Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/proceedings2015/Higgs_Ginzburg_2HDM.pdf Äàòà èçìåíåíèÿ: Fri Nov 6 18:25:42 2015 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:30:45 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

Two-Higgs-doublet model in terms of observable quantities and problems of renormalization
I. F. Ginzburg Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia

We found a minimal and a comprehensive set of directly measurable quantities defining the most general twoHiggs-doublet model (2HDM), we call these quantities observables. The potential parameters of the model are expressed explicitly via these observables (plus nonphysical parameters which are similar to gauge parameters). The model with arbitrary values of these observables can, in principle, be realized (up to general enough limitations). Our results open the door for the study of Higgs models in terms of measurable quantities only. The experimental limitations can be implemented here directly, without complex, often model-dependent, analysis of the Lagrangian coefficients. The principal opportunity to determine all parameters of the 2HDM from the (future) data meets strong practical limitation. It is the problem for a very long time.

1

Introduction

The essential part of this report is based on results [1]. The recent discovery of a Higgs-like particle with M 125 GeV at the LHC [2] hints that the spontaneous electroweak symmetry breaking is most probably realized by the Higgs mechanism. The minimal realization of the Higgs mechanism introduces a single scalar isodoublet with the Higgs potential VH = -m2 ( )/2 + ( )2 /2. This model is usually called "the Standard Model" (SM). The experimental results favor the realization of that minimal scenario [3] (SM-like scenario [4], or SM alignment limit [5]). Nevertheless, many variants of extended Higgs models are not ruled out. The two Higgs doublet model (2HDM) presents the simplest extension of the standard Higgs mechanism [6]. This name unites a group of models in which the standard Higgs doublet is supplemented by an extra hypercharge-one doublet. It offers a number of phenomenological scenarios with different physical content realized in different regions of the model parameter space (see e.g. in [7]). After electroweak symmetry breaking the 2HDM contains three neutral Higgs bosons h a h1,2,3 and charged Higgs boson H ± with masses Ma , M± respectively. In the SM, parameters of the Higgs potential can be treated as measurable quantities. These are the 2 mass of the Higgs boson Mh and the Higgs self-coupling parameter = Mh /v2 , where v = 246 GeV is the vacuum expectation value of the Higgs field. Physical problems in this model can be equally discussed in terms of parameters of the potential or in terms of these observables. The 2HDM contains two fields with identical quantum numbers. Therefore, its description in terms of original fields or in terms of their linear superpositions are equivalent. This freedom makes clear that the study in terms of the Lagrangian may be likened to discussion of electrodynamical effects in a certain gauge defined by some particular gauge-fixing conditions. The discussion of the 2HDM in terms of only well-measurable quantities seems preferable. Here we present solution of this problem [1].

· The 2HDM describes a system of two scalar isospinor fields 1 , 2 with hypercharge Y = 1. The most


Ginzburg@math.nsc.ru

1


1

INTRODUCTION

general form of the 2HDM potential is V= 1 2 2 2 + 2 + 3 1 1 2 2 + 4 1 2 2 1 2 11 22 2 + 5 1 2 + 6 1 1 1 2 + 7 2 2 1 2 + h.c. 2 m2 m2 m2 11 1 1 - 22 2 2 - 12 1 2 +h.c. 2 2 2

(1)

-

Its coefficients are restricted by the requirement that the potential be positive at large quasiclassical values of i (positivity constraints).

· The model contains two doublets of fields with identical quantum numbers. Therefore, it can be described either in terms of the original fields 1 , 2 , which enter (1), or in terms of fields 1 , 2 , which ^ are obtained from k by a global unitary reparameterization (RPA) transformation F of the form
1 2 ^ =F
gen

( , , )

1 , 2

^ F

gen

=e

-i

0

cos ei/2 - sin e-i( -

/2)

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

)

.

(2)

This transformation induces a transformation of the parameters of the Lagrangian i i in such a way that the new Lagrangian, written in fields i , describes the same physical content. We refer to these different choices as different RPa bases. Transformation (2) is parameterized by angles , , and 0 . The parameter 0 describes an overall phase transformation of the fields, and since it does not affect the parameters of the potential, we do not consider this degree of freedom. In the potential (1), parameters 1-4 , m2 and m2 are real while 5-7 , m2 are generally complex. So it 22 12 11 takes 14 real quantities to fully define the scalar part of the 2HDM. Since the three remaining parameters of the RPa transformation cannot influence description of physical phenomena, the actual number of physically relevant parameters of the potential is 14 - 3 = 11.

