unparticle-higgs field mixing: mikheyev-smirnov-wolfenstein resonances, seesaw mechanism, and...

Unparticle-Higgs field mixing: Mikheyev-Smirnov-Wolfenstein resonances, seesaw mechanism,and spinodal instabilities

D. Boyanovsky,1,* R. Holman,2,† and Jimmy A. Hutasoit2,‡

1Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA2Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA(Received 27 December 2008; revised manuscript received 1 April 2009; published 23 April 2009)

Motivated by slow-roll inflationary cosmology we study a scalar unparticle weakly coupled to a Higgs

field in the broken symmetry phase. The mixing between the unparticle and the Higgs field results in a

seesaw type matrix and the mixing angles feature a Mikheyev-Smirnov-Wolfenstein (MSW) effect as a

consequence of the unparticle field being noncanonical. We find two (MSW) resonances for small and

large spacelike momenta. The unparticlelike mode features a nearly flat potential with spinodal

instabilities and a large expectation value. An effective potential for the unparticlelike field is generated

from the Higgs potential, but with couplings suppressed by a large power of the small seesaw ratio. The

dispersion relation for the Higgs-like mode features an imaginary part even at ‘‘tree level’’ as a

consequence of the fact that the unparticle field describes a multiparticle continuum. Mixed unparticle-

Higgs propagators reveal the possibility of oscillations, albeit with short coherence lengths. The results are

generalized to the case in which the unparticle features a mass gap, in which case a low energy MSW

resonance may occur for lightlike momenta depending on the scales. Unparticle-Higgs mixing leads to an

effective unparticle potential of the new-inflation form. Slow-roll variables are suppressed by seesaw

ratios and the anomalous dimensions and favor a red spectrum of scalar perturbations consistent with

cosmic microwave background data.

DOI: 10.1103/PhysRevD.79.085018 PACS numbers: 12.60.�i, 14.80.�j, 98.80.Cq

I. INTRODUCTION

As evidence for physics beyond the standard modelaccumulates, exploration of extensions is becoming morefocused by the possibility of constraining them with forth-coming collider experiments. A recent proposal by Georgi[1] suggests that a conformal sector with a nontrivialinfrared fixed point coupled to the standard model mightbe a possible extension with a wealth of phenomenologicalconsequences, some of which may be tested at the LargeHadron Collider [2–4]. Early work by Banks and Zaks [5]provides a realization of a conformal sector emerging froma renormalization flow towards the infrared below an en-ergy scale � through dimensional transmutation, andsupersymmetric QCD may play a similar role [6]. Belowthis scale there emerges an effective interpolating field, theunparticle field, that features an anomalous scaling dimen-sion [1].

Recently various studies recognized important phe-nomenological [1,2,7] (for important caveats see [8]), as-trophysical [9–11], and cosmological [12–18] con-sequences of unparticles, including Hawking radiationinto unparticles [19] and potentially relevant phenomenol-ogy in CP-violation [20], flavor physics [21], and lowenergy parity violation [22].

A deconstruction program describes the unparticle fromthe coupling of a particle to a tower of a continuum ofexcitations [23]. Although this is an interesting interpreta-tion of unparticles, the physics of anomalous dimensionsarising from the exchange of massless (conformal) excita-tions is an ubiquitous feature in critical phenomena, a fieldthat was already well developed before the advent ofunparticle physics. Anomalous dimensions in the fermionpropagators in gauge theories had been understood in themid 40’s–50’s [24,25], where multiple emissions and ab-sorptions of massless quanta leads to anomalous dimen-sions. This is also well known within the context of QCD[26].Critical phenomena associated with second order phase

transitions provide a natural realization of unparticle phys-ics. Indeed, a scalar order parameter, such as the magneti-zation in a three dimensional Heisenberg ferromagnet,features anomalous scaling dimensions at a critical point.That the multiple exchange of massless excitations leads toanomalous scaling exponents at critical points has beenknown since the 70’s and well understood via renormal-ization group or large N resummations by the 80’s [27,28].Scale invariance appears as a consequence of renormaliza-tion group flow towards an infrared fixed point and thecorrelation functions of the order parameter scale withanomalous scaling dimensions. This, of course was theoriginal motivation behind the Banks-Zaks suggestionwithin the context of non-Abelian gauge theories [5]. Incritical phenomena, integrating out degrees of freedombelow a cutoff scale (in condensed matter systems deter-

*[email protected]†[email protected]‡[email protected]

PHYSICAL REVIEW D 79, 085018 (2009)

1550-7998=2009=79(8)=085018(11) 085018-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.79.085018

mined by the lattice spacing) down to a renormalizationscale � yields an effective field that describes long-wavelength phenomena below this scale. The effectivepotential of the unparticle field is the Landau-Ginzburgfree energy, a functional of the order parameter, whosesecond derivative with respect to the (scalar) order parame-ter (the unparticle field) vanishes at the critical point; theeffective potential becomes flat, reflecting the underlyingscale invariance emerging at the infrared fixed point[27,28].

All of these predate the ‘‘deconstruction’’ interpretation,in some cases by decades. Although the practitioners of‘‘unparticle physics’’ in the literature may prefer the de-construction interpretation, we would like to emphasizethat unparticle fields are ubiquitous in critical phenomena,and that the deconstruction interpretation is one, but by nomeans the only one.

