Mu to e gamma in the 2 Higgs Doublet Model: an exercise in EFT

0.28 MB·

by  Sacha Davidson

µ eγ in the 2HDM: an exercise in EFT
→
Sacha Davidson
∗
IPNL, CNRS/IN2P3, 4 rue E. Fermi, 69622 Villeurbanne cedex, France; Université Lyon 1, Villeurbanne;
Université de Lyon, F-69622, Lyon, France
Abstract
The 2 Higgs Doublet Model (2HDM) of type III has renormalisable Lepton Flavour-Violating couplings, and its
oneandtwo-loop(“Barr-Zee”)contributionstoµ→eγ areknown. Inthedecouplinglimit,wherethemassscaleM
of thesecond doubletis muchgreater thantheelectroweak scale, themodel canbeparametrised withan Effective
6 FieldTheory(EFT)containingdimensionsixoperators. The1/M2 termsoftheexactcalculationarereproducedin
1 theEFT,providedthatthefour-fermionoperatorbasisbelowtheweakscaleisenlargedwithrespecttotheSU(2)-
0 invariant Buchmuller-Wyler list. It is found that the dominant two-loop “Barr-Zee” contributions arise mostly in
2 two-loop matching and running,and that dimension eight operators can be numerically relevant.

r
p
A
1 Introduction
6
2
This exercise was born from a puzzle: experiments that search for µ e flavour change constrain a long list of
↔
] QCD QED invariant four-fermion operators, some of which turn out to be dimension eight when SU(2) invariance
h ×
is imposed. But it is common, when describing New Physics from above m with Effective Field Theory(EFT)[1],
p W
- to use the SU(2) invariant basis of dimension 6 operators given by Buchmuller and Wyler[2] and pruned in [3]. To
p
explore when it is justified to neglect the additional four-fermion operators below m , we were looking for a model
W
e
where they might give relevant contributions. The first model we tried was the 2 Higgs Doublet Model (2HDM). It
h
[ turns out that the additional four-fermion operators (not in the Buchmuller-Wyler list) must be included below mW
to correctly reproduce the (1/M2) terms in the µ eγ amplitude of the 2HDM.
3 O →
The exercise takes place in a Type III 2HDM in the decoupling limit, where the one-loopand two-loop“Barr-Zee”
v
9 contributionstotheµ eγ amplitudeareknown[4]. Theaimistotoextractthenumericallydominantcontributions,
→
4 and identify where they arise in an EFT description.
9 Section 2 reviews the 2HDM of Type III in the decoupling limit, and the calculation in this model of the µ eγ
1 →
amplitude by Chang,Keung andHou[4](CHK). Type III 2HDMs include charged-leptonflavour-changingcouplings;
0
for simplicity, only a µ e flavour-changing interaction is allowed. The decoupling limit is taken, by requiring the
1. mass scale M of the sec↔ond doublet to be > 10v, to ensure that EFT can give a reasonable approximation to the
0 µ eγ amplitude in this model. In the fina∼l subsections, the (1/M2) and (1/M4) parts of the µ eγ amplitude
6 → O O →
are compared, for the various classes of diagrams.
1
: Section 3 sets up an EFT formalism, based on dimension six operators and one-loop RGEs, which correctly
v
reproducesallthe ([αlog]n/M2)termsoftheCHKcalculationinthefullmodel. Howtoobtaintheremainingterms
i O
X of the CHK result is outlined in Appendix B.
r Section 4 summarises the less trivial aspects of the calculation, and the location in the EFT of the Barr-Zee
a
diagrams. It should be readable independently of the more technical sections 2 and 3. There are two-and-a-half
issues with the results obtained with dimension six operators and one-loop RGEs: the four-fermion operator basis
below m only needs to be invariant under QCD and QED, so it should be enlarged with respect to the list[3] of
W
SU(2)-invariant dimension six operators. Second, two-loop effects are important, because the one-loop contribution
is suppressed by the square of the muon Yukawa coupling. Finally, dimension eight terms can be enhanced, both by
logs and by unknown couplings in the 2HDM (encapsulated in tanβ).
This exercise overlaps with several studies. Pruna and Signer[5] studied µ eγ in an EFT consisting of the SM
→
extended by a complete set of SU(2)-invariant operators, but they focused on the electroweak running above m ,
W
rather than the matching at m , so did not extend the operator basis below m . In the context of B physics,
W W
Alonso, Grinstein and Camalich [6], and Aebischer etal[7] calculated the coefficients of the enlarged operator basis
below m , given a selection of SU(2) invariant operators above m . This exercise only agrees approximatively with
W W
[6], as discussed in section 4.
∗E-mailaddress: [email protected]
1
2 µ eγ in the type III 2HDM
→
2.1 Review of the 2HDM type III in the decoupling limit
The 2HDM is a minimal extension of the Standard Model, including one extra Higgs doublet with new unknown
interactions to the Standard Model fermions and Higgs — for a review, see [8]. In the 2HDM considered here, the
extra doublet is taken to be heavy — this is the decoupling limit, and should be describable with EFT. To allow for
LFV, consider a “type III” model, where there is no discrete symmetry that distinguishes the Higgs, so no “symmetry
basis” inwhichtowritetheLagrangian(soalsonounambiguousdefinitionoftanβ). Forsimplicity,theHiggspotential
is taken CP invariant.
