# The Higgs Decay Width in Multi-Scalar Doublet Models

###### Abstract

We show that there are regions of parameter space in multi-scalar doublet models where, in the first few hundred inverse femtobarns of data, the new charged and neutral scalars are not directly observable at the LHC and yet the Higgs decay rate to is changed significantly from its standard model value. For a light Higgs with a mass less than , this can cause a large change in the number of two photon and Higgs decay events expected at the LHC compared to the minimal standard model. In the models we consider, the principle of minimal flavor violation is used to suppress flavor changing neutral currents. This paper emphasizes the importance of measuring the properties of the Higgs boson at the LHC; for a range of parameters the model considered has new physics at the TeV scale that is invisible, in the first few hundred inverse femtobarns of integrated luminosity at the LHC, except indirectly through the measurement of Higgs boson properties.

^{†}

^{†}preprint: CALT 68-2660

## I Introduction

Experiments at the LHC will directly probe physics at the weak scale. Most physicists believe that there is new physics at this energy scale, beyond what is in the minimal standard model (SM). This belief is motivated to a large extent by the hierarchy puzzle and by the fact that the scalar sector of the standard model has yet to be directly probed by experiment. The single doublet in the SM is the simplest example of a scalar sector but many extensions have been studied. Amongst the most widely considered are two Higgs doublet models ^{1}^{1}1See Reina:2005ae () for a review.. A problem that immediately arises in such models is the possibility of flavor changing neutral current (FCNC) effects that are unacceptably large. In particular, when the SM fermion fields couple to both doublets, and the couplings are
arbitrary, FCNC effects are possible at tree level. Glashow and Weinberg gave a simple prescription for how to avoid such effects through imposing a discrete symmetry Glashow:1976nt (). One can also suppress FCNC effects by adopting an ansatz suppressing the coupling of the new doublet Cheng:1987rs (); Luke:1993cy (); Antaramian:1992ya (); Atwood:1996vj (). In this paper, we use the principle of minimal flavor violation (MFV) Chivukula:1987py (); Hall:1990ac (); D'Ambrosio:2002ex (); Cirigliano:2005ck (); Buras:2003jf (); Branco:2006hz () which causes
tree level FCNC to vanish (or at least be suppressed by small mixing angles) in multi doublet models in a natural way.

We are interested in the possible effects of new physics on the properties of the Higgs boson and we characterize the impact of new physics on the Higgs through an operator analysis Manohar:2006gz (); Grinstein:2007iv (); Kile:2007ts (); Graesser:2007yj (); Mantry:2007sj (); Pierce:2006dh (); Fox:2007in () . We assume that the new physics mass scale is much larger than the Higgs boson mass and add higher dimension operators that are invariant under the symmetry of the standard model. Since the operators give small corrections to the SM, one does not expect them to influence standard model processes that are unsuppressed. For example, the Higgs coupling to two W bosons is an unsuppressed tree level coupling and new physics contributions to it should be negligible. However the dominant Higgs production mechanism through gluon fusion, occurs at leading order in perturbation theory through a top quark loop. Hence, in the SM it is suppressed and new physics can easily compete with the standard model contribution Manohar:2006gz (). Similar remarks hold for the decay amplitude. Some of the tree level couplings of the standard model Higgs are also very small. For example, the Higgs to Yukawa coupling is of order^{2}^{2}2Here is the vacuum expectation value that spontaneously breaks the weak gauge group down to the electromagnetic gauge group and the Higgs to b-quark Yukawa coupling is of order . New physics characterized by higher dimension operators can also compete with the standard model in the Mantry:2007sj () and decay amplitudes.