In this article, we are motivated to study unparticlephysics by the similarity between the effective field theoryof single field slow-roll inflation as a paradigm for cosmo-logical inflation whose predictions are in remarkableagreement with WMAP data [29,30], and a nearly criticaltheory. On the one hand, single field, slow-roll inflation isbased on the dynamics of a scalar field, the inflaton, whoseevolution is determined by a fairly flat potential. The powerspectrum of inflaton fluctuations is nearly Gaussian andscale invariant [31], and nonlinear couplings are small. InRef. [32] it was argued that an effective field theory de-scription a la Landau-Ginzburg provides a compellingdescription of single field slow-roll inflation in which thehierarchy of slow-roll parameters emerges as a systematicexpansion in the number of e-folds and the nonlinearcouplings emerge as seesaw like ratios of two widelydifferent scales, the Hubble scale during inflation and thePlanck scale. The nearly Gaussian and scale invariantspectrum of fluctuations, the flatness of the potential, nec-essary to allow at least 60 e-folds, with the concomitantsmallness of the mass of the inflaton field, and the small-ness of the relevant couplings all suggest that perhaps ahidden scale (or conformal) invariant sector is underlyingthe successful paradigm of slow-roll inflation (see also[33]).

In Ref. [32], this observation led to the suggestion thatperhaps, slow-roll inflation is described by an effectivefield theory near a low energy fixed point, thus unparticlephysics may provide a framework for inflationary cosmol-ogy. Indeed, recently in [34] some of the roles of unpar-ticles in inflationary cosmology were studied, along withthe intriguing possibility that the unparticle field itself maybe the inflaton.

On the other hand, however, inflation cannot be de-scribed by an exactly scale invariant theory: the powerspectrum of inflaton fluctuations is not exactly scale in-variant, and the inflaton potential cannot be completelyflat, since inflation must end, eventually merging withstandard hot big bang cosmology, which in turn implies

that the inflaton mass cannot vanish, thus preventing theinflaton from being described by a conformally invariantunparticle field [34].Thus a mechanism that would lead to a small breaking of

conformal invariance of the unparticle sector is sought. InRef. [6], it was recognized that if a scalar unparticle fieldcouples to a Higgs field, a nonvanishing expectation valueof the Higgs field leads to the explicit breakdown of theconformal symmetry in the unparticle sector. Althoughslow-roll inflation is not exactly scale invariant, it is nearlyso, so that the explicit breaking of scale invariance must besuch so as to lead to a nearly flat inflaton potential, leadingin turn to a nearly scale invariant spectrum of fluctuations[31]. This reasoning suggests that we should study thecoupling between the unparticle and the Higgs field ofthe seesaw form, just as in the case of neutrino mixing[35–37] (see [38] for some work on this topic, and [39]where the unparticle and Higgs particle are taken ascomposites).In this article, we focus on studying in detail the mixing

between a scalar unparticle and a Higgs field in a sponta-neously broken phase in flat space-time as a prelude todealing with inflationary cosmology. In particular we ad-dress the following issues.(i) What are the consequences of mixing fields of differ-

ent scaling dimensions? More specifically, using thelanguage of neutrino mixing, how dowe compute themixing angles and ‘‘mass eigenstates’’? Althoughunparticle-Higgs mixing has been studied in theliterature [4,6], to the best of our knowledge theseissues have not been addressed (although see the firstreference in [38]).

(ii) Consider a seesaw mass matrix between two ca-nonical scalar fields, one massless and one massive,namely,

0 m2

m2 M2

� �: (1.1)

For M2 � m2, it follows that there is one eigen-value�M2 corresponding to the massive scalar andanother eigenvalue ��m4=M2 corresponding tothe lighter state. However, this latter eigenvaluedescribes an instability. Since a canonical scalarfield is a special case of unparticle, it is natural toask the questions if and how the fact that the un-particle field has noncanonical scaling dimensionalters the seesaw mechanism and whether there is aninstability as in the case of canonical fields.It is interesting to note that in the absence of under-lying symmetries, the vanishing matrix element cor-responding to the lighter scalar would be modifiedby radiative corrections. However, it is precisely theunderlying conformal symmetry in the unparticlesector that guarantees the vanishing of that matrixelement.

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What we find in this study is quite interesting. First wesee that the mixing between the unparticle and the Higgsfield enjoys a number of similarities with the Mikheyev-Smirnov-Wolfenstein (MSW) phenomenon of neutrinomixing in a medium [35–37,40], namely, the mixing angledepends on the energy. This is a direct consequence of thenoncanonical nature of the unparticle fields, with the hid-den sector that lends the multiparticle nature to the unpar-ticle interpolating field acting as a ‘‘medium.’’ We find thepossibility of two MSW resonances one at low and one athigh energy.

We also show that the combined unparticle-Higgs sys-tem exhibits spinodal instability as well as a nearly flatpotential. The propagator of the diagonal field closest tothe unparticle field exhibits a pole for spacelike momenta(in Minkowski space-time) which is exactly the signal ofspinodal instability. For small unparticle-Higgs mixing weshow that this instability implies that the field correspond-ing to the unparticle develops a large expectation value andits potential is nearly flat. This instability is a remnant ofthe instability described by the seesaw matrix (1.1). Theunparticlelike field develops a potential with self-interaction which is suppressed by a high power of theseesaw ratio m=M.

Even when the unparticle-Higgs mixing is described bya linear coupling between them, namely, / UH, the propa-gator of the Higgs-like field features a complex pole, theimaginary part (in Minkowski space-time) of which de-scribes the decay of the Higgs-like degree of freedom intounparticlelike degrees of freedom. The fact that this decaycan happen even for linear coupling reflects the fact thatthe unparticle field describes multiparticle states and theHiggs couples to a continuum described by the spectraldensity of the unparticle.

We generalize the above results for the case when sym-metry breaking in the Higgs sector induces a mass gap inthe unparticle sector.

We find that unparticle-Higgs coupling leads to an un-particle effective potential of the new-inflation form, withcoefficients that are suppressed by the seesaw ratios andfurther suppression from the anomalous dimensions for theresulting slow-roll parameters.