The Lagrangiancanbe written in the “Higgs basis”,defined suchthat H =0, and H =0, sothe doublets are
1 2
written 1 h i6 h i
G+ H+
H = , H = , (1)
1 1 v+H0+iG0 2 1 H0+iA
(cid:18) √2 1 (cid:19) (cid:18) √2 2 (cid:19)
wheretheGsaregoldstones. Thisshow(cid:0)sthatthemass(cid:1)eigenstatesAandH (cid:0)aretheC(cid:1)P-oddandchargedcomponents
±
of the vev-less H , but there can be some mixing between H and H in the CP-even scalars h,H. The convention
2 1 2
H =v/√2 implies v =246 GeV,
1
h i
4G 2 1
F = , √2G = . (2)
√2 v2 F v2
In this “Higgs basis”, in the notation of [9], the potential parameters are written in upper case, so the potential is:
V = M121H1†H1+M2H2†H2−[M122H1†H2+h.c.] (in Higgs basis)
+21Λ1(H1†H1)2+ 21Λ2(H2†H2)2+Λ3(H1†H1)(H2†H2)+Λ4(H1†H2)(H2†H1)
+ 12Λ5(H1†H2)2+ Λ6(H1†H1)+Λ7(H2†H2) H1†H2+h.c. , (3)
n (cid:2) (cid:3) o
The angle β α can be defined from the Higgs potential of type a III model, unlike β and α. It rotates between
−
the mass basis of h,H and the Higgs basis:
h = H0 s +H0c ,
1 β α 2 β α
− −
H = H0 c H0s , (4)
1 β−α− 2 β−α
so that what is here called β α is independent of the angle β that will later be defined from the Yukawas. If the
−
potential is CP-invariant, there is a simple relation for c [9]:
β α
−
Λ v2
6
cos[(β α)]sin[(β α)] = − (5)
− − m2 m2
H − h
The masses of the scalars are related to potential parameters in the Higgs basis as [9] :
v2
m2 = M2+ Λ
H± 2 3
v2
m2 m2 = (Λ Λ )
A− H± − 2 5− 4
m2 +m2 m2 = +v2(Λ +Λ )
H h− A 1 5
(m2 m2)2 = [m2 +(Λ Λ )v2]2+4Λ2v4 (6)
H − h A 5− 1 6
and the couplings to W+W are igm C gµν with
− W φWW
C =s , C =c , C =0 (7)
hWW β α HWW β α AWW
− −
In the decoupling limit [10] where Λ v2 M2, the exact relations (6) can be expanded in v2/M2 to obtain:
i
≪
m2 m2 +Λ v2 (decoupling limit)
H − A ≃ 5
v4 (Λ Λ )2
m2 +v2Λ Λ + 5− 1 (decoupling limit) (8)
h ≃ 1− M2 6 4
(cid:18) (cid:19)
and then approximating s 1 in eqn (5), gives
β α
− ≃
Λ v2 Λ v2
6 1
cos[(β α)] − 1+ +... (decoupling limit) (9)
⇒ − ≃ M2 M2
(cid:18) (cid:19)
1NeglectingthephaseambiguityχdiscussedateqnA.11of[9]
2
which confirms that in decoupling limit, h is mostly H , and H is mostly H .
1 2
TheYukawacouplingsintheHiggsbasisfortheHiggses,andthemasseigenstatebasisforthe u ,d ,e ,d ,e ,
R R R L L
{ }
are:
−LY = QjH1Ki∗jYiUUi+QiH1YiDDi+LiH1YiEEi
(cid:16)+QiHe2[K†ρU]ijUj +QiH2[ρD]ijDj +LiH2[ρ(cid:17)E]ijEj +h.c., (10)
whereQ,L areSU(2) doublets,E,U,D aresinglets,SU(2) indices areimplicit (LH =ν¯H++e¯H0), H =iσ H ,the
e 1 1 1 i 2 i∗
generation indices are explicit, and K is the CKM matrix. The Y matrices are flavour diagonal and equal to the SM
Yukawas: e
mP
[YP] =√2 j δ (11)
ij ij
v
as a result of being in the fermion mass eigenstate bases.
The ρP matrices can be flavour-changing. To obtain µ eγ without other flavour-changing processes, I neglect
→
all off-diagonal elements except ρ ρ = 0. To obtain a predictive model, and make contact with other 2HDM
µe eµ
≃ 6
literature,the diagonalelements followthe patternof one ofthe types of 2HDMwhich havea discrete symmetry that
ensures flavour conservation:
Model type ρU ρD ρE
Type I YUcotβ YDtanβ YEtanβ
− −
Type II YUtanβ YDtanβ YEtanβ (12)
− − −
Type X YUcotβ YDcotβ YEtanβ
−
Type Y YUtanβ YDcotβ YEtanβ
− −
This requires a definition of tanβ which is common to all the fermions. It can for instance be defined from the τ
Yukawa: in the mass eigenstate basis for τ ,τ , define H to be the linear combination of Higgses to which couples
R L τ
the τ, and β as the angle in Higgs doublet space between H (the vev) and H :
1 τ
H = H sinβ+H cosβ
1 2
⊥
H = H cosβ H sinβ (13)
τ 1 2
e − e
Sincetanβ isdefinedfromtheleptonYukawas,ρE tanβ inallTypesof2HDMlistedabove. Thisisunconventional,
∝
but should include the same predictions provided that tanβ is allowed to range from 1/50 50.
