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2HDM in terms of observable quantities and problems of renormalization
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia In essential part based on the paper IFG, K. Kanishev, ArXive:1502.06346 [hep-ph], (to appear in Phys, Rev. D, July)

I. F. Ginzburg


Two Higgs doublet model 2HDM
The 2HDM describes a system of two scalar isospinor elds 1, 2 with hypercharge Y = 1. The most general form of the 2HDM potential is
1 2 + 2 ( )2 + ( )( ) + ( )( ) V = (11) 2 3 1 1 2 2 4 1 2 2 1 2 22 [ ] 5 + (12)2 + 6(11)(12) + 7(22)(12) + h.c. 2 [ ] 2 2 2 m m m - 11(11) - 22(22) - 12 (12) +h.c. 2 2 2

Its coecients (brawn are real, blue are complex) are restricted by the requirement that the potential be positive at large quasiclassical values of i (positivity constraints). This potential is described by 14 real parameters


The model contains two doublets of scalar elds with identical quantum numbers it can be described either in terms of the original elds 1, 2 or in terms of elds , , which are obtained from k by a 1 2 ^ global unitary reparameterization transformation F
(

1 2 (

^ Fgen = e-i

0

with corresponding transformation of the parameters i i. We refer to these dierent choices as dierent RPa bases. This transformation is described by 3 angles , , and 0, the parameter 0 don't inuence for co ecient Model is described by 11 relevant parameters.
!

^ = Fg en(, , ) 1 , 2 ) i/2 i( -/2) cos e sin e . -i( -/2) -i/2 - sin e cos e

)

(

)


We develop a method for nding the minimal and a comprehensive set of directly measurable quantities dening the 2HDM and have built simple example of such set. We call these quantities observables and call the chosen complete set the basic set of observables. This basic set is subdivided naturally into two subsets, dened below. We have found simple explicit expressions for the parameters of potential of the model via these observables (and non-physical parameters, xing RPa basis). Fortunately, the obtained description appeared to be simple enough.
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Extrema of the potential satisfy the stationarity equations V / i|1= 0 (i = 1, 2). The most general solution that describ es the S U (2) в U (1)Y U (1)E M symmetry breaking is expressed via two p ositive numbers vi and the relative phase factor ei as:
1 1 = 2 v1 = v cos ,
(

1

,2=

0 v1

The ground state of potential (the vacuum) is the extremum with the lowest energy, and its vacuum expectation value (v.e.v.) is v = 246 GeV. The elds i are then decomposed into their v.e.v.'s and the quantized component elds, their linear combinations describe Goldstone modes G±, G0, charged Higgses H ± with mass M± and neutral Higgses h1,2,3 with masses M1,2,3
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1 0 , 2 = , v2ei 2 2 2 v2 = v sin , v = v1 + v2 .

)

(

)


We consider couplings of each neutral Higgs P boson to a fundamental particle P by ga (P = {V (W, Z ), f = q(t, b, c, ... P and similar couplings of the standard Higgs boson of SM gSM. We use the relative couplings
Relative couplings.

), ( , µ

PP P = ga /gSM . a

Besides, we introduce dimensionless relative couplings:
H W a
+ -

g (H +W -ha) H -W a = MW /v

(

+

)

;

2 ± = g (H +H -ha)/(2M±/v ) . a

The neutrals ha generally have no denite CP parity. Couplings V and a ± are real due to Hermiticity of Lagrangian, while other couplings a are generally complex.
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We analyze the model with known vacuum using the special RPA transformed basis with v2 = 0 the Higgs (or Georgi) basis. It is obtained from mentioned basis with known v.e.v.'s by RPa transformation: substitution = , = - . remaining RPh freedom in the choice of the Higgs basis that is, independence of the physical picture from the choice of relative phase i, the RPh phase. Vise versa, any form of the potential can be obtained from the Higgs ^ ^- basis form with the transformation, FH 1 = Fgen( = - , = + ) B with -, 0 -0. We do not x in this denition the RPh phase and the irrelevant parameter 0.
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Higgs basis
(

1 ^ ^ = FH B 1 , FH B = e-i0 2 2 ^ ^ (Here FH B is obtained from Fgen by The phase factor e±i/2 represents the

(

)

(

)

cos ei/2 sin ei(/2- ) - sin e-i(/2- ) cos e-i/2

)


The potential obtained can be rewritten in the form (capital letters for elds and parameters in Higgs basis)
( ) ) 2 22 2 (~ ) + 1 -v + 2 (~ )2 + -v (~ ) ~ 2 ~2 VH B = M± 2 ~2 3 1 1 2 2 2 112 2 2 [ ( ) 2 5 v ~ ~ ~ ~ ~~ ~ ~ +4(12)(21) + (12)2 + 6 11- (12) 2 2 ] ~ ~ ~ ~ +7(22)(12) + + Evac, Ev ac = - 1 v 4. (

h.c. 8 In the Higgs basis, the decomposition of elds v.e.v. has simple form
1 = v + 1 + iG0 , 2

G+

2 = 2 + i3 . 2
&

H+


To arrive to the description in terms of physically observable elds, one should start by substituting these expressions into the potential. By choosing the unitarity gauge for the gauge elds, we omit the Goldstone modes Ga from now on. Now the potential takes the form:
Mij 2 V = M± H +H - + ij + v Ti H +H -i + v Tij k 2 Bij + - +C H +H -H +H - + H H ij + Qij kl ij k 2
ij k

l .