II. UNPARTICLE-HIGGS MIXING

The unparticle field describes a low energy conformal(or rather, a scale invariant) sector [1,2,5]. The unparticlefield scales with an anomalous scale dimension that can beinterpreted as a nonintegral number of ‘‘invisible’’ parti-cles [1]. This situation is akin to that of a scalar orderparameter at a nontrivial (Wilson-Fisher) infrared fixedpoint in critical phenomena [27,28]. At this critical pointthis sector becomes scale invariant and correlation func-tions of the unparticle field scale with anomalous dimen-sions. The anomalous scaling dimension reflects the natureof the multiparticle intermediate states and the unparticle

propagator features a dispersive representation with a spec-tral density that features anomalous scaling exponents anddescribes branch cut singularities for timelike momenta.We consider the following Euclidean-space Lagrangian

for unparticle-Higgs mixing

L ¼Z

d4xd4y1

2½UðxÞFðx� yÞUðyÞ þ�ðxÞð�hÞ

� �4ðx� yÞ�ðyÞ� þZ

d4x½g�UðxÞ�2ðxÞ þ Vð�Þ�;(2.1)

where � has dimensions of mass (or momentum) and is ascale that characterizes the unparticle field [1,2,5]. Theunparticle field U emerges as a composite interpolatingfield that describes the infrared fixed point below this scale[1,2,5], and g � 1 is a dimensionless coupling.The scalar potential Vð�Þ features a symmetry breaking

minimum at ’, therefore we write

� ¼ ’þH; (2.2)

and in order to study the mixing between the unparticle andthe Higgs sector we keep only up to quadratic terms in Uand H in the above Lagrangian,

Lð2Þ ¼Z

d4xd4y1

2½UðxÞFðx� yÞUðyÞ

þHðxÞDðx� yÞHðyÞ�þZ

d4x½hUðxÞ þm2UðxÞHðxÞ�; (2.3)

where

h ¼ g�’2; m2 ¼ 2g�’: (2.4)

The vacuum expectation value of the Higgs field breaksexplicitly scale invariance in the unparticle sector [6].We note that the Lagrangian density (2.1) can be ex-

tended to include a term of the form U2ðxÞ�2ðxÞ whichwould result in an explicit mass term M2

UU2ðxÞ for the

unparticle field when the Higgs acquires an expectationvalue. This explicit mass term, which obviously breaksconformal symmetry, moves the unparticle threshold inthe spectral density to M2

U. Indeed the same massless‘‘hidden sector’’ that gives the unparticle its multiparticlenature yields a spectral density that now features a thresh-old at the value of the conformal breaking massM2

U [6]. Wewill consider this case explicitly in Sec. V.The linear term in U in (2.3) leads to a tadpole contri-

bution to the unparticle field that requires renormalization[4]. In what follows we will neglect this term and focus onthe quadratic form to study the mixing.The Euclidean space-time Fourier transforms of the

nonlocal kernels F and D, denoted by F ðpÞ and DðpÞ,respectively, are given by [1,6]

UNPARTICLE-HIGGS FIELD MIXING: MIKHEYEV- . . . PHYSICAL REVIEW D 79, 085018 (2009)

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F ðpÞ ¼ p2

�p2

�2

��; (2.5)

D ðpÞ ¼ p2 þM2H; (2.6)

where 0 � �< 1.1

We will consider weak unparticle-Higgs mixing and thatthe unparticle scale � � ’ [1,2]. These result in thefollowing hierarchy of mass scales:

m � MH � �: (2.7)

ForMH � m the mixing matrix will be of the seesaw formwhich is our primary interest for inflationary cosmology asdiscussed above. Furthermore, consistent with the unpar-ticle interpretation, the symmetry breaking scale of theHiggs sector must be below the unparticle scale so thatthe interpolating unparticle field is a suitable description ofthe hidden sector in the broken symmetry phase.

In the absence of unparticle-Higgs interaction, theEuclidean propagator for the unparticle field

GUðpÞ ¼ F�1ðpÞ ¼ 1

p2ðp2

�2Þ��(2.8)

is normalized so that

GUð�Þ ¼ 1

�2(2.9)

as is usually done in critical phenomena. This normaliza-tion fixes the wave function renormalization constant at thescale � [27]. Alternatively the field can be normalized sothat dG�1

U ðp2Þ=dp2jp2¼�2 ¼ 1 which differs from the pre-

vious normalization by a finite wave function renormaliza-tion constant for �< 1. This field normalization differsfrom the usually adopted one in the literature [1]. We notethat the unparticle field U normalized in this mannerfeatures engineering mass dimension 1 just like an ordi-nary scalar field, but scaling mass dimension 1� � asbefits a conformal field with anomalous dimension �.Therefore the engineering mass dimensions of h and mare 3 and 1, respectively. It is convenient to pass on toFourier transforms (in Euclidean space-time) introducingthe Fourier transforms of the fields as ~U, ~H, respectively, interms of which the quadratic part of the action (2.3) be-comes

Sð2Þ ¼Z

d4p1

2ð ~Uð�pÞ ~Hð�pÞÞ F ðpÞ m2

m2 DðpÞ� �

~UðpÞ~HðpÞ

� �:

(2.10)

The Lagrangian is diagonalized in the basis of ‘‘mass’’eigenstates (borrowing from the language of neutrino mix-ing) � and �, related to the unparticle and Higgs fields as

~UðpÞ~HðpÞ

� �¼ CðpÞ SðpÞ

�SðpÞ CðpÞ� �

�ðpÞ�ðpÞ

� �; (2.11)

where

CðpÞ ¼ 1ffiffiffi2

p�1þ DðpÞ �F ðpÞ

½ðDðpÞ �F ðpÞÞ2 þ 4m4�1=2�1=2

;