→
From eqn (10) the couplings to fermions are
1
Y = d YD(PR+PL)sβ α+(ρDPR+ρD†PL)cβ α dh
−L √2 − −
h i
1 i
+d YD(PR+PL)cβ α (ρDPR+ρD†PL)sβ α dH + d(ρDPR ρD†PL)dA
√2 − − − √2 −
h i
1
e YE(PR+PL)sβ α+(ρEPR+ρE†PL)cβ α eh
√2 − −
h i
1 i
+e YE(PR+PL)cβ α (ρEPR+ρE†PL)sβ α eH + e(ρEPR ρE†PL)eA
√2 − − − √2 −
h i
1
+u YU(PR+PL)sβ α+(ρUPR+ρU†PL)cβ α uh
√2 − −
h i
1 i
+u YU(PR+PL)cβ α (ρUPR+ρU†PL)sβ α uH u(ρUPR ρU†PL)uA
√2 − − − − √2 −
h i
+ u KρDPR ρU†KPL dH++h.c. + ν ρEPR eH++h.c. . (14)
−
This gives Feynman rulesnofhthe form iFφ,XP i, where: o (cid:8) (cid:2) (cid:3) (cid:9)
− ij X
YP [ρP ] YP [ρP]
Fh,L = ij s + † ijc , Fh,R = ij s + ijc (15)
ij √2 β−α √2 β−α ij √2 β−α √2 β−α
YP [ρP ] YP [ρP]
FH,L = ij c † ijs , FH,R = ij c ijs
ij √2 β−α− √2 β−α ij √2 β−α− √2 β−α
[ρU ] [ρU]
FA,L =i † ij , FA,R = i ij
uiuj √2 uiuj − √2
[ρP ] [ρP]
FA,L = i † ij , FA,R =i ij ,for P E,D ,
ij − √2 ij √2 ∈
where the possible forms for ρP are given in eqn 12.
3
2.2 µ eγ in the 2HDM
→
The decay µ eγ can be parametrised by adding the dipole operator to the Standard Model Lagrangian. In the
→
notation of Kuno and Okada[11]
4G
= Fm A µ σαβe F +A µ σαβe F (16)
meg µ R R L αβ L L R αβ
L − √2
(cid:0) (cid:1)
which gives
BR(µ eγ)=384π2(A 2+ A 2)<5.7 10 13 (17)
R L −
→ | | | | ×
where the upper bound is from the MEG experiment [12]. (The amplification factor of 384π2 = 3(16π)2 is because
2
µ eγ is a 2-body decay, and is compared to the usual three-body muon decay.). A , A are dimensionless, and
R L
→ | | | |
if A = A , then A <8.6 10 9.
R L X −
| | | | | | ×
γ γ
φ t,b W
µ
x
µ Fµφµ Feφµ e γ,Z φ γ,Z φ
µ Fφ e µ Fφ e
γ eµ eµ
Figure 1: One and two-loop diagrams contributing to µ eγ in the 2HDM, in the presence of a flavour-changing
→
Yukawa coupling F (see eqns (15)).
µe
The decay µ eγ has been extensively studied in the 2HDM [2, 13, 14], particularily in connection [16] with the
→
recent LHC[17, 15] excess in h τ µ . Chang,Hou and Keung [4](CHK) calculate the contributions to µ eγ of
± ∓
→ →
neutral Higgs bosons with flavour-changingcouplings. Their calculation can be divided into four classes of diagrams,
illustrated in figure 1: the one-loop diagrams, then three classes of two loop diagrams, that is, those with a t loop,
with a b loop and with a W loop. I use their results, and later, in matching onto SU(2) invariant operators, assume
that the chargedHiggs contribution ensures the SU(2) invariance of the results. The results of CHK are refered to as
“full-model” results in the following.
In the appendix is given a translation dictionary between the notation of CHK and here. The amplitude given by
2
CHK [4] for the one-loop diagram of figure 1, with an internal µ, is
em Fφ,LFφ,L m2 3 ∗
2√2G m A1 loop = µ µµ eµ ln µ + (18)
− F µ L −32π2 φ=h,H,A m2φ m2φ 2!
X
 
em Fφ,RFφ,R m2 3 ∗
2√2G m A1 loop = µ µµ eµ ln µ +
− F µ R −32π2 φ=h,H,A m2φ m2φ 2!
X
 
Notice that these amplitudes are suppressed by two leptonic Yukawas and an additional muon mass insertion to flip
the chirality. In the decoupling limit, with φ H,A and µ in the loop, eqn(18) for A gives
L
∈{ }
2√2G m AH,A,1 loop emµ [ρE] YE Λ6v2 [ρE]µµΛ v2 ∗ln m2µ (19)
− F µ L ≃ −32π2M2 µe µµ2M2 − 2M2 5 M2
(cid:20) (cid:21)
where m m M wasused inthe logarithm,andterms suppressedby more than M 4 were dropped. The result
H A −
≃ ≃
for A is obtained by replacing [ρE] [ρE] in the above. The one-loop diagram with the light Higgs h and a µ
L ij → ∗ij
(also in the the decoupling limit), gives
em Λ v2 Λ v2 2 ∗ m2
2√2G m Ah,1 loop µ [ρE] YE 6 +2[ρE] 6 ln µ . (20)
− F µ L ≃ −32π2m2h µe"− µµ2M2 µµ(cid:18)2M2(cid:19) # m2h
2Theconstant factors givenbyOmura et.al. [15]differ:-4 forh,H and-5 forA. However, the constant isirrelevant here because the
3 3
aim here is only to reproduce the log in EFT. Also as noted by Omura et.al., there are doubtful signs and typos/missing terms in the
Barr-Zeeformulaein[14],whichdifferfromtheformulaehere.