· Extrema of the potential satisfy the stationarity equations V /i |1 = 1 ,2 = 2 = 0 (i = 1, 2). The most general solution that describes the SU (2) â U (1)Y U (1) E M symmetry breaking is expressed via two positive numbers vi and the relative phase factor ei as:
1 1 = 2 0 v1 , 1 2 = 2 0 v2 e
i

,

v1 = v cos ,

v2 = v sin ,

v=

v2 + v2 . 2 1

(3)

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 fields i are then decomposed into their v.e.v.'s and the quantized component fields, 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

· Relative couplings. We use the relative couplings for each neutral Higgs boson h a :
P a = P ga , P gSM

± = a

g( H + H - h a ) , 2 2 M± / v



H+ W - a

=

g( H + W - h a ) . MW /v

(4)

P The quantities a are the ratios of the couplings of h a with the fundamental particles P = V (W , Z ), q = t, b, ..., = , ... to the corresponding couplings for the would be SM Higgs boson ± with Mh = Ma . The other relative couplings describe interaction of h a with charged Higgs boson Hb .

+- The quantity ±b describes interaction Hb Hb h a , the quantity a ± Hb W h a . (Below we omit the adjective "relative".)

+ Hb W - a

describes off-diagonal interaction

The neutrals h a generally have no definite CP parity. Couplings V and ±b are real due to Hermiticity a a f f of Lagrangian, while other couplings are generally complex. The Re( a ) and I m( a ) are responsible for the interaction of fermion f with CP-even and CP-odd components of h a respectively.

2


3

QUADRATIC TERMS OF POTENTIAL. FIRST SUBSET OF OBSERVABLES

2

Higgs basis. Basic equations

Any RPa basis can be used for solving physical problems. Some of them are more suitable than others when solving specific problems. In particular, when the system possesses an additional symmetry, the preferable RPa basis is the one in which this symmetry is made obvious. We find it useful here to analyze the model with known vacuum (the ground state of the potential) using the basis with v2 = 0. This basis is called the Higgs (or Georgi) basis [8]. This basis is obtained from any given basis with known v.e.v.'s by transformation (2) with
1 2

^ =F

HB

1 ^ ,F 2

HB

^ =F

gen

( = , = - ).

(5)

The phase factor e±i/2 represents the remaining rephasing (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, ^ -1 ^ F H B = F gen ( = - , = + ) with -, 0 -0 . Again, we do not fix in this definition the RPh phase and the irrelevant parameter 0 . The potential obtained has the same form as (1). To distinguish its parameters in the Higgs basis from a generic basis, we use the capital letters , for parameters and fields. Using the extremum conditions, one can rewrite the potential in the simple form [9] VH B = M
2 ±

2 + 2

v2 1 v2 2 2 + 4 1- + 2 2 + 3 1- 2 2 1 1 1 2 2 2 2 2 5 v 2 + 1 2 + 6 1- 2 + 7 2 2 + h.c. . 2 1 1 1 2 2

2

2

2

1

(6)

In the Higgs basis the decomposition of fields around v.e.v. has form G+ 1 = v + 1 + i G 2
0

,

H+ 2 = 2 + i 2


3

.

(7)

To arrive at the description in terms of physically observable fields, one should start by substituting these expressions into the potential (6). Also, by choosing the unitarity gauge for the gauge fields, we omit the Goldstone modes G a from now on. As a result, the potential (6) takes the form in which coefficients are expressed via parameters of (6) (here and below, the usual convention of summation over repeated indices is adopted):
2 V = M± H + H - +

Mi j + v Ti H + H - i + v Ti jk i j 2 ij Bi j + - +C H + H - H + H - + H H i j + Q i j k l i j k l . 2

k

(8)

3

Quadratic terms of potential. First subset of observables

2 In Eq. (8), the coefficients Mi j form the neutral scalar mass matrix (here N = M± /v2 + 4 ):





1

Mi j = v 2 R e 6 -Im

6

R e 6 N + R e 5 2 - I m 5 /2

-Im

6

(9)

- I m 5 /2 . N - R e 5 2

The physical neutral Higgs states h a are such superpositions of fields i that diagonalize this matrix: h a = Ria i , i = Ria h a ; Mi j i j /2 = a M2 h2 /2, aa 3 Mi j = Ria R a M2 . a j (10)


4

OTHER TERMS OF POTENTIAL. SECOND SUBSET OF OBSERVABLES

The mixing matrix Ria 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 fields, i.e. it is irrelevant. The overall sign of this matrix is insignificant, we fix R1 > 0. 1 The trace of the mass matrix is invariant under transformations (10). Therefore we obtain a sum rule 2 v 2 ( 1 + 4 ) = a M 2 - M± . a One of the advantages of the Higgs basis as compared to other RPa bases is the fact that elements of the rotation matrix are directly related to the couplings (4), which are, in principle, measurable:
a V = R1 , a



H+ W - a



H- W + a

a a = R2 + i R3 .