A light Higgs with a mass less than is likely to be detected first through its decay to two photons despite the fact that the branching ratio for this process is quite small, i.e. of order . Early detection through its decay to , which has a branching ratio around , may also be possible. The dominant decay mode is to pairs. However this decay mode is much harder to observe because of large theoretical uncertainties^{3}^{3}3See Rainwater:2007cp () for a recent review on production and detection of the Higgs and Benedetti:2007sn () for a recent study on the SM background. on the cross section of the irreducible SM background . An integrated luminosity of order may be required to observe the standard model Higgs in the channel. If the rate for decay is changed by a factor from its standard model value (but other properties of the Higgs are left unaltered) then the number of or decay events observed at the LHC is changed by a factor where

(1) |

and is the SM branching fraction of . Even though decay will not be directly observable until there are many years of LHC data, it’s rate is of crucial importance for all measurable properties of the low mass Higgs.

In this paper, we consider multi-doublet scalar models.^{4}^{4}4New physics in the form of a second scalar doublet with an unbroken symmetry can also provide a component of dark matter and the effect of such a second doublet on the SM
Higgs was recently examined in Cao:2007rm (). For simplicity we restrict our attention to two scalar doublets and . Here denotes the usual SM Higgs doublet with a mass less than GeV. is a new scalar doublet with a mass TeV and has the same quantum numbers as the SM Higgs doublet . We demonstrate that there are regions in parameter space for which the new doublet will be invisible at the LHC, at least in the first few hundred inverse femtobarns of data. The largest observable effect of this new doublet will be order one shifts in the rate which in turn will affect the branching ratios for all light Higgs decay channels. It is straightforward to generalize our results to a scenario with more than two scalar doublets.

## Ii The Two Doublet Model

The scalar potential for the model we consider is given by ^{5}^{5}5We thank Lisa Randall for pointing out the typo in the sign of in Eqn.(2) in the previous version Randall:2007as ().

(2) |

We assume the phase of is adjusted so that is real. appears with a positive mass term and acquires a vacuum expectation value only through it’s coupling to , which undergoes the usual electroweak symmetry breaking. Since is much greater than the weak scale , the neutral component gets a vacuum expectation value that is much smaller than

(3) |

In addition to the SM Yukawa couplings of the doublet , the doublet has the following Yukawa couplings to the quarks,

(4) |

We make use of the principal of minimal flavor violation. This results in small loop level FCNC through the appearance of Yukawa coupling matrices and in the loops. We assume that possible multiple insertions of the Yukawa matrices in Eq. (4) are suppressed. In this model, the physical quark masses are the result of the sum of contributions from the coupling of quarks to and . Thus in the mass eigenstate basis, the Yukawa matrices do not satisfy the usual relation to physical quark masses. In the SM, . In the down quark sector, in the mass eigenstate basis, the couplings of the heavy scalar doublet are

(5) |

Here is the CKM matrix, and is related to the physical down quark mass by

(6) |

We assume that the constant in Eq. (4) is very small so that the coupling to the up-type quarks can be neglected. When the production of via it’s coupling to the top quarks is suppressed.

On the other hand, we take to be large, and for simplicity we choose it to be real. Since the down type quark Yukawa couplings in the matrix are very small, the effective coupling is still perturbative. The choice of makes the coupling of quarks stronger to compared to resulting in a large shift in once is integrated out. Thus, with , , and an almost invisible can be produced at the LHC and will leave it’s footprint through large shifts in the rate.

## Iii FCNC Constraints

Even though we imposed MFV eliminating the possibility of tree level FCNC, in our two doublet model, there are at one loop corrections to standard model FCNC processes. In particular, we want to check that the choice of is consistent with constraints from FCNC. Consider the weak radiative b decay with two doublets Grinstein:1987pu (). Calculating the Feynman diagrams in Fig.(1) we find that charged exchange induces at one loop the effective Hamiltonian,

(7) |

where is the electron charge.

For inclusive decay, this does not interfere with the standard model contribution from Grinstein:1987vj () since the strange quarks have opposite chirality. Hence, we find

(8) |

Taking , , and in Eq. (8) gives , which is much too small to be observed^{6}^{6}6See Misiak:2006zs (); Misiak:2006ab () for the latest calculation of BR() at NNLO in QCD and its comparison to the results from Babar Aubert:2005cua (); Aubert:2006gg () and Belle Koppenburg:2004fz () as averaged by the Heavy Flavor Averaging Group Barberio:2006bi (). The remaining uncertainties in theory and experiment preclude a exclusion of our model based on constraints for the parameter space of interest.. For exclusive decays there can be interference between the standard model contribution and the contribution of the effective Hamiltonian in Eq. (7) but there hadronic uncertainties cloud our ability to constrain the new physics Becirevic:2006nm (); Ball:2006eu ().