4
As noted long ago by Bjorken and Weinberg[18], there are two-loop diagrams which can be relevant for µ eγ.
→
Some examples are illustrated to the right in figure 1; a more complete set of diagrams with broken electroweak
symmetry can be found in [19]. The resulting two-loop contributions to µ eγ can be numerically larger than the
→
one-loopcontributions,becausetheHiggsattachesonlyoncetothe leptonline,viathe flavour-changingcoupling,and
otherwise couples to a W,b or t loop. The result for the top loop (neglecting the diagrams with internal Z exchange,
see figure 1), is
eα 1 m2 m2 ∗
2√2G m At loop = 3Q2 Fφ,LFφ,Lf( t)+FA,LFA,Lg( t )
− F µ L 16π3m t  eµ tt m2 eµ tt m2 
t φ=h,H φ A
X
eαm  Λ v2 m2  m2
t [ρE] 3Q2 [ρU] +YU 6 ln2 t 2YUΛ f( t) (21)
≃ 32π3M2 µe t tt tt 2M2 M2 − tt 6 m2
(cid:20)(cid:18) (cid:19) h (cid:21)
wherethe log2 termsofthe approximateequalityarethesumofthe heavyH andAcontributions,andthe thirdterm
is due to the light h. The approximate equality is in the decoupling limit, so uses m m M in the log, neglects
H A
≃ ≃
terms suppressed by v2/M2, and uses that, for small z [4],
z
f(z) g(z) ln2z , h(z) zlnz , f(z) g(z) z(lnz+2) . (22)
≃ ≃ 2 ≃ − ≃
(ThefunctionhwillappearintheW loopcontribution.) Thefunctionsf,gandharegiveninCHK,areslowlyvarying
near z 1 and at z = m2 /m2 0.4 they are f 0.7,g 0.9 and h 0.5. The formula for At loop is obtained by
∼ W h ≃ ∼ ∼ ∼ R
replacing FH,L FH,R in the coefficient of f, and ρF ρF in the coefficient of g. The top-loopamplitude is m ,
ij → ij † → ∝ t
as expected because a top mass insertion is required both to have an even number of γ-matrices in the loop, and to
provide the Higgs leg of the dipole operator.
Ab-quarkloopshouldbedescribedbythesameformulaasthetoploop(againneglectingtheZ-exchangediagrams),
but the A-exchange contribution will subtract from H-exchange, because of the sign difference in the couplings of A
to bs and ts (see eqn (15)):
eα 3Q2 m2 m2 ∗
2√2G m Ab loop = b Fφ,LFφ,Lf( b)+FA,LFA,Lg( b )
− F µ L 16π3 m  eµ bb m2 eµ bb m2 
b φ=h,H φ A
X
eαm  v2 m2 v2  Λ2v2 m2
b 3Q2ρE YDΛ ρDΛ ln2 b YDΛ ρD 6 ln2( b) (23)
≃ 64π3M2 b µe M2 bb 6− bb 5 M2 − m2 bb 6− bb M2 m2
(cid:20) h (cid:18) (cid:19) h (cid:21)
(cid:0) (cid:1)
where the approximate equality is in the decoupling limit, and uses the approximations eqn (22). The last term is
the light higgs contribution. Notice that the (1/M2) contributions of H and A cancelled against each other in the
O
decoupling limit. The formula for A is obtained by replacing FH,L FH,R in the coefficient of f, and ρF ρF in
R ij → ij † →
the coefficient of g.
Finally, there is a two-loopcontribution involving a W-loop; the third diagramof figure 1 is one of the many that
contribute. In the CP-conserving 2HDM considered here, A does not couple to the W or the goldstones, so cannot
appear in these diagrams. CHK compute separately the diagrams with either photon or Z exchange between the
lepton and W (see figure 1). The Z-mediated amplitude is proportional to 1 4sin2θ , and estimated by CHK
W
−
to be about 10% of the photon-mediated amplitude, so only the the γ-amplitude is considered here. The H and h
contributions individually are:
eα Λ vρE 3 f(z ) g(z )
2√2G m AW loop = 6 µe 3f(z )+5g(z )+ [g(z )+h(z )]+ φ − φ (24)
− F µ L ±32π3 M2 √2 (cid:18) φ φ 4 φ φ 2zφ (cid:19)
where z =m2 /m2, and - is for φ=H, +is for φ=h. The sum of the contributions, in the decoupling limit, is
φ W φ
eα Λ vρE m2 m2 35 m2 3 m2
2√2G m AW loop 6 µe W ln W ln W + +ln W +2
− F µ L ≃ −64π3 M2 √2 M2 M2 4 M2 2 M2
(cid:20)(cid:18) (cid:18) (cid:19) (cid:19)
3 f(z ) g(z )
h h
2 3f(z )+5g(z )+ [g(z )+h(z )+ − ] (25)
h h h h
− 4 2z
(cid:18) h (cid:19)(cid:21)
where the limiting forms of eqn (22) were used. The first line of the approximate equality is the contribution of H,
and the last line is the contribution of h where z = m2 /m2, and the parenthese evaluates to 7. The heavy H
h W h ∼
exchangedbetweenthe leptonlineandagoldstoneloopgeneratesan (log/M2)term,whichiscarefullydiscussedby
O
CHK, because the 1/M2 suppressionarisesfrom the decoupling limit of c , givenin eqn (9). CHK take the mixing
β α
−
angle c as a free parameter, so refer to this term as a “non-decoupling” contribution.