(2.12)

SðpÞ ¼ 1ffiffiffi2

p�1� DðpÞ �F ðpÞ

½ðDðpÞ �F ðpÞÞ2 þ 4m4�1=2�1=2

(2.13)

are effectively the cosine (CðpÞ) and sine (SðpÞ) of the‘‘mixing angle’’ between the unparticle and Higgs fields.In terms of ‘‘mass eigenstate’’ fields �, �, we can write

the action as

S¼Z

d4p1

2ð�ð�pÞ�ð�pÞÞ G�1

� ðpÞ 0

0 G�1� ðpÞ

!�ðpÞ�ðpÞ

!;

(2.14)

where

G�1� ðpÞ ¼ 1

2ðF ðpÞ þDðpÞ� ½ðDðpÞ �F ðpÞÞ2 þ 4m4�1=2Þ; (2.15)

G�1� ðpÞ ¼ 1

2ðF ðpÞ þDðpÞþ ½ðDðpÞ �F ðpÞÞ2 þ 4m4�1=2Þ: (2.16)

III. MSW EFFECT: RESONANCES, MIXING, ANDOSCILLATIONS

The similarity with the MSW effect of neutrinos in amedium [40] can be established by defining a ‘‘self-energy’’ for the unparticle field

�UðpÞ ¼ p2

��p2

�2

�� 1

�; (3.1)

which for� � 1 is reminiscent of the one-loop self-energyof a scalar order parameter at an infrared nontrivial criticalpoint renormalized at a scale �, namely, �UðpÞ ’��p2 lnðp2=�2Þ [27]. The action (2.10) can be written as

S ¼Z

d4p1

2ð ~Uð�pÞ ~Hð�pÞÞ

��p2 þM2

H

2

�I

þ 1

2½M4

H þ 4m4�1=2 � cosð2�Þ sinð2�Þsinð2�Þ cosð2�Þ

!

þ �UðpÞ 0

0 0

!� ~UðpÞ~HðpÞ

!; (3.2)

where I is the identity 2� 2 matrix and

1The anomalous dimension � defined in this manner is twicethe critical exponent for the two point correlation function in acritical theory [27].


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cosð2�Þ ¼ M2H

½M4H þ 4m4�1=2 ; (3.3)

sinð2�Þ ¼ 2m2

½M4H þ 4m4�1=2 : (3.4)

This action can be written in terms of a ‘‘mixing angle inthe medium’’ �mðpÞ as follows:

L ¼Z

d4p1


��p2 þM2

H

2þ�UðpÞ

2

�I

þ 1

2½ð�UðpÞ �M2

HÞ2 þ 4m4�1=2

� � cosð2�mðpÞÞ sinð2�mðpÞÞsinð2�mðpÞÞ cosð2�mðpÞÞ

� ��~UðpÞ~HðpÞ

� �;

(3.5)

with

cosð2�mðpÞÞ ¼ M2H ��UðpÞ

½ð�UðpÞ �M2HÞ2 þ 4m4�1=2 ; (3.6)

sinð2�mðpÞÞ ¼ 2m2

½ð�UðpÞ �M2HÞ2 þ 4m4�1=2 : (3.7)

This form makes manifest that the multiparticle contribu-tion that defines the unparticle can be associated with a‘‘medium effect,’’ namely, the degrees of freedom thathave been integrated out leading to the anomalous scalingdimension of the unparticle field, and leads to a MSWphenomenon [40], namely, a dependence of the mixingangle on the energy (here the Euclidean momentum).From the expressions (3.6),(3.7) it is clear that CðpÞ andSðpÞ of (2.12), (2.13) are identical to cosð�mðpÞÞ andsinð�mðpÞÞ, respectively.

There is an MSW resonance phenomenon when thecondition

�UðpÞ ¼ M2H (3.8)

is fulfilled. It is convenient to define

x � p2

�2(3.9)

in terms of which the resonance condition becomes

x½x�� 1� ¼ M2H

�2: (3.10)

The function xðx�� 1Þ is depicted in Fig. 1. From thisfigure it becomes clear that there are twoMSW resonanceswhenever

M2H

�2<�ð1� �Þð1��Þ=�: (3.11)

Consider the case �2 � M2H such that the condition for

MSW resonances (3.11) is fulfilled: there is a low energy

resonance (p2 � �2) at

p2 ’ M2H

�M2

H

�2

��=ð1��Þ

(3.12)

and a high energy resonance (p2 ��2) at

p2 ’ �2

�1� 1

2�

M2H

�2

�: (3.13)

Upon analytically continuing from Euclidean toMinkowski momenta p2 ! �ðp2

0 � ~p2Þ � i0þ we note

that the low energy resonance occurs near the light conebut for slightly spacelike momenta (in the limitM2

H � �2)whereas the high energy resonance occurs at large space-like momenta.Unlike the case of neutrinos wherein a single particle

Fock representation is a suitable description of the quan-tum mechanics of mixing and oscillations, the fact that theunparticle field is an effective field that describes multi-particle states prevents a similar analogy. However, we canlearn about mixing and oscillation phenomena by studyingcorrelation functions of the unparticle and Higgs fields.This is best achieved by introducing sources JU;� conju-

gate to the respective fields and a generating functionalZ½JU; J�� that yields the correlation functions throughfunctional derivatives. It is straightforward to obtain thegenerating functional by inverting the quadratic form in theaction (2.10), we find

lnZ½JU; J�� ¼ 1

2

Zd4pðJUð�pÞJ�ð�pÞÞ

� G�ðpÞG�ðpÞ

DðpÞ �m2

�m2 F ðpÞ

!!