There are contributions from the interactions of the new doublet to the Wilson coefficient that are proportional to and are not suppressed by . However, because we have focused on the region of parameter space where is very small these have been omitted. Our assumption of very small greatly diminishes the constraint that places on the model.

## Iv Effects on light Higgs decays

So far we have described the general features of the two doublet model we are considering and demonstrated the compatibility of a large with constraints. Next we investigate how this simple extension of the SM affects light Higgs decay to quarks by integrating the heavy doublet out of the theory to induce the effective operator

(9) |

Including the effects of this operator and using Eqs.(6) and (3) we find that the rate is modified relative to the SM as

(10) |

Note we have included terms suppressed by more powers of in Eq. (10) which are accompanied by the large factor . However, we can still consistently ignore the effects of dimension 8 operators contributing to since their contributions start at order .

At TeV, the parameter choices of or give
the rate for which is 1.6 times it’s SM value. An even more dramatic effect is seen for the parameter choices of
and which give a rate that is 121 and 0.008 times the SM value respectively.
Thus, the presence of an additional scale scalar doublet with a coupling to quarks about ten times the SM value can have dramatic changes in the decay width and branching fractions of a light Higgs. For example, with the branching ratio for the experimentally promising modes of and will be down by a factor of . Such a scenario would make detection of the light Higgs very difficult. On the other hand for in which case the rate is 0.008 it’s SM value. In this case, the branching fractions for and will increase by the factor for GeV. This would apply for Higgs searches at the Tevatron as well Acosta:2005bk (). ^{7}^{7}7New physics in the form of a massive fourth generation neutrino Belotsky:2002ym () or additional scalar singlets BahatTreidel:2006kx () have also been shown to
effect the possibility of the detection of the Higgs at LHC.

One can also generalize the above analysis to the lepton sector and induce a corresponding effective operator

(11) |

which contributes to the decay of the Higgs to charged leptons . By choosing a large value for one can similarly induce order one shifts in the decay rate to . Such order one shifts can be seen at the LHC in the experimentally promising channel of . The effect of the operator in Eq.(11) was recently studied in Mantry:2007sj () where naturalness criteria were used to constrain the size of the Wilson coefficient. It was shown that order one shifts are indeed possible and compatible with experimental constraints. The branching ratio for can be influenced both by the effect of the operator in Eq. (11) on the rate for and by the effect of the operator in Eq. (9) on the total Higgs width.

Order one shifts in the rate for can also affect the total width of the light Higgs since it’s branching ratio is not negligible. For example, if GeV then the branching ratio in the SM for is about . As order one corrections are possible to partial decay widths for both and , the relative impact of the two decays on the total width can be changed dramatically. This can make the total decay width even more sensitive to the effects of . However, for the sake of simplicity we will assume in this paper that is small so that effects on the width of the Higgs from the coupling of to leptons are negligible.

The number of observed events can also be effected by higher dimension operators that induce a direct coupling between and and between and as discussed in Manohar:2006gz (). The latter effects the Higgs production rate by gluon fusion. New physics effects of this form are distinguishable from a change in the total width as the new physics effects on the total width will cancel in the ratios of the number of expected events for different Higgs production mechanisms and decay channels.

## V Production and Decay of the new scalar doublet

We now study the production and decay of the new scalar doublet and discuss the possibility for it’s observation at the LHC. The doublet contains new neutral and charged scalars with masses approximately equal to . If the LHC has enough energy to produce these states. However, we will show that for a range of parameters their production rates are quite small and that the dominant decay channels have poor experimental signatures making them invisible at the LHC, at least for the first few hundred inverse femtobarns of data.