β α
−
5
2.2.1 The relative importance of the (1/M2) and (1/M4) terms
O O
Since EFT is an expansioninoperatordimension, itis interesting to know under whatconditions the (1/M2) terms
O
give a good approximation to the full answer. These terms should arise in a relatively simple EFT using dimension
six operators; a more extended EFT would be required to reproduce the (1/M4) terms. So this section compares
O
the magnitude of the (1/M2) and (1/M4) terms in the four classes of diagrams (one-, b-, t- and W-loop), with
O O
little attention to signs and 2s.
For the one-loop diagrams,eqns (19,20) give the ratio of the (1/M4) parts to the (1/M2) part as
O O
v2 2[ρE]µµΛ26+ YµEµΛ6−[ρE]µµΛ5 (1+lnMm2h2/lnmm2h2µ) (26)
(cid:18)M2(cid:19) (cid:2) 2YµEµΛ6 (cid:3)
whereIusedv2/m2 4. Recallthatinthis paper,tanβ isdefinedfromthe leptons,seeeqn(13),so[ρE] /[YE]
tanβ. Therefore, thhe≃ (1/M2) parts are larger than the (1/M4) contributions, provided that 3 µµ µµ ≡
O O
v2 Λ v2 1 1
5
Λ tanβ <1 , tanβ <1 ( )> ( ) (27)
6 M2 4Λ M2 ⇒O M2 O M4
6
If 1/50<tanβ <50, v2/M2 <0.01, and Λ Λ <1, then the (1/M2) terms give the correct order of magnitude,
5 6
∼ O
but the (1/M4) terms can be required to get two significant figures for large tanβ.
O
For the top loop, the ratio of (1/M4) over (1/M2) terms is
O O
v2 YUΛ
tt 6 (28)
(cid:18)M2(cid:19)[ρU]tt−YtUtΛ6/ln2 Mm2t2
where f(m2t) 1 was used. This shows that the (1/M2) terms are always larger (neglecting the possible can-
m2h ≃ O
cellation of [ρU]ttln2 Mm2t2 against Λ6), however, in 2HDMs where the top couples mostly to the SM Higgs such that
[ρU]ttln2 Mm2t2 ≪ Λ6, the dimension eight contribution is only suppressed by Mm2t2 ln2 Mm2t2 which is ≃ .2 for Mm2t2 = 0.01.
So the dimension eight contribution can be numerically relevant in some areas of parameter space.
For the b loop, the ratio of (1/M4) over (1/M2) terms is
O O
v2 4Λ6ρDbb†+ YbDb −ρDbbΛΛ56 1+ln2 Mm2h2/ln2(mm2h2b (29)
M2 h (cid:16) 4Y(cid:17)D(cid:16) (cid:17)i
(cid:18) (cid:19) bb
So the (1/M2) parts are larger than the (1/M4) contributions, provided that
O O
Λ ρD v2 ρD v2 1 1
5 bb <1 , Λ bb <1 ( )> ( ) (30)
4Λ YD M2 6YD M2 ⇒O M2 O M4
6 bb bb
where ρDbb 35 in 2HDMs where ρD 1. As in the case of the one-loop contribution, the logarithm multiplying the
YD ∼ bb ≃
bb
(1/M4) heavy Higgs exchange diagrams runs from M m rather than from m m , which could give a factor
b h b
O → →
2.
For the W loop, the ratio of (1/M4) over (1/M2) terms is
O O
mM2W2 lnmM2W2 345lnmM2W2 − 32 m2W ln2 m2W (31)
m(cid:16)2 (cid:17) ∼ M2 M2
ln W 12
M2 −
wheretheestimateneglectscancellations,andshowsthatthe (1/M4)contributionisonlymildlysuppressed,because
zln2z does not decrease rapidly. O
So in summary, provided that
v2 v2
Λ Λ , cotβ <1 , tanβ <1 (32)
6 ∼ 5 M2 M2
in all four types of 2HDM, then the (1/M2) terms give the correctorder of magnitude, but the (1/M4) terms can
O O
be required to get two significant figures in some areas of parameter space.
3ThereisanO(1/M4)termfromtheheavyHiggsexchangediagramsthatbenefitsfromanadditionallogenhancement—itrunsfrom
M →mµ ratherthanfrommh→mµliketheO(1/M2)lighthiggscontribution. However,thisisinsignificant,becauseM/mh≪mh/mµ.
6
2.2.2 The relative size of the four classes of diagram
The previous section suggested that it would be reasonable to use an EFT with dimension six operators. So the next
question is to determine the loop order to which the matching and running should be performed in the EFT, in order
toreproducethenumericallydominant (1/M2)terms. Sobelowarecomparedthemagnitudesofthe (1/M2)terms
O O
of the four classes of diagrams.