� JUðpÞJ�ðpÞ

!; (3.14)

where the Green’s functions of ‘‘mass eigenstates’’G�;�ðpÞ are given by Eqs. (2.15) and (2.16). For conve-

FIG. 1 (color online). The function xðx�� 1Þ vs. x for � ¼0:2, 0.4, 0.6, 0.8.


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nience of notation, we introduce

�ðpÞ ¼ 1

2ðF ðpÞ þDðpÞÞ ¼ p2 þM2

H

2þ �UðpÞ

2; (3.15)

�ðpÞ ¼ 12½ðF ðpÞ �DðpÞÞ2 þ 4m4�1=2

¼ 12½ð�UðpÞ �M2

HÞ2 þ 4m4�1=2; (3.16)

in terms of which G�1� ðpÞ ¼ �ðpÞ þ �ðpÞ, G�1

� ðpÞ ¼�ðpÞ � �ðpÞ. The diagonal and off diagonal correlationfunctions in the unparticle-Higgs basis are given by

hUðpÞUð�pÞi ¼ DðpÞ�2ðpÞ � �2ðpÞ ; (3.17)

h�ðpÞ�ð�pÞi ¼ F ðpÞ�2ðpÞ � �2ðpÞ ; (3.18)

hUðpÞ�ð�pÞi ¼ ��ðpÞ sinð2�mðpÞÞ�2ðpÞ � �2ðpÞ

¼ 1

2sinð2�mðpÞÞ

�1

G�ðpÞ �1

G�ðpÞ�:

(3.19)

The off diagonal propagator (3.19) is exactly of the sameform of the mixed propagators for two neutrinos in theflavor basis [41] whereG�1

�;� correspond to the propagators

of mass eigenstates. In the case of neutrino mixing, thetransition probability emerges directly from the off diago-nal correlation function as follows. Analytically continuingto Minkowski space-time, each propagator has a simplepole at the values of p0 ¼ E1ð ~pÞ, E2ð ~pÞ, respectively. Thenthe time evolution of the off diagonal correlator is obtainedby performing an inverse Fourier transform in time, whichin turn yields the time dependence for the off diagonalcorrelator

/ sinð2�mðpÞÞ½eiE1ð ~pÞt � eiE2ðpÞt�:The transition probability is obtained from the absolutevalue squared of the off diagonal correlator [41] P ðtÞ /sin2ð2�mðpÞÞsin2½ðE1ð ~pÞ � E2ð ~pÞÞt=2�.

Hence the off diagonal correlation function (3.19) de-scribes the generalization of mixing and oscillations forunparticle-Higgs mixing. For weak unparticle-Higgs cou-pling, we expect the propagators to feature singularitiesnear those corresponding to the unparticle cut beginning atp2 ¼ 0 and Higgs pole at p2 ¼ M2

H, respectively, inMinkowski space-time. Therefore, unlike the case of al-most degenerate neutrinos, the unparticle-Higgs oscillationwill not be coherent over long space-time intervals.Instead, dephasing occurs on space-time scales of the orderof the Compton wavelength of the Higgs-like mode�1=MH and is further suppressed by the decay of thismode (see below).

IV. SINGULARITY STRUCTURE: POLES ANDCUTS

The singularity structure of the propagators for the‘‘mass eigenstates’’ is obtained from the conditions

G�1� ðpÞ ¼ 0; G�1

� ðpÞ ¼ 0: (4.1)

Both conditions can be combined into

F ðpÞDðpÞ ¼ m4: (4.2)

For m2 � M2H � �2 we expect to find singularities near

the Higgs ‘‘mass shell’’ p2 ��M2H and near the beginning

of the massless threshold for the unparticle p2 � 0.(i) Higgs-like pole: To find the position of the singular-

ity near the Higgs mass shell we write

p2 þM2H � M2

H�; � � 1: (4.3)

The condition (4.2) yields

� ’ � m4

M4H

��M2H

�2

�� þO

�m8

M8H

��M4H

�4

��: (4.4)

Upon analytic continuation to Minkowski space-time p2

E ! �p2M, M2

H ! M2H � i0þ the pole be-

comes complex with an imaginary part

�H ’ m4

2M3H

�M2

H

�2

��sinð��Þ: (4.5)

This is a complex pole of G�ðpÞ given by (2.15).

From the expressions (2.12) and (2.13) with p2 ��M2

H, DðpÞ � 0, and F 2ðpÞ � m4, it follows thatCðpÞ � 1, SðpÞ � 0, and the complex pole in G�ðpÞdescribes a Higgs-like unstable particle. What isremarkable in this result is the fact that theunparticle-Higgs coupling is linear in both fields,and the decay width emerges at ‘‘tree level.’’ Thisis in marked contrast with the decay rate from non-linear couplings studied in Ref. [4]. If the unparticlefield were canonical, such coupling would not resultin an imaginary part in the propagator of the Higgs-like mode at tree level. However, the unparticle fieldis an interpolating field that describes a multiparticlecomposite with a continuum spectral density. Sincethe threshold begins at p2 ¼ 0, it follows that uponcoupling the fields, the Higgs pole on the positivereal axis in the p2 plane (in Minkowski space-time)is actually embedded in the continuum of statesdescribed by the unparticle field, resulting in themotion of this pole off the physical sheet into asecond or higher Riemann sheet. The imaginarypart just describes the decay of the Higgs at ‘‘tree’’level.We note that the real part of the pole receives a finitemass renormalization, and the real part is lightlike,


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therefore far away from the MSW resonance regionwhich occurs for spacelike momenta.

