions. Through this vertextwo new diagrams can be drawn that contribute to thedipole moment shown on the second line of Fig. 1. Weevaluate these new contributions in Appendix B. The loopdiagrams are finite but their values depend on the regu-larization procedure. Evaluating the integrals in 4D onefinds,
ΔXð∂hÞ2μ ≃ −
1
48π2mμmχ
f2cLc�R
ΔX∂hhμ ≃ 1
24π2mμ
fðcLλ�R − λLc�RÞ; ð11Þ
valid within the same approximations as above.Before analyzing the explicit model above let us discuss
the general structure of the result. The chiral structure ofdipole moments is identical to the one of mass terms. As aconsequence, the group theoretical structure, controlled bythe global symmetries of the theory, is also similar in thetwo cases. To leading order the dipole moment must beproportional to the product of the mixings of left and rightchirality of the muon. The Higgs dependence can bedetermined using a spurion analysis. To do this one shouldassign the elementary fields to a representation of the globalsymmetry and write all the invariants under the unbrokengroup using the GB matrix, see [6] for more details. Onefinds,
ΔXμ ¼XA;i;j
xijAyiLy
jRðlLÞiUPij
AU†ðμRÞj ð12Þ
where ðlLÞi and and ðμRÞj denote the embedding of theelementary fields into G representations riL and rjR and Pij
Aare the projectors over the irreducible H representationscontained in the product of riL × rjR. The coefficients xijAcontain the dynamical information.When a single invariant exists, ΔXμ will always be
proportional to the muon mass because the Yukawacouplings have an identical expansion as Eq. (12). Forthe model in Eqs. (4), (6) this can be realized whenyL4
¼ yL1and yR4
¼ yR1(other possibilities are yL1
¼yR1
¼ 0 or yL4¼ yR4
¼ 0). In this case one finds,
ΔXμ ∼κ
16π2m2
μ
f2ð13Þ
where κ depends solely on the parameters of the compositesector and can be complex only if the composite sectorviolates CP. When elementary fields couple to more thanone state as in (6) or several invariants arise in thedecomposition of rL × rR, ΔXμ will not be proportionalto mμ but will depend explicitly on the mixing parameters.In particular it can be complex even if the composite sectorrespects CP.
III. RESULTS
We now apply the tools described in the previous sectionto the model given by the eqs. (4), (6). The relevantcouplings of the muon to the heavy fermion resonances canbe extracted from Appendix A. Using the formulas abovewe find,
MUON MAGNETIC MOMENT AND THE PSEUDO-GOLDSTONE … PHYSICAL REVIEW D 90, 065016 (2014)
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ΔXμ ≃ m2μ
16π2f2þ mμ
16π2
�1ffiffiffi2
pm4
yL4yR4
−c�Lm1
yL1yR4
−cRm1
yL4yR1
þffiffiffi2
p c�LcRm4
m21
yL1yR1
�shch
þ mμ
24π2
��cLm4
−cRm1
�yL4
yR1þ�c�Rm4
−c�Lm1
�yL1
yR4
�shch
þ mμ
24ffiffiffi2
pπ2
�m4cRc�Lm2
1
yL1yR1
−m1cLc�Rm2
4
yL4yR4
�shch ð14Þ
to leading order in the mixings. On the first line there arethe contributions from nonderivative interactions mediatedby the Higgs and the Z boson, respectively. The contribu-tion of W loops is zero due to a cancellation between thediagrams with doubly charged and neutral heavy fermion inthe loop. In the second and third lines we show thecontributions from the derivative Higgs interactions.In Fig. 2 we plot a scan over the parameters of the model
assuming real parameters and cL ¼ cR ¼ c. A sizablecontribution to Δaμ can be generated and the effect doesnot require extremely light fermions. Δaμ tends to growwith c but larger values of c may lead to tension withbounds from S parameter, see discussion below. We shouldnote that ΔXμ is in general complex, even for realcomposite sector parameters. This implies strong boundsif a similar contribution is induced for the electron [14].An interesting special case is obtained when the left and
right chirality of the muon couple to a single operator of thestrong sector. This can be realized for yL1
¼ yL4and yR1
¼yR4
and it is the scenario effectively realized in extra-dimensional constructions (deviations form this relationcorrespond to nonminimal terms studied in [9]). For cL ¼cR ¼ c and real (CP and parity conserving compositesector) one finds,
ΔXμ ≃ m2μ
16π2f2
�1þ ðm1 −
ffiffiffi2
pcm4Þ2
m1ðm1 −m4Þ
þ 8
3ffiffiffi2
p c −2ðm2
1 þm1m4 þm24Þ
3m1m4
c2�: ð15Þ
As expected ΔXμ is expressed in terms of the muon massand composite sector parameters and it is real, contributingonly to the magnetic dipole moment. In Fig. 3 we show acontour plot of Δaμ as a function of m4=m1 and c. Δaμ isenhanced for a small splitting between the quadruplet andsinglet masses and grows with c. The light green regiongives a contribution to Δaμ in agreement with the exper-imental value at 1σ.