The production of the charged is suppressed compared with the neutrals and the pseudoscalar does not have a significant branching ratio to the most promising detection channels and . Hence we present in detail a discussion of the neutral scalar^{8}^{8}8Similar conclusions hold for the charged scalar and neutral pseudoscalar.. Expanding and about their vacuum expectation values we write

(12) |

The fields and mix and the resulting mass eigenstate fields and are approximately given by^{9}^{9}9We assume that the parameters in the scalar potential are real so there is no mixing.

(13) |

The production rate for is very small. The dominant production mode is through
. In this process the initial and each come mostly from collinear gluon splitting and the remaining spectator quarks have very low transverse momentum to be observed in the final state. The large logarithms associated with collinear gluon splitting into light quark pairs leads to an enhancement of the rate by one or two orders of magnitude Dawson:2005vi (); Dittmaier:2003ej (); Dawson:2003kb () over ^{10}^{10}10This result is based on the NLO QCD calculation for production in Dawson:2002tg (); Reina:2001sf (); Beenakker:2002nc (); Beenakker:2001rj (). where the
scalar is radiated off one of the final state -quarks. This is because the final state -quark which radiates the scalar is far offshell before emission and thus the rate does not receive the enhancement of large logarithms associated with collinear gluon splitting.
The cross-section for at leading log takes the form

(14) |

where and are the quark and antiquark parton distribution functions respectively and is the center of mass energy squared. The large logs from collinear gluon splitting are summed into the parton distribution functions by choosing . As seen from Eq.(14), the cross section receives an additional enhancement by a factor of compared to the production of a SM Higgs with the same mass. This production cross section as a function of the mass is shown in Fig.(2) as the solid black curve. This curve was generated for the choice of and . We see that at TeV the cross section is about fb. Thus, for 100 fb of data one can expect the production of about 1000 neutral scalars from fusion. Note that this dominant production mechanism doesn’t exist for the heavy charged scalars.

The next largest production mode of is through gluon fusion and is given by a direct modification of the SM cross section Dawson:1994ri ()

(15) |

where we have used the functions

(16) |

and for which is given by Bergstrom:1985hp ()

(17) |

Here denotes the gluon parton distribution function. As seen in Eq. (15) this production channel receives significant contributions from bottom and top loops. The bottom loop has a significant contribution due to the direct coupling of which involves . Although the direct coupling of to the top quark is negligible for , the top loop still gives a significant contribution due to the mixing of with the Higgs . With the same parameters used as in Fig.(2) at TeV one can expect the production via gluon fusion of only about 8 neutral scalars for 100 fb of data.

Other production mechanisms are similarly small. For example Higgs production (via vector-boson fusion) in association with massless jets, , where , is dominated by the Higgs being radiated off a virtual or boson. So

(18) |

The pattern of possible decays of the new scalars in the model depends on the mass splittings between the various states. To simplify our discussion of the spectrum we neglect the mixing and assume that the coupling constants in the scalar potential are real. Then there is no mixing and the mass spectrum is,

(19) |

We focus on the region of parameter space where the lightest scalar is . For its decays it is important to include the effects of mixing. The most important decay modes of have the partial rates

(20) | |||||

(21) | |||||

(22) | |||||

(23) | |||||

(24) | |||||

(25) |

A plot of the branching fractions of as a function of the mass is shown in Fig. (3) for , and . The dominant decay channels are and . The channel is known to have a large SM background. As we will discuss later, even the channel can be difficult to observe. The final states where the decays to gauge bosons and at least one of the gauge bosons decays to electrons and/or muons have a cleaner experimental signature. Note that for the parameters used in Fig. (3) the total width of an scalar of mass is only 3 . For comparison, note that the width of a standard model Higgs with a mass of is about 700 Djouadi:2005gi (). This is because of the small vacuum expectation value of the heavy doublet which suppresses the coupling of to two gauge bosons.