(ii) Near unparticle threshold: The singularity near theunparticle threshold at p2 ¼ 0 can be found bysetting p2 ¼ 0 in DðpÞ and corresponds to a singu-larity in G�ðpÞ given by Eq. (2.15).Defining x ¼ p2=�2 � 1 the condition (4.2) yields

x1�� ’ m4

M2H�

2) p2 ’ m4

M2H

�m4

M2H�

2

��=ð1��Þ

:

(4.6)

For � ¼ 0 one recovers the negative eigenvalue�m4=M2

H of the seesaw mass matrix (1.1).Furthermore it is clear that although this pole isspacelike, it is well below the position of the lowenergy MSW resonance (3.12).This pole on the positive real axis in Euclidean p2

describes an instability since upon analytic continu-ation back to Minkowski space-time p2 ! �p2 ¼�ð!2 � k2Þ corresponding to frequencies

!ðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 �M2

p;

M2 ¼ m4

M2H

�m4

M2H�

2

��=ð1��Þ

:

(4.7)

These become imaginary in the band of spinodallyunstable modes [42]

0< k2 � M2 (4.8)

corresponding to spinodal instabilities [42].Thus we see that the instability obtained from the seesaw

mass matrix (1.1) is reproduced but with a coefficient thatdepends on the ratio of scales and the anomalous dimen-sion. For this pole the mixing angles CðpÞ � 1, SðpÞ � 0and the � field is identified with an unparticlelike mode,the pole for p2 < 0 in Minkowski is on the opposite side ofthe branch cut singularity for p2 > 0 and is isolated fromthe unparticle continuum.

The spinodal instabilities signal that the ‘‘potential’’associated with the � (unparticlelike) mode features aminimum away from the origin and the instabilities reflectthe ‘‘rolling’’ of the expectation value of the � fieldtowards the minimum [43].

The unparticle-Higgs mixing leads to a potential for theunparticlelike field �. Consider the Higgs self-interaction�H4 from (2.11)

�H4 ¼ �ðCðpÞ�ðpÞ � SðpÞ�ðpÞÞ4 ’ �S4ðpÞ�4ðpÞ;(4.9)

where we have focused on the direction along which � ¼0. For the unstable pole we can set p ’ 0 in the expressionfor the mixing angle (2.13) or (3.7) and we find

SðpÞ � �mðp� 0Þ ’ m2

M2H

; (4.10)

combining this result with Eq. (4.7) leads to the effectiveunparticle potential for the � mode

V ð�Þ ’ V ð0Þ �M2

2�2 þ �

m8

M8H

�4 (4.11)

which is of the form describing new inflation. Thus we seethat the effective self-coupling for the�mode (unparticle-like) is a seesaw ratio of scales, which for m � MH con-sistent with small breaking of conformal invariance, entailsthatV ð�Þ is very shallow. Even for �� 1, the quartic self-coupling for the unparticle field is a large power of theseesaw ratio m=MH very similarly to the effective fieldtheory approach to slow-roll inflation discussed inRef. [32].The expectation value of� is obtained by balancing the

quadratic term whose coefficient is determined by theunstable pole (4.6) and the quartic term (4.11); we find

h�i ’ M3H

�m2

�m2

MH�

��=ð1��Þ

: (4.12)

It is interesting to note that the spinodal instabilities existindependently of the specific form of the Higgs potential.Even if we replace the Higgs and its �H4 self-interactionwith a scalar field with no potential, these instabilitiespersist. In this latter case, the unparticle field will acquirea potential by the virtue of the Coleman-Weinberg mecha-nism [44] and the instabilities reflect the rolling toward theminimum of this potential.

V. UNPARTICLE MASS GAP

In the study presented in the previous section, we haveneglected a mass gap for the unparticle field.As mentioned in Sec. II, a term U2�2 in the unparticle-

Higgs Lagrangian yields an explicit mass term for theunparticle field, which manifestly breaks conformal invari-ance. The emission and absorption of massless excitationsof the ‘‘hidden’’ conformal sector that leads to the multi-particle nature of unparticles, results in that this mass termmoves the threshold away from p2 ¼ 0 to p2 ¼ M2

U in thecomplex p2 plane.In Ref. [6], a simple manner to introduce a mass scale to

break conformal invariance in the unparticle sector wasintroduced by modifying the spectral representation of theunparticle propagator. Such a modification was also used inthe study in Ref. [4] where it is argued that the mass gap inthe spectral representation arises from unparticle-Higgscoupling. The introduction of an unparticle mass gap MU

results in a spectral density that features a branch cutbeginning at a threshold p2 ¼ �M2

U in Euclidean momen-tum. The spectral density featuring a branch discontinuitywith threshold M2

U resulting in anomalous scaling dimen-sions is reminiscent of the Bloch-Nordsieck [24] resum-mation of the emission and absorption of nearly collinear(soft) photons in quantum electrodynamics [25] where the


085018-7

fermion propagator acquires an anomalous dimensionfrom infrared threshold divergences [25].

In Ref. [45] it was shown that the renormalization groupresummation of the infrared divergences arising from theemission and absorption of soft massless quanta by amassive field is equivalent to the Bloch-Nordsieck resum-mation and yields precisely a spectral density with a branchcut beginning at a threshold given by the mass of theparticle that emits and absorbs the soft quanta.

Motivated by these cases and the following Refs. [4,6],we include an unparticle mass gap MU by modifying theF ðpÞ in Eq. (2.5) to

F ðpÞ ¼ ðp2 þM2UÞ�p2 þM2

U

�2

��: (5.1)

This is the type of inverse propagator obtained by a renor-malization group or alternatively a Bloch-Nordsieck re-summation of infrared divergences arising from emissionand absorption of soft massless quanta [45]. Consistentwith a small breaking of conformal invariance, we focuson the case in which MU � MH � �.