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The coecients


Mij

form the neutral scalar

mass matrix

1 2 Re 6 Mij = v -I m 6

The physical neutral i that diagonalize this mass matrix:
i ha = Rai ,

R e 6 -I m 6 N + R e 5 -I m 5/2 ; 2 N - Re5 -I m 5 / 2 2 Higgs states ha are such



2 N = M±/v 2 + 4

superpositions of elds
aa 2 Mij = Ri Rj Ma .

2 ij Ma h2 a a i = Ri ha Mij = , a 2 2




is a real-valued orthogonal matrix determined by the parameters of the mass matrix. It can be parameterized with three Euler angles. One of them is responsible for rephasing transformation of elds, i.e. it is irrelevant. The trace of the mass matrix is invariant under orthogonal transformations. 2 ( + ) = M 2 - M 2 . Therefore v 1 4 aa ± One of the advantages of the Higgs basis as compared to other RPa bases is the fact that the elements of rotation matrix are directly related to the relative couplings, which are, in principle, measurable: ( )
The mixing matrix

a Ri

a V = R1, a

+- H W a

H -W + a a a = R2 + iR3.

a a The phases of quantities H +W - , i.e. the ratios R3/R2 cannot be a xed because of the rephasing freedom of potential in the Higgs basis, but their relative phases for dierent ha are determined unambiguously.




The orthogonality of the mixing matrix means that its elements obey a set of relations:

i ab Ri Ri = ab ,

in particular:
i = Ra

aa Ri Rj = ij ; a V2 V |2 + |H + W - |2 = 1 . (a ) = 1 , |a a a



V 1 - V V 12 1 - (V 2 V 3 1 - (V 2

V 2


)2 )2

1-(V )2 2 0

V 3 -V V 23 1 - (V 2 - V 1 1 - (V 2

T,

)2 )2

T = 0

1

0 0 cos sin 0 - sin cos




Finally, one can read the mass matrix, written above, as expressions of some 's via elements of the mass matrix and then, withto express them via the masses of Higgs bosons and their couplings to gauge bosons:
2 2 2 v 21 = (V )2Ma ; v 24 = Ma - M± - v 21; a a a H +W - 2 2 2 = ( 2 = V H + W - M 2 . v5 ) Ma ; v 6 a aa a a a

The observables entering into this equation form a rst subset of the basic set of observables. Couplings H +W - are expressed via V . The a a phase freedom in the denition of these couplings is reproduced as a similar freedom in phases of 5, 6.
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The second subset of basic set form 3 triple Higgs couplings and one quartic coupling. Each triple Higgs vertex depends on 3, Re7, I m7, in addition to the parameters of the rst subset. The cubic interaction of neutral and charged scalars can be written as vTi H +H -i, with Ti = (3, Re7, -I m a After transformation to physical states i = Ri ha, we obtain the a corresponding couplings: g(H +H -ha) = vRi Ti. This expression is easy to solve for Ti by inverting the rotation matrix:
2 3 = (2M±/v 2)

7)i

.


a

V ±; aa

2 = (2M±/v 2) 7


a

H - W + ± . a a

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The parameter 2 can only be extracted from quartic couplings. (Each quartic Higgs vertex depends on parameter 2 in addition to parameters determined from mass terms and triple Higgs couplings). We use for basic set the vertex H +H -H +H - . The H +H -H +H - vertex enters Lagrangian in a very simple form 2 H +H -H +H -, and 2 its observation oers the simplest way to measure 2:
2 = 2g (H +H -H +H -) .

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Possible strong interaction in the Higgs sector.

The fact that free parameters of the potential naturally fall into three very distinct categories, oers a new opportunity which was absent in the SM. Before the Higgs discovery, the large coupling constant was, in principle, possible within SM. In this case, the Higgs boson would be very heavy and wide, and it could not be seen as separate particle. Instead, its dynamics would be governed by the strong interaction in the Higgs sector, which would manifest itself in the form of resonances in the WLWL, WLZL, ZLZL scattering in the 1-2 TeV energy range. In the SM this opportunity is closed by the discovery of the Higgs boson with M 125 GeV. Our analysis shows that, within 2HDM, the reasonably low values of all Higgs masses are well compatible with large 3, |7|, 2, i.e. with
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the strong interaction in the Higgs sector. To-day I like to see for opportunity that "observable"(at 3 level now) in WW, ZZ, hh production at M2 TeV, observed by both CMS and ATLAS could be signal of such strong interaction.


For rst summary
The observables of the basic set are measurable quantities, independent of each other. The models with arbitrary values of these observable parameters can in principle be realized. In some special variants of 2HDM, additional relations between these parameters may appear (for example, in the CP conserving case V = ± = 0). 3 3 Our results open the door for the study of Higgs models in terms of measurable quantities alone. It allows to remove from the data analysis the widely spread intermediate stages with complex, often model-dependent, analysis of coecients of Lagrangian. The principal possibility to determine all parameters of 2HDM from the (future) data meet strong practical limitations (which can be hidden in other approaches). In the best case, it looks the problem for a very long time.
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Renormalization scheme
The standard calculation of the radiative corrections (RC) in the model is based on the parameters of Lagrangian which are RPa dependent. This RPa ambiguity can be removed, for example, by using the renormalization procedure xing parameters of the basic set. In the modern approach the calculation of any physical eect should be supplemented by calculation of renormalized values of masses and other parameters of basic set which should be taken into account in the data analysis. For example, in some particular variant of MSSM the value of triple Higgs coupling with RC looks essentially dierent from its tree form in the SM. However, within the same approximation the using of the renormalized mass M1 makes the result close to the SM value.
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