The (1/M2) part of the one-loop contribution arises from h exchange:
O
em
2√2G m Ah,1 loop µ[ρE] YEΛ (33)
F µ L ∼ πM2 µe µµ 6
where v2/m2 4, and 1 lnm2µ =0.998 1.
h ≃ 16π m2h ≃
Normalising the (1/M2) part of the two-loop diagrams to eqn (33) gives
O
αm2 3Q2 m2
t loop t t [ρU] ln2 t Λ ) (34)
∼ 32π2m2 Λ tt M2 − 6
µ 6 (cid:20) (cid:21)
αm2 m2
b loop b 3Q2ln2( b) (35)
∼ 16π2m2 b m2
µ h
α m2 m2
W loop t ln W 12 (36)
∼ 64π2m2 M2 −
µ (cid:20) (cid:21)
where f(m2/m2) 1 was used. Since α m2t 32, this reproduces the well-known dominance of the top and W
t h ≃ 64π2m2µ ≃
loops. Interestingly, only the top loop can be enhanced (or suppressed) by the angle β — the relative magnitude of
the other terms is mostly controlled by Standard Model parameters.
The two-looptopand b contributionscontainlog2 terms, whichshouldariseatsecondorderin the one-loopRGEs
of dimension six operators. However,there are significant terms of the top and W contributions without a log, which
presumablyariseintwo-loopmatching. AndthelogtermoftheW amplitudeshouldbegeneratedbytwo-loopRGEs.
So we see that a two-loop analysis would be required to reproduce the dominant (1/M2) terms.
O
3 The EFT version
Theaimofthissectionistoobtain,inEFT,the“leading” ([αlog]n/M2)partsoftheµ eγ amplitude. AppendixB
O →
discusses where some other parts of the full-model calculation would arise. EFT is transparently reviewed in [1], and
the EFT construction here attempts to follow the recipe given there. For simplicity, the EFT has only three scales:
the heavy Higgs scale M, the weak scale m (taken m ,m ,m ) and a low scale m (taken m ). Between M
W Z h t µ b
≃ ≃
and m , the theory and operatorsare SU(3) SU(2) U(1) invariant;below m , they are SU(3) U(1) invariant.
W W
× × ×
The EFT constructed here contains dimension six operators, which should reproduce the (1/M2) terms of the
O
µ eγ amplitude. In eqns (39)-(42) are listed the dimension six, SU(2)-invariant operators required above m .
W
→
Eqn (47) gives some additional SU(3) U(1)-invariant, dimension six operators which are required below m (but
W
×
which wouldhave been dimension eight in the SU(2)-invariantformulationappropriate abovem ). As expected, the
W
operatorbasisbelowm shouldincludeallfour-fermionoperatorsthatareofdimensionsixinaQED QCD-invariant
W
×
theory. If instead, the basis is restricted to SU(2)-invariant “Buchmuller-Wyler” operators, one cannot obtain all the
([αlog]n/M2) terms of the µ eγ amplitude (only the top loop is included).
O →
The EFT studied here is at “lowest order” in the loop expansion: tree-level matching of operator coefficients, and
one-loop RGEs. This should reproduce the (αnlogn) terms 4 in the amplitude. I do not calculate the two loop
O
matching, or the two-loop RGEs, which would allow to reproduce the dimension six contributions of the top and W.
3.1 Setting up the EFT calculation
The aim ofthis top-downEFT calculationis to reproducethe amplitude for µ eγ in the 2HDM. So the firststep is
→
tomatchoutthe heavyHiggsesH andAatthe“NewPhysicsscale” M. TheH areneglectedbecausethefull-model
±
calculation(to which the EFT is compared)includes only neutralHiggses. Presumeably the H contributionensures
±
SU(2) invariance. The 2HDM from above M is matched onto the (unbroken) Standard Model, with its full particle
content, and a selection of SU(3) SU(2) U(1) invariant dimension six operators. A list of possible operators
× ×
was given by Buchmuller and Wyler [2], and slightly reduced in [3]. Here, only those operators which are required
to reproduce the (1/M2) part of the µ eγ amplitude are selected. In matching out, the operator coefficients of
O →
the effective theory are assigned so as to reproduce the tree-level Greens functions of the full theory, at zero external
momentum.
4Since the dipoleoperator has only twofermion legs but anexternal photon, the anomalous dimensionmixingfour-fermion operators
intothedipoleis∝1/e. Soincountingpowers ofαlog,sometimesoneshouldmultiplybye.
7
The secondstep is to run the operatorcoefficients down to the scale m . This running should be performedwith
W
electroweak RGEs, which are given in [5, 20].However, since CHK separate photon and Z diagrams, and only the
photon contributions were retained in the previous section, the running from M m is performed with the RGEs
W
→
of QED. The operator coefficients evolve[21] with scale µ as
∂ α
µ C~ = emC~Γ (37)
∂µ 4π
wheretheoperatorcoefficientshavebeenorganisedintoarowvectorC~,and αemΓistheanomalousdimensionmatrix.
4π
The algorithmto calculate Γ is given, for instance, in [22]. For a squareΓ, this equationcan be perturbatively solved
to give the components of the vector C~(µ) at a lower scale µ:
α(µ) M α2(µ) M
C (M) δ [Γ] ln + [ΓΓ] ln2 +.. =C (µ) (38)
A AB− 4π AB µ 32π2 AB µ B
(cid:18) (cid:19)
At m , the W,Z,h and t should be matched out. The theory below m should be QED and QCD for all the
W W
SM fermions except the top, augmented by a complete set of QED QCD-invariant dimension six operators. For
×
simplicity, I consider only QED for the b,µ and e, plus those dimension-six operators required in matching onto the
tree-level Greens functions of the SU(2)-invariant EFT from above m .