(11)

It can be seen easily after writing the kinetic term of the Higgs Lagrangian with definitions (7) and (10). The absolute values of the real quantities V are directly measurable in the decays h a W W (or a W -fusion process), etc.
H a a The phases of quantities a W , i.e. the ratios R3 / R2 , cannot be fixed because of the rephasing freedom of potential in the Higgs basis . Their relative phases for different h a are determined unambiguously. We fix the RPh basis by the condition R2 = 0. 3 b The orthogonality of the mixing matrix means that its elements obey a set of relations Ria Ri = ab , i
+ -

Ria R a = i j , which can be rewritten as very useful Sum Rules: j
a


a

( V )2 = 1 , a

| V |2 + | a

H+ W - 2 | a

= 1.

(12)

This orthogonality allows to express all elements Ria via couplings of different Higgs neutrals h a to gauge bosons V ( is arbitrary phase, determined RPh freedom): a
V 1



V 2 V2 2)



V 3 V 3

0 cos - sin 0 sin cos

- V V 12 1 - ( V ) 2 Ria = V 3 V 1 - ( 2 )

1-(
2

V - 2

0
2

1 V )2 1 - ( 2 0 V 0 - 1 V 1 - ( 2 )2

(13)

with limitation, given by the first sum rule (12). Finally, one can read (9) as expressions of some 's via elements of the mass matrix and then, with the aid of (10), express them via the masses of Higgs bosons and their couplings to gauge bosons: v2 1 = ( V )2 M 2 ; a a
v2 5 = (
a a H + W - )2 a 2 v 2 4 = M 2 - M± - v 2 1 ; a a

M2 ; a

v2 6 = V a
a

H+ W - a

M2 . a

(14)

These equations describe parameters 1 , 4 , 5 , 6 of Lagrangian via observables of the first subset, i.e. masses of all Higgs bosons M1,2,3 , M± , vacuum expectation value of Higgs field v = 246 GeV and the couplings V of any two (of three) chosen neutrals to the gauge bosons ­ 7 quantities. Below a we assume that h1 is the discovered Higgs boson with M1 125 GeV. The final equations also contain H+ - the couplings a W , expressed via V with the aid of Eq. (13). This subset determines explicitly all a quadratic (mass) terms of potential (6).

4

Other terms of potential. Second subset of observables

The Higgs boson masses and couplings to the gauge bosons do not depend on 2 , 3 , 7 . In turn, these parameters are necessary to determine triple and quartic Higgs boson vertices. The triple and quartic Higgs vertices of the potential (6) can be determined completely only if one supplements the parameters of the first subset with additional information. In turn, to form the second subset, one needs to use triple 4


6

DISCUSSION

and quartic Higgs self-interactions. For this goal we use three triple couplings H+ H- ha (quantities ± ) a and one quartic coupling g( H + H - H + H - ) ­ 4 quantities. The analysis is simple but cumbersome. The parameters of the first subset plus three couplings ± determine all triple Higgs couplings. The a coefficients 3 , 7 of the Lagrangian are expressed simply via these three couplings and observables of the first subset. 2 2 H- + 3 = ( 2 M± / v 2 ) V ± ; 7 = ( 2 M± / v 2 ) a W ± . (15) aa a
a a

The description of quartic interactions of Higgs particles demands adding one more observable 2 = 2 g ( H + H - H + H - ) . (16)

· Certainly, the second subset of observables can be constructed with other triple and quartic couplings.
Using processes involving charged Higgses looks preferable for two reasons. First, with charged Higgses, this procedure requires the fewest calculations, improving accuracy and reducing uncertainties. Second, the amplitudes of the processes e+ e- H + H - h a , H + H - ha , + e- H + H - H + H - , H + H - H + H - at ILC/CLIC [10] are directly proportional to the core responding couplings, without any nonrelevant diagrams interfering.

· The obtained equations for parameters of the Lagrangian in the Higgs basis contain one irrelevant parameter: the RPh phase related to a rephasing freedom in the Higgs basis . In order to switch to another RPa basis, which could be more useful for some special reasons, one should use two parameters tan and , which are determined by the RPa basis choice. Once these parameters are determined from problem-specific conditions, the transition to this RPa basis is performed with the aid of the back ^ -1 rotation F H B (5). The final equations for parameters i , m2j are constructed from measurable quantities i discussed above and RPa basis-choice parameters , , .

5

Notes about 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 fixing parameters of the basic set. In the modern approach the calculation of any physical effect 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. The development of such scheme looks important problem.