A. Bounds
The phenomenology of partially composite leptons wasdiscussed in [16], (see also [4]). Due to the smallness oftheir masses the compositeness of SM leptons is typicallysmall leading to very mild constraints from modifiedcouplings and compositeness bounds. For example thecorrection to the coupling of left-handed muons in themodel discussed above is
�2 �1 0 1 2�1. 10�8
5. 10�9
0
5. 10�9
1. 10�8
c
a Μ
FIG. 2 (color online). New physics contribution to Δaμ forcL ¼ cR ¼ c (real) and f ¼ 800 GeV. The scan is performedby choosing y ⊂ ½−0.1; 0.1� and m1;4 ⊂ ½300; 3000� GeV. Bluepoints corresponds to fermionic contribution to the S parameterΔS < 0.5 assuming 3 degenerate generation partners (we use theformulas with finite terms of Ref. [15]). The green bandrepresents the experimental value for Δaμ within 2σ.
2 1 0 1 20.6
0.8
1.0
1.2
1.4
c
m4
m1
6 10 9
2 10 9
2 10 9
6 10 9
FIG. 3 (color online). Contribution to Δaμ in the scenario withyL1
¼ yL4and yR1
¼ yR4for f ¼ 800 GeV. The 2σ experimental
value is reproduced in the region between the dashes lines. Thewhite horizontal strip corresponds to yL;R > 0.1. Some of theallowed regions are not sampled in the scan corresponding toFig. 2 due to the finite number of points.
ANTIPIN et al. PHYSICAL REVIEW D 90, 065016 (2014)
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δgZμLμLgSMZμLμL
≃ −v2
1 − 2s2W
�y2L1
2m21
þ y2L4
2m24
−ffiffiffi2
pcyL1
yL4
m1m4
�ð16Þ
while the coupling of μR does not receive corrections at treelevel. Large effects can only be obtained if one chirality ofleptons is strongly composite. The most important indirectconstraint arises from the S parameter. As we have seen, thederivative coupling proportional to c is a key ingredientto obtain a sizable contribution to ΔXμ, unless the reso-nances are almost degenerate. The same parameter alsoinduces a calculable correction to S from loops ofcomposite fermions [8,15],
ΔS ≃ 2
π
v2
f2ð1 − 2c2Þ log Λ
2
m24
þ finite terms ð17Þ
where Λ is an UV cutoff and finite terms depend on theregularization scheme. In the formula above we included amultiplicity factor for 3 generations. Indeed, realizingminimal flavor violation (MFV) in these models requiresa degenerate spectrum and couplings across differentgenerations [17]. In Fig. 2 red points correspond to afermionic contribution ΔS > 0.5 and are therefore disfa-vored from the experimental bound. Other contributions toS could however compensate for this effect.Direct searches from LHC exclude composite partners
only up to 300–400 GeV. The most significant differencefrom other models of vectorlike leptons concerns Higgscouplings. The mass spectrum and, as a consequence, thecoupling of the Higgs to muons (hμμ) does not depend oncL;R,
hμμhSMμμ
≃ 1 −3
2
v2
f2: ð18Þ
The modification of the Higgs coupling to fermions is infact universal to leading order, depending only on therepresentation. With a phenomenologically plausible valuef ¼ 800 GeV or larger, hμμ does not place a significantbound on our scenario. This removes the correlationbetween ΔXμ and the Higgs couplings found in renorma-lizable models [2,3]. In those references the contribution toΔXμ needed to reproduce the experimental anomaly wouldimply an order 5-10 modification of the decay rate of theHiggs to muons, that is on the verge of being excluded byLHC measurements. Moreover in a complete flavor picturerealizing MFV an identical modification of the τ couplingto the Higgs would be generated that is grossly excluded byLHC measurements.ΔXμ in (14) is in general complex so that the imaginary
part contributes to the muon EDM. When only twocouplings exist the phase is different from zero if thecomposite sector violates CP (cL;R complex) and parity
(cL ≠ cR). At present this does not provide a constraint forthe muon but an analogous contribution for the electron istightly constrained [14]: the imaginary part should besuppressed by a factor 10−3 relative to Δae.