W
Finally, the operator coefficients are run down to m , where the dipole coefficient can be used to calculate the
µ
µ eγ amplitude. In principle, the RGEs of QED and QCD should be used. However, QCD is neglected because it
→
is not included in the full-model calculation of CHK.
3.2 Matching at M and one-loop running to m
W
The operator for µ eγ is given in eqn (16). It is convenient to rescale the coefficient as
→
YE YE
2√2G m Aij(e σαβP e )F = µµCij (L H σαβP E )F µµCij Oij (39)
F µ R i R j αβ M2 D,R i 1 R j αβ ≡ M2 D,R D,R
so that the dimensionful operator coefficient is suppressed by the heavy Higgs scale M. Above m , there is a
W
hyperchargedipole andan SU(2) dipole; here only the linearcombinationcorrespondingto the photon dipole is used,
because only the QED part of the SU(2) running is included. To obtain the ([αlog]n/M2) parts of the µ eγ
O →
amplitude, only three additional operators from the pruned Buchmuller-Wyler list are required between M and m :
W
Oeµtt = (LAE )ǫ (QBU )
LEQU e µ AB t t
Oµett = (LAE )ǫ (QBU ) (40)
LEQU µ e AB t t
Oeµtt = (LAσµνE )ǫ (QBσ U )
T,LEQU e µ AB t µν t
Oµett = (LAσµνE )ǫ (QBσ U ) (41)
T,LEQU µ e AB t µν t
OeeHµ = H1†H1LeH1Eµ
OeµHe = H1†H1LµH1Ee (42)
where σ is the anti-symmetric tensor i[γ,γ], and ǫ provides an SU(2) contraction. These operators appear in the
2
Lagrangianas
Cijmn
δ = Z Oijmn+h.c. (43)
L − M2 Z
operators
X
so the coefficients C are dimensionless, and the four fermion operators are normalised such that the Feynman rule is
iC/M2.
−
The lastpairofoperators,eqn(42)willgivethe flavour-changinglight-Higgsinteractionrequiredfor the one-loop,
b-loop and W-loop amplitudes. As can be seen from the right diagram of figure 2, matching at the scale M of the
tree-level Greens functions of the full theory onto those of the SM + dimension six operators gives coefficients
Ceµ [ρE] Λ Cµe [ρE] Λ
eH = eµ 6 , eH = µe 6 . (44)
M2 − M2 M2 − M2
The running of these coefficients between M and m is neglected, because it is not required to reproduce the CHK
W
results.
The first four operators of the list above will generate the top loop contribution. Matching at M via the diagram
given on the left in figure 2 gives coefficients for the scalar LEQU operators
CLeµEtQtU [ρE]eµ[ρU†]tt CLµEetQtU [ρE]∗µe[ρU†]tt
= , = , (45)
M2 − M2 M2 − M2
8
t t
R L
ρU
tt
H1
H2 H1
Λ
6
H2 H1
ρE ρE
eµ eµ
− − − −
µ e µ e
R L R L
Figure 2: The left diagram generates the dimension six (Q U )(L E ) operator, by matching-out the heavy doublet
3 t e µ
Higgs H2 (dashed line). The right diagram generates the dimension six H1†H1(LeH1Eµ) operator.
where the negative sign is from the scalar propagator. Then, between the scales M and m , the RGEs of QED mix
t
the scalar operator Oeµtt to the tensor Oeµtt , and the tensor to the dipole, such that the coefficient of Oeµ is
LEQU T,LEQU D,L
Ceµ Ceµtt eα M eα m2
m D,L = LEQU [(2Q )(8N Q m )]log2 = 3Q2m [ρE] [ρU ] log2 t (46)
− µ M2 − M2 128π3 t c t t m 32π3M2 t t eµ † tt M2
t
where in the brackets after the first equality is the product of the scalar tensor and tensor dipole elements of the
anomalous dimension matrix Γ. This agrees with the (1/M2) part of→eqn (21) that is ge→nerated by heavy Higgs
O
exchange.
3.3 Matching at m and one-loop running to m
W µ
Attheweakscale m ,theh,W,andtshouldbematchedoutofthetheory,ontoabasisofdimensionsixoperators.
W
≃
These operators should respect the gauge symmetries below m , which, in the absence of the h,W, and Z, are QCD
W
and QED. So there is no reason to impose SU(2) on the operator basis below m .
W
1. If the matching is performed at tree or one loop, then matching out the top leaves only the contribution to the
dipole operator given in eqn (46). The top-loop contribution with a light higgs, which is (1/M2), could be
O
included by matching at two-loop.
2. InmatchingouttheHiggsh,theone-loop,andb-loop (1/M2)-contributionscanbeobtainedbymatchingonto
O
the scalar and tensor operators:
Oeµbb =(eP µ)(bP b) Oµebb =(µP e)(bP b)
S R R S R R
Oeµbb =(eσP µ)(bσP b) Oµebb =(µσP e)(bσP b)
T R R T R R
Oeµµµ =(eP µ)(µP µ) Oµeµµ =(µP e)(µP µ)
S R R S R R
Oeµµµ =(eσP µ)(µσP µ) Oµeµµ =(µσP e)(µσP µ) (47)
T R R T R R
If SU(2) were imposed, these operators would be of dimension eight (for instance, the first operator could be
written as (8)Oeµbb = (L HE )(Q HD )). However, they are of dimension six in the QCD QED-invariant
LEQD e µ 3 b ×
EFTbelowm ,andarerequiredinthe2HDMtocorrectlyreproducethe (αnlogn/M2)termsthatdimension-
W
O
six, one-loop EFT should obtain. The operators of eqn (47) are not included in the EFT analysis of µ eγ
→
performed by Pruna and Signer [5], who restrict to dimension six SU(2)-invariant operators.