The analysis presented in the previous sections can befollowed by replacing the unparticle self-energy (3.1) by

�UðpÞ ¼ ðp2 þM2UÞ��

p2 þM2U

�2

�� 1

�; (5.2)

and the action (3.5) by

L ¼Z

d4p1


�"�

p2 þM2H þM2

U

2þ �UðpÞ

2

�I

þ 1

2½ð�UðpÞ þM2

U �M2HÞ2 þ 4m4�1=2

� � cosð2�mðpÞÞ sinð2�mðpÞÞsinð2�mðpÞÞ cosð2�mðpÞÞ

!#~UðpÞ~HðpÞ

!; (5.3)

with the ‘‘in medium’’ mixing angles determined by

cosð2�mðpÞÞ ¼ M2H �M2

U � �UðpÞ½ð�UðpÞ þM2

U �M2HÞ2 þ 4m4�1=2 ;

(5.4)

sinð2�mðpÞÞ ¼ 2m2

½ð�UðpÞ þM2U �M2

HÞ2 þ 4m4�1=2 :(5.5)

The condition for an MSW resonance now becomes

�UðpÞ ¼ M2H �M2

U � �M2: (5.6)

Upon introducing the variable x ¼ ðp2 þM2UÞ=�2 this

condition becomes

x½x�� 1� ¼ �M2

�2; (5.7)

which is again satisfied with two resonances for

0<�M2

�2<�ð1� �Þð1��Þ=�: (5.8)

For � � M2H � M2

U the resonances occur for

p2 ’ �M2U þ �M2

��M2

�2

��=ð1��Þ

; (5.9)

and

p2 ’ �M2U þ�2

�1� 1

2�

�M2

�2

�: (5.10)

Upon analytically continuing from Euclidean toMinkowski momenta p2 ! �ðp2

0 � ~p2Þ � i0þ we note

that whereas the high energy resonance (5.10) occurs forspacelike momenta there is the tantalizing possibility thatthe low energy resonance (5.9) could occur at lightlikemomenta. Obviously whether or not this possibility isrealized depends on the details of the scales.The results for mixing and oscillations obtained in

Sec. III remain the same with obvious modifications inthe mixing angles and propagators.The singularities in the propagators lead to the same

condition (4.2) as for M2U ¼ 0.

(i) Higgs-like pole: We write again

p2 þM2H � M2

H�; � � 1 (5.11)

and the condition (4.2) now yields

� ’ � m4

M2H�M

2

��M2

�2

��: (5.12)

Upon analytic continuation to Minkowski space-time p2

E ! �p2M; M2

H ! M2H � i0þ the pole be-

comes complex with an imaginary part

�H ’ m4

2MH�M2

��M2

�2

��sinð��Þ�ðM2

H �M2UÞ:(5.13)

The�ðM2H �M2

UÞ in (5.13) results from the fact thatnow the continuum spectral weight for the unparticlehas a threshold atM2

U. Thus, ifMH <MU the pole inthe bare Higgs propagator along the real axis in theMinkowski p2 plane, lies below the unparticle con-tinuum and the Higgs-like particle is stable. ForMH >MU this (bare) pole is in the unparticle con-tinuum and moves off into an unphysical sheet in-dicating a decay width for the Higgs-like particle.

(ii) Near unparticle threshold: Defining x ¼ðp2 þM2

UÞ=�2 � 1 the condition (4.2) yields


085018-8

x1�� ¼ m4

�M2�2) p2 ’ m4

�M2

�m4

�M2�2

��=ð1��Þ

:

(5.14)

For M2H >M2

U this pole along the real axis inEuclidean space, again indicates spinodal instabil-ities. The band of spinodally unstable wave vectorsis the same as in Eq. (4.8) but with

M 2 ¼ m4

�M2

�m4

�M2�2

��=ð1��Þ

: (5.15)

If M2U >M2

H the value of p2 becomes complex,indicating the possibility of the decay of the unpar-ticle mode. While this possibility is interesting anddeserves to be explored in its own right, we areprimarily interested in the case M2

H >>M2U be-

cause this case corresponds to the seesaw mecha-nism with two widely different mass scales that maybe of relevance for inflationary cosmology.

Therefore we conclude that in the relevant case M2H >

M2U the results obtained are a straightforward generaliza-

tion of the case M2U ¼ 0 with the same features, namely, a

band of spinodally unstable wave vectors, a shallow effec-tive potential for the unparticle mode and a width for theHiggs-like particle indicating its decay into unparticlemodes as a consequence of the multiparticle continuumdescribed by them.

Furthermore the effective unparticle potential is of thesame form as (4.11) but with the replacement M2

H ! �M2

with similar conclusions.

VI. POTENTIAL CONSEQUENCESFOR SLOW-ROLL INFLATION

Although the WMAP data [29,30] rules out a purelyquartic inflaton potential, a systematic analysis combinedwith Markov-chain Monte Carlo of the available data fromcosmic microwave background and large scale structureshows that a new-inflation type potential with a nonvanish-ing mass term, just as that given by Eq. (4.11) fits the dataremarkably well [46]. Therefore identifying the unparticlewith the inflaton field [34], with the effective inflatonpotential given by (4.11), we can advance some importantpreliminary consequences for slow-roll inflation. Bothlowest order slow-roll parameters

v ¼ M2Pl

2

V 0

V

!2

; �v ¼ M2Pl

V 00

V(6.1)

involve M2 for slow-roll initial conditions (�� 0). Wesee from (4.7) that the slow-roll parameters are determinedM2 (for a general dependence see [46]). This term issuppressed by the small seesaw ratio m2=M2