Decay Channel | |||
---|---|---|---|

1.4 | 0.58 | 47 | |

4.6 | 2.0 | 160 | |

3.0 10 | 12 | 1.0 10 |

Decay Channel | |||
---|---|---|---|

34 | 14 | 26 |

In Table. 1 we show the number of expected events for 100 fb of data at the LHC when decays to gauge bosons and TeV. For example with , and we find that , where we have summed over leading to about five events with 100 fb of data. For these parameters, the heavy scalar will not be detected at the LHC in the first few hundred femtobarns of integrated luminosity. In fact, within much of the region of parameter space where the coupling of the new S-doublet to charge -quarks is suppressed (i.e., very small) the heavy scalar degrees of freedom associated with the doublet are difficult to detect at the LHC as shown in in the first two columns of Table. 1. However, as seen in the third column of Table. 1, even with small, there are regions of parameter space that are more promising for detection at LHC. For example, for , and we find that and detection of the new heavy scalar at the LHC with a few hundred inverse femtobarns of integrated luminosity is more likely.

In Table. 2 we show the number of expected events when decays to a pair of light Higgses and one of them decays to the experimentally favored or channels. As seen in the table, for the parameters chosen detection is unlikely. The number of events in the last column are suppressed because for these parameters the Higgs decay rate to is enhanced by a factor of about 100 which reduces the Higgs branching ratio to and by a similar factor.

## Vi Concluding Remarks

We have demonstrated that for regions of parameter space in multi doublet models the states of the new doublets are impossible to directly detect at LHC, using the first few hundred inverse femtobarns of data, and yet the effect of the new doublet on the total width of the light Higgs is very significant. In the simple two doublet model we considered in detail, the promising and signals at the LHC for detecting a light Higgs could be significantly enhanced or suppressed. This demonstration emphasizes the importance of determining the properties of the Higgs boson in the presence of new physics that is difficult to directly detect at LHC.

We thank Marat Gataullin for many helpful comments. This work was supported in part by the DOE grant number DE-FG03-92ER40701 and DE-FG03-97ER40546.