IV. DISCUSSION
In this paper we computed the anomalous magneticmoment of the muon in theories with GB Higgs andpartially composite fermions. Some new features arisecompared to renormalizable theories studied in the liter-ature. In particular, interactions associated to the GB natureof the Higgs give extra contributions that can enhance Δaμand new diagrams with Higgs derivative interactions existthat can give a sizable effect. Our results show that it isplausible in certain regions of parameters to obtain acontribution that would account for the experimentalanomaly. This depends crucially on the model dependentcoupling c that controls the interactions of the Higgs withthe composite fermions.We should note that, working within a nonrenormaliz-
able effective field theory, our results should be interpretedas an estimate of the size of Δaμ in this type of theories.Certainly we also expect UV contributions to the muonmagnetic moment that are uncalculable in our framework.In particular composite sector operators such as,1
1
ΛΨi
4LσμνΨj
4RðTaÞijðfþμνÞa þ H:c: ð19Þ
contribute to the magnetic moment of the muon. Assumingthat dipoles are suppressed by a loop of the strongdynamics (as for example in weakly coupled 5D realiza-tions of our framework) we find that their typical size is
ΔaUVμ ∼1
16π2m2
μ
f2ð20Þ
which is an order of magnitude smaller than required toreproduce the anomaly for f ¼ 800 GeV. The IR contri-bution from loops of light degrees of freedom would be inthis case dominant. Nevertheless, we cannot a prioriexclude that larger UV contributions are present.It is interesting to cast our results into the broader flavor
picture of partially composite Higgs models, see [16,17] fora detailed discussion. The hypothesis of partial compos-iteness can suppress flavor transitions beyond the SM.Nevertheless, severe bounds exist especially in the leptonsector. For example Br½μ → eγ� < 5 × 10−13 hints to ascale of compositeness Λ > 50 TeV much larger thanthe value expected for these models if they are relevant
1We define fμν ≡ U†FμνU ¼ ðfþμνÞaTa þ ðf−μνÞaTa≡fþμν þ f−μν.
MUON MAGNETIC MOMENT AND THE PSEUDO-GOLDSTONE … PHYSICAL REVIEW D 90, 065016 (2014)
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to the hierarchy problem. Tension with flavor constraintscan be eliminated if the theory realizes MFV. In fact, partialcompositeness allows us to elegantly realize this hypoth-esis: this requires that the composite sector possesses flavorsymmetries that are only broken by mixings proportional tothe SM Yukawa couplings. This can be realized if left-handed or right-handed fermions have equal degree ofcompositeness. One interesting prediction is that the con-tribution to the ðg − 2Þ of the electron is related to the oneof the muon as,
ΔaeΔaμ
¼ m2e
m2μ
ð21Þ
that could be of interest in future experiments [14].Moreover, contributions to EDMs are automatically zeroat 1-loop if the strong sector also respects CP.Our results can be extended in various directions. First,
models with different representations of composite fer-mions or different patterns of symmetry breaking can bestudied with the techniques described in this paper andother dipole moments relevant for composite Higgs modelscan be computed. One obvious generalization is forexample the computation of chromo-magnetic operatorsin the quark sector. The same type of effects studied herealso appears in models with extra-dimensions that corre-spond to an infinite number of resonances with derivativecouplings determined by the metric. Finally the contribu-tion of composite spin-1 resonances could also be studiedalong the lines described in this paper.
ACKNOWLEDGMENTS
The work of O. A. and M. R. is supported by theDOEMIUR-FIRB Grant No. RBFR12H1MW. M. R.would like to thank Roberto Contino and GiulianoPanico for discussions.