From the diagrams illustrated in figure 3, one obtains
Ceµµµ [YE] [ρE] Λ v2/2 Cµeµµ [YE] [ρE] Λ v2/2
S = µµ eµ 6 S = µµ µe 6 (48)
M2 m2M2 M2 m2M2
h h
Ceµbb [YD] [ρE] Λ v2/2 Cµebb [YD] [ρE] Λ v2/2
S = bb eµ 6 S = bb µe 6 (49)
M2 m2M2 M2 m2M2
h h
so the coefficients are (1/M2) because v2/m2 4.
O h ≃
9
− − − −
b b µ µ
R L R L
YD YE
bb µµ
x x x x
µ− ρEeµΛ6/M2 e− µ− ρEeµΛ6/M2 e−
R L R L
Figure3: TherightdiagramgeneratestheQCD QEDinvariant,dimensionsixoperator(eP µ)(µP µ),bymatching-
R R
×
out the light Higgs h (dotted line), which has a SU(2)-invariant dimension-six LFV vertex represented by the grey
circle. The free Higgs legs attach to the vev. The left diagram generates a similiar operator (bP b)(eP µ) involving
R R
b quarks.
Then, between the scales m and m , the RGEs of QED mix the scalar operator Oeµbb to the tensor Oeµbb,
W µ S T
whichmixestothedipole,exactlyasinthe previouslydiscussedcaseoftops. Sowiththeanomalousdimensions
as in eqn (46), the dipole coefficient
Ceµ Ceµbbeαm m2 eαm [YD] [ρE] Λ v2 m2
m D,L = S b3Q2log2 µ = b 3Q2 bb eµ 6 log2 µ (50)
− µ M2 − M2 32π3 b m2 −64π3M2 b m2 m2
h h h
is generated, in agreement with the (1/M2) part of the b-loop contribution to µ eγ, given in eqn (23).
O →
In matching out the Higgs h, the flavour-changing operator Oeµ can also generate the dimension six operator
eH
Oeµbb = (L E )(D Q ) with coefficient ρE Λ v2/(m2M2). However, this operator is not useful for gen-
LEDQ e µ b b ∝ µe 6 h
erating µ eγ via one-loop RGEs, because there is no tensor operator for it to mix to, on the way to the
dipole. Th→e reason there is no tensor, is that σ and σγ are related: σ = iε σαβγ , which implies that
5 µν 2 µναβ 5
(eσαβγ µ)(bσ γ b)=(eσ µ)(bσµνb) or (eσαβP µ)(bσ P b)=0.
5 αβ 5 µν L αβ R
The scalar operator with three muons, Oeµµµ, mixes directly to the dipole via a penguin diagram, so the
S
coefficient of Oeµ is
D,L
Ceµ α 1 m em [YE] [ρE] Λ v2 m2
m D,L =Cµeµµ − log h = µ µµ µe 6 log µ (51)
− µ M2 S 16π2 e m 64π2M2 m2 m2
(cid:20) (cid:21) µ h h
where in the brackets is the scalar dipole element of the anomalous dimension matrix Γ. This agrees with the
→
(1/M2) part of the one-loop, light higgs diagrams eqn (20).
O
3. In matching out the W at tree or one-loop,none of the W-loop contributionto µ eγ is included, because the
→
light higgs part arises in 2-loop matching. In the full-model result of eqn (25), there is also an (1/M2) heavy
O
higgs part, but it is only enhanced by one log, and presumeably arises in the 2-loop RGEs.
4 Discussion
TheCP-conserving2HDMstudiedhereisaminimalextensionoftheStandardModel—ithasonlyoneLFVcoupling
[ρE] [ρE] (see eqn14),the magnitude ofthe othernew flavouredcouplingsis controlledby the singleparameter
µe eµ
≃
tanβ, defined in eqn (13), and the only new mass scale is the heavy doublet mass M, taken > 10v. The one and
two-loop contributions to µ eγ of the neutral Higgses were calculated by Chang, Hou and∼Keung[4], and their
→
result, in the decoupling limit, is given in eqns (20), (19), (21), (23) and (25).
Section 3 tries to reproduce the amplitude for µ eγ, in a simple EFT with three scales: M,m ,m . At the
W µ
→
scale M, the heavy Higgses were “matched out” onto dimension six, SU(2)-invariant operators. Between M and m ,
W
the operatorcoefficientsshouldrunaccordingto electroweakRGEs, butIusedthoseofQED(becauseI compareonly
to the photon diagrams of [4]). At m , the W,Z,h and t are matched out, so below m is an EFT containing all
W W
Standard Model fermions but the top, interacting via QCD, QED and various four-fermion operators. Finally the
operator coefficients run to m according to the RGEs of QED.
µ
There were three issues in reproducing the amplitude for µ eγ in EFT:
→
10