H, but alsofor a nontrivial unparticle anomalous dimension 0<�<1 it is further suppressed by the factor

�m2

MH�

��=ð1��Þ

;

making the slow-roll parameters smaller. To lowest orderin slow-roll parameters, the scalar index of the powerspectrum is ns ¼ 1þ 2�v � 6v [29,30]. We note thatwhereas v is manifestly positive, the sign of �v deter-mines whether the power spectrum is red or blue tilted. Theeffective unparticle potential (4.11) distinctly yields a redtilt which is consistent with the results from WMAP[29,30].We would like to emphasize that here, we have used

slow-roll parameters for a field with canonical kineticterms. This is done to give an indication that simply basedon the potential, an unparticle field description of theinflaton can yield a nearly scale invariant power spectrum.However, as the kinetic term of the unparticlelike field isnot canonical, a deeper understanding of the slow-rollconditions in this case is required. This will be exploredin a future work. It is conceivable that the next generationof cosmic microwave background observations may yieldinformation on the unparticle anomalous exponent.

VII. CONCLUSIONS

Motivated by the successful and predictive paradigm ofslow-roll inflation, we explored the possibility that unpar-ticle physics may yield to an underlying understanding ofthe main features of slow-roll inflation: a fairly flat poten-tial, nearly Gaussian and scale invariant spectrum of fluc-tuations. We identify these as hallmarks of a nearly criticaltheory. Thus, unparticle physics, describing a conformallyinvariant theory appears as a natural candidate for anunderlying theory of slow-roll inflation. However, confor-mal invariance must be explicitly broken but in a ‘‘small’’manner, since inflation must end. Coupling unparticle andHiggs fields in a broken symmetry phase of the Higgssector leads to an explicit breaking of conformal symmetryin the unparticle sector [6]. However, we seek a mechanismthat yields small breaking of this symmetry in the form of aweak coupling between unparticle and Higgs sectors and aseesaw type mixing matrix between them. Thus we study amodel of a scalar unparticle weakly coupled to a Higgsfield in a broken symmetry phase. The expectation value ofthe Higgs leads to linear Higgs-unparticle coupling and aseesaw type mixing matrix. We find a wealth of phe-nomena possibly relevant to inflationary cosmology butalso of intrinsic interest.As a consequence of the unparticle field being defined

by noncanonical scaling dimensions we find that the mix-ing angles depend on the momentum four vector leading toan MSW effect: the hidden sector that is integrated out todefine the interpolating unparticle field acts as a medium.We find two MSW resonances, one at low and one at highenergy, respectively. For low momentum we find isolatedpoles in the unparticlelike mode away from the branch cuts


085018-9

that characterize the unparticle spectrum. These polesdescribe spinodal instabilities and indicate that the unpar-ticle field acquires an expectation value. Indeed, the Higgspotential generates an effective potential for the unparticle-like field because of the mixing. We find that even for astrongly self-coupled Higgs sector, the self-couplings ofthe unparticlelike mode are suppressed by large powers ofthe seesaw ratios. The instability and small self-couplingsboth entail a nearly flat potential for the unparticlelikefield, a hallmark of slow-roll inflation. We also find theremarkable result that the propagator for the Higgs-likemode features a complex pole, whose imaginary part de-termines the decay rate. We emphasize that this is a ‘‘tree-level’’ effect, not a result of nonlinear couplings and is aconsequence of the continuum in the spectral representa-tion of the unparticle field. The pole in the bare Higgspropagator becomes embedded in the continuum of theunparticle field upon their coupling, even for a linearcoupling.

Unparticle-Higgs mixing also leads to oscillation phe-nomena just as in the case of neutrino mixing. However,because of the large difference in scales, oscillations deco-here on short space-time scales of the order of the Comptonwavelength of the Higgs particle.

The results obtained for a massless scale invariant un-particle field were then generalized to the case in whichthere is an unparticle mass gap MU. A seesaw mechanismconsistent with a slow-roll picture requires that the unpar-ticle mass gap be much smaller than the Higgs mass, inwhich case all of the results of the massless unparticle casetranslate with minor modifications to the case of a massivethreshold for the unparticles.

A major result of this exploration is that an unparticlefield weakly coupled to a Higgs particle yields a remark-

able similarity to the main features of slow-roll inflationand could possibly provide an underlying justification forthe slow-roll paradigm. Unparticle-Higgs seesaw type cou-pling yields an effective unparticle potential of the new-inflation form with coefficients that are suppressed byseesaw ratios. Simply based on this potential, the furthersuppression by the anomalous dimension of the unparticlefield might lead to smaller departures from scale invarianceand to a red spectrum consistent with the WMAP results onthe scalar spectral index.Since the unparticle field features noncanonical kinetic

terms, we envisage these models as possible alternatives toDirac-Born-Infeld inflationary proposals [47]. An impor-tant follow-up will be to understand the influence of thenoncanonical kinetic terms for the unparticlelike field so asto establish a firmer correspondence with the dynamics ofslow-roll inflation. Noncanonical kinetic terms in othercontexts have been studied in Refs. [48] and we willexplore these aspects along with loop corrections in futurework.We focused our study in Minkowski space-time as a

prelude to a more comprehensive exploration of inflation-ary cosmology. The results obtained here are most cer-tainly encouraging and suggest that further explorationwithin the inflationary context is worthwhile.

ACKNOWLEDGMENTS

D.B. acknowledges support from the U.S. NationalScience Foundation through Grant No. PHY-0553418.R.H. and J. H. acknowledge support from the DOE throughGrant No. DE-FG03-91-ER40682. J. H. is also supportedby the De Benedetti Family.

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unparticle-higgs field mixing: mikheyev-smirnov-wolfenstein resonances, seesaw mechanism, and...

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