APPENDIX A: RELEVANT FORMULAS
In the CCWZ formalism one introduces the GB matrix,
U ¼ eiffiffi2
pf π
aTa ðA1Þwhere Ta are the broken generators, and constructs theMaurer-Cartan form
U†½Aμ þ i∂μ�U ¼ iU†DμU ¼ idaμTa þ ieaμTa: ðA2Þ
Explicitly for SOð5Þ=SOð4Þ this is given by,
daμ ¼ffiffiffi2
p �1
f−sin π=f
π
�~π ·Dμ~π
π2πa þ
ffiffiffi2
p sin π=fπ
Dμπa
eaμ ¼ −Aaμ þ 4i
sin2ðπ=2fÞπ2
~πTtaDμ~π ðA3Þ
with ta the SOð4Þ generators in 4x4 matrix form and
Dμπa ¼ ∂μπ
a − iAaμðtaÞabπb: ðA4Þ
For the model in Sec. II the Lagrangian can be writtenexplicitly as,
L ¼ Lkinetic − ðΘLM−1ΘR þ N LMNNR þ H:c:Þ −m4E−2E−2
þ gffiffiffi2
p ½N LgWNL WþΘL þ NRgWN
R WþΘR þ E−2LgWCL W−ΘL þ E−2RgWC
R W−ΘR þ H:c:�
þ gcW
½ΘLgZLZΘL þ ΘTRg
ZRZΘR� þ i
cLfΘLR∂hΘL þ i
cRfΘRR∂hΘR ðA5Þ
where we have defined the fields,
ΘL;R ¼
0BBB@
μ
E
E−1
~E
1CCCA
L;R
N L ¼�
ν
N
�L
ðA6Þ
the mass matrices
M−1 ¼
0BBBBB@
0 yL4f 1þch
2yL4
f 1−ch2
yL1f shffiffi
2p
−yR4f shffiffi
2p m4 0 0
yR4f shffiffi
2p 0 m4 0
yR1fch 0 0 m1
1CCCA
MN ¼�yL4
f
m4
�ðA7Þ
and the couplings
ANTIPIN et al. PHYSICAL REVIEW D 90, 065016 (2014)
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gZL ¼
0BBBBB@
− 12þ s2W 0 0 0
0 − ch2þ s2W 0 −cL
sh2
0 0 ch2þ s2W −cL
sh2
0 −c�Lsh2
−c�Lsh2
s2W
1CCCCCA
gZR ¼
0BBB@
þs2W 0 0 0
0 − ch2þ s2W 0 −cR
sh2
0 0 ch2þ s2W −cR
sh2
0 −c�Rsh2
−c�Rsh2
s2W
1CCCA
gWNL ¼
�1 0 0 0
0 1þch2
1−ch2
cLsh
�gWNR ¼
�0 1þch
21−ch2
cRsh�
gWCL ¼
�0 1−ch
21þch2
−cLsh
�gWCR ¼
�0 1−ch
21þch2
−cRsh
�
R ¼
0BBB@
0 0 0 0
0 0 0 1
0 0 0 −10 −1 1 0
1CCCA: ðA8Þ
The relevant couplings used in the paper (denoted with a“hat”) are obtained rotating to the physical mass basisdefined by Eq. (A7) (Higgs Yukawa couplings are given byλ ¼ dM−1=dhhi). Explicit formulas are easily derived tofirst order in the mixings sufficient for the analysis in thispaper.