6

Discussion

· We have found the minimal complete set of measurable quantities (named observables) that determines all parameters of the 2HDM Lagrangian ­ the basic set of observables. · 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, provided that the positivity constraints are satisfied and the couplings V are not too large, in order not to violate the sum a rule (12). In some special variants of the 2HDM, additional relations between these parameters may ± V appear (for example, in the C P conserving case 3 = 3 = 0).
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 coefficients of the Lagrangian.

· Possible strong interaction in the Higgs sector. The fact that free parameters of the potential naturally fall into three very distinct categories offers a new opportunity that was absent in the SM. Before the Higgs discovery, the large coupling constant was, in principle, possible within the SM. In this case, the Higgs boson would be very heavy and wide, and it could not be seen as a 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 WL WL , WL ZL , ZL ZL 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.

5


REFERENCES

REFERENCES

Our analysis shows that, within the 2HDM, the reasonably low values of all Higgs masses are well compatible with large 3 , |7 |, 2 , i.e. with the strong interaction in the Higgs sector. A signal of this feature can be observed in the multi-Higgs final states or (for 3 , |7 |) in the anomalously large two-photon width of some neutral Higgs boson. Moreover, this strong interaction can coexist even with moderate values of triple Higgs couplings as it could be driven exclusively by the large value of a single parameter 2 .

· The principal possibility to determine all parameters of the 2HDM from the (future) data meets strong practical limitations (which can be hidden in other approaches). This will remain a problem for a very long time.
Indeed, the modern data on the Higgs boson couplings, the analysis of many particular models (see, e.g., [11]), and the using of sum rules (12) allow us to conclude that the discovery of new Higgs bosons h2,3 , H ± is a difficult problem for the LHC and e+ e- colliders [12]. If these h2,3 are discovered, the inaccuracies in the measuring of their masses and couplings are not expected to be small. The measuring of triple and quartic interactions of Higgs bosons looks more difficult problem. So it is natural to expect that these measurements will be made later and with bigger inaccuracy. The general limitations for the model, similar to the positivity constraint, contain parameters of the first and second subsets simultaneously. Thus, there are few chances that such restrictions can be verified in the near future.

Acknowledgements
This work was partially supported by Grants of Russian Foundation for Basic Research, No RFBR 1502-05868 (Russian Federation), of Russian Support of scientific schools No NSh-3003.2014.2 (Russian Federation) and grant No NCN OPUS 2012/05/B/ST2/03306 (2012-2016). We are thankful to I.P. Ivanov, K.A. Kanishev and M. Krawczyk for discussions.

References
[1] I.F. Ginzburg, K.A. Kanishev, Phys. Rev. D 92 (2015) 015024; arXiv:1502.06346 [hep-ph] [2] ATLAS Collaboration, Phys. Lett. B 716, 1 (2012); CMS Collaboration, Phys. Lett. B 716, 30 (2012); M. Flechl. arXiv:1503.00632 [3] G. Belanger, B. Dumont, U. Ellwanger, J.F. Gunion and S. Kraml, Phys. Rev. D 88, 075008 (2013); B. Dumont, arXiv:1305.4635 [4] I. F. Ginzburg, M. Krawczyk and P. Osland, "AIP Conf. Proc. 578,304 (2001). [5] P.S.B. Dev and A. Pilaftsis. arXiv:1503.09140 [6] T.D. Lee, Phys. Rev. D8, 1226 (1973); J.F. Gunion, H.E. Haber, G. Kane and S. Dawson, The Higgs Hunter 's Guide (Addison-Wesley, Reading, 1990); G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rep. 516, 1 (2012) [7] I.F. Ginzburg, JETP Lett. 99 (2014) 742-751; ArXiv: 1410.0873 [hep-ph]; these Proceedings [8] H. Georgi and D. V. Nanopoulos, Phys. Lett. B 82, 95 (1979). [9] I. F. Ginzburg and K. A. Kanishev, Phys. Rev. D 76, 095013 (2007). [10] M. Hashemi and I. Ahmed, arXiv:1401.4841. [11] B. Holdom and M. Ratzlaff, Phys. Rev. D 91, 035031 (2015); S. Kanemra, K. Tsumura and H. Yokoya, Phys. Rev. D 90, 035021 (2013); N. Craig, J. Galloway and S. Thomas, arXiv:1305.2424; B. Dumont, J.F. Gunion, Y. Jiang and S. Kraml, Phys. Rev. D 88, 055010 (2013); C. Lange, on behalf of the ATLAS and CMS collaborations, arXiv:1411.7279; D. Cline, X. Ding and J. Lederman, arXiv:1204.6700; J. Baglio, O. Eberhardt, U. Nierste and M. Wiebusch, Phys. Rev. D 90, 015008 (2014). [12] I.F. Ginzburg, arXiv:1502.07197.

6