## References

- (1) L. Reina, (2005), hep-ph/0512377.
- (2) S. L. Glashow and S. Weinberg, Phys. Rev. D15, 1958 (1977).
- (3) T. P. Cheng and M. Sher, Phys. Rev. D35, 3484 (1987).
- (4) M. E. Luke and M. J. Savage, Phys. Lett. B307, 387 (1993), hep-ph/9303249.
- (5) A. Antaramian, L. J. Hall, and A. Rasin, Phys. Rev. Lett. 69, 1871 (1992), hep-ph/9206205.
- (6) D. Atwood, L. Reina, and A. Soni, Phys. Rev. D55, 3156 (1997), hep-ph/9609279.
- (7) R. S. Chivukula and H. Georgi, Phys. Lett. B188, 99 (1987).
- (8) L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 (1990).
- (9) G. D’Ambrosio, G. F. Giudice, G. Isidori, and A. Strumia, Nucl. Phys. B645, 155 (2002), hep-ph/0207036.
- (10) V. Cirigliano, B. Grinstein, G. Isidori, and M. B. Wise, Nucl. Phys. B728, 121 (2005), hep-ph/0507001.
- (11) A. J. Buras, Acta Phys. Polon. B34, 5615 (2003), hep-ph/0310208.
- (12) G. C. Branco, A. J. Buras, S. Jager, S. Uhlig, and A. Weiler, (2006), hep-ph/0609067.
- (13) A. V. Manohar and M. B. Wise, Phys. Lett. B636, 107 (2006), hep-ph/0601212.
- (14) B. Grinstein and M. Trott, (2007), arXiv:0704.1505 [hep-ph].
- (15) J. Kile and M. J. Ramsey-Musolf, (2007), arXiv:0705.0554 [hep-ph].
- (16) M. L. Graesser, (2007), arXiv:0704.0438 [hep-ph].
- (17) S. Mantry, M. J. Ramsey-Musolf, and M. Trott, (2007), arXiv:0707.3152 [hep-ph].
- (18) A. Pierce, J. Thaler, and L.-T. Wang, (2006), hep-ph/0609049.
- (19) P. J. Fox, Z. Ligeti, M. Papucci, G. Perez, and M. D. Schwartz, (2007), arXiv:0704.1482 [hep-ph].
- (20) D. Rainwater, (2007), hep-ph/0702124.
- (21) D. Benedetti et al., J. Phys. G34, N221 (2007).
- (22) Q.-H. Cao, E. Ma, and G. Rajasekaran, (2007), arXiv:0708.2939 [hep-ph].
- (23) L. Randall, (2007), arXiv:0711.4360 [hep-ph].
- (24) B. Grinstein and M. B. Wise, Phys. Lett. B201, 274 (1988).
- (25) B. Grinstein, R. P. Springer, and M. B. Wise, Phys. Lett. B202, 138 (1988).
- (26) M. Misiak et al., Phys. Rev. Lett. 98, 022002 (2007), hep-ph/0609232.
- (27) M. Misiak and M. Steinhauser, Nucl. Phys. B764, 62 (2007), hep-ph/0609241.
- (28) BABAR, B. Aubert et al., Phys. Rev. D72, 052004 (2005), hep-ex/0508004.
- (29) BaBar, B. Aubert et al., Phys. Rev. Lett. 97, 171803 (2006), hep-ex/0607071.
- (30) Belle, P. Koppenburg et al., Phys. Rev. Lett. 93, 061803 (2004), hep-ex/0403004.
- (31) Heavy Flavor Averaging Group (HFAG), E. Barberio et al., (2006), hep-ex/0603003.
- (32) D. Becirevic, V. Lubicz, and F. Mescia, Nucl. Phys. B769, 31 (2007), hep-ph/0611295.
- (33) P. Ball, G. W. Jones, and R. Zwicky, Phys. Rev. D75, 054004 (2007), hep-ph/0612081.
- (34) CDF, D. Acosta et al., Phys. Rev. D72, 072004 (2005), hep-ex/0506042.
- (35) K. Belotsky, D. Fargion, M. Khlopov, R. Konoplich, and K. Shibaev, Phys. Rev. D68, 054027 (2003), hep-ph/0210153.
- (36) O. Bahat-Treidel, Y. Grossman, and Y. Rozen, JHEP 05, 022 (2007), hep-ph/0611162.
- (37) S. Dawson, C. B. Jackson, L. Reina, and D. Wackeroth, Mod. Phys. Lett. A21, 89 (2006), hep-ph/0508293.
- (38) S. Dittmaier, M. Kramer, and M. Spira, Phys. Rev. D70, 074010 (2004), hep-ph/0309204.
- (39) S. Dawson, C. B. Jackson, L. Reina, and D. Wackeroth, Phys. Rev. D69, 074027 (2004), hep-ph/0311067.
- (40) S. Dawson, L. H. Orr, L. Reina, and D. Wackeroth, Phys. Rev. D67, 071503 (2003), hep-ph/0211438.
- (41) L. Reina and S. Dawson, Phys. Rev. Lett. 87, 201804 (2001), hep-ph/0107101.
- (42) W. Beenakker et al., Nucl. Phys. B653, 151 (2003), hep-ph/0211352.
- (43) W. Beenakker et al., Phys. Rev. Lett. 87, 201805 (2001), hep-ph/0107081.
- (44) CTEQ, H. L. Lai et al., Eur. Phys. J. C12, 375 (2000), hep-ph/9903282.
- (45) C. W. Bauer, Z. Ligeti, M. Luke, A. V. Manohar, and M. Trott, Phys. Rev. D70, 094017 (2004), hep-ph/0408002.
- (46) W.-M. Yao et al., Journal of Physics G 33, 1+ (2006).
- (47) S. Dawson, (1994), hep-ph/9411325.
- (48) L. Bergstrom and G. Hulth, Nucl. Phys. B259, 137 (1985).
- (49) A. Djouadi, (2005), hep-ph/0503172.
- (50) CMS Collaboration, G. Bayatian et al., CERN/LHCC 2006-021 (2006).