APPENDIX B: DIPOLE MOMENTS
In this appendix we present the relevant formulas fordipole moments in theories with GB Higgs. At 1-loop only
states with charge -2, -1, 0 (χ−2;−1;0) contribute. Weconsider the following interaction terms,
Lint¼ ½Vμ0g
V0
L μLγμχ−1LþVμþg
VþL μLγμχ−2LþVμ
−gV−L μLγμχ0L
−λLμLhχRþ iCL
fμL∂hχLþðL→RÞ�þH:c: ðB1Þ
1. Nonderivative interactions
With the couplings on the first line one finds thefollowing contributions to the muon magnetic moment,
ΔXV0μ ¼ m2
μ
8π2m2V0
�ðjgV0
L j2 þ jgV0
R j2ÞFV0ðxÞ þ gV0
L ðgV0
R Þ�GV0ðxÞmχ
mμ
�
ΔXV−μ ¼ m2
μ
16π2m2V−
�ðjgV−
L j2 þ jgV−R j2ÞFV−
ðxÞ þ gV−L ðgV−
R Þ�GV−ðxÞmχ
mμ
�
ΔXVþμ ¼ m2
μ
16π2m2Vþ
�ðjgVþ
L j2 þ jgVþR j2Þð4FV0
ðxÞ þ FV−ðxÞÞ þ gVþ
L ðgVþR Þ�ð4GV0
ðxÞ þ GV−ðxÞÞmχ
mμ
�
ΔXhμ ¼
m2μ
16π2m2h
�ðjλLj2 þ jλRj2ÞF hðxÞ þ λLλ
�RGhðxÞ
mχ
mμ
�ðB2Þ
respectively for diagrams with V0, V� and h in the loop. Here mχ the mass of the heavy fermion. The loop functions aregiven by
MUON MAGNETIC MOMENT AND THE PSEUDO-GOLDSTONE … PHYSICAL REVIEW D 90, 065016 (2014)
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FV0ðxÞ ¼ −5x4 þ 14x3 þ 18x2 log x − 39x2 þ 38x − 8
12ðx − 1Þ4ðB3Þ
GV0ðxÞ ¼ x3 − 6x log xþ 3x − 4
2ðx − 1Þ3 ðB4Þ
FV−ðxÞ ¼ 4x4 þ 18x3 log x − 49x3 þ 78x2 − 43xþ 10
6ðx − 1Þ4ðB5Þ
GV−ðxÞ ¼ −x3 − 6x2 log xþ 12x2 − 15xþ 4
ðx − 1Þ3 ðB6Þ
F hðxÞ ¼x3 − 6x2 þ 6x log xþ 3xþ 2
6ðx − 1Þ4 ðB7Þ
GhðxÞ ¼x2 − 4xþ 2 log xþ 3
ðx − 1Þ3 ðB8Þ
with x ¼ m2χ=m2
V0;Vþ;h.
2. Derivative interactions
The contribution of the diagram with two Higgs deriva-tive interactions is formally given by,
ΔXð∂hÞ2μ ∼
Z1
0
uduZ
d4lð2πÞ4
Al2 þ Bðl2 − ΔÞ3
A ¼ ð2 − 3uÞ½m2μðjCLj2 þ jCRj2Þ −mμmχCLC�
R�B ¼ 2f½m4
μðu2 − u3Þ −m2μm2
χu2�ðjCLj2 þ jCRj2Þ−m3
μmχu3CLC�Rg
Δ ¼ uðu − 1Þm2μ þ ð1 − uÞm2
h þ um2χ : ðB9Þ
Naively the integral over momenta is logarithmicallydivergent and needs to be regularized. One can see thatupon integration over u the result is finite but it depends onthe regulator chosen. The different results correspond to theaddition of UV local operators such us (19) to the effectiveaction. For our estimates we perform the integral in 4D.Neglecting the muon mass relative to mh and mχ we find,
ΔXð∂hÞ2μ ¼ −
m2μ
16π2f2
�ðjCLj2 þ jCRj2ÞF ð∂hÞ2ðxÞ þ CLC�
RGð∂hÞ2ðxÞmχ
mμ
�
F ð∂hÞ2ðxÞ ¼−2x4 − 12x3 þ 6ð2x − 1Þx2 log xþ 27x2 − 16xþ 3
6ðx − 1Þ4
þm2μ
m2χ
3x4 þ ð24x4 − 12x3Þ log xþ 10x3 − 18x2 þ 6x − 1
12ðx − 1Þ5
Gð∂hÞ2ðxÞ ¼2x3 − 6x2 log xþ 3x2 − 6xþ 1
3ðx − 1Þ3 −m2
μ
m2χ
2x3 − 6x2 log xþ 3x2 − 6xþ 1
6ðx − 1Þ3 : ðB10Þ
The diagram with one derivative interaction and a Yukawa coupling has very similar features. In this case one finds,
ΔX∂hhμ ¼ −
mμ
16π2f½ðC�
LλL þ CRλ�RÞF ∂hhðxÞ þ ðCLλ
�R − λLC�
RÞG∂hhðxÞ�
F ∂hhðxÞ ¼mμ
mχ
6x3 log x − 11x3 þ 18x2 − 9xþ 2
3ðx − 1Þ4
G∂hhðxÞ ¼x3 − 6x2 log xþ 6x2 − 9xþ 2
3ðx − 1Þ3
þm2μ
m2χ
7x4 þ 12ð2x4 − 2x3 þ x2Þ log xþ 12x3 − 36x2 þ 20x − 3
12ðx − 1Þ5 ðB11Þ
ANTIPIN et al. PHYSICAL REVIEW D 90, 065016 (2014)
065016-8
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