On spontaneous versus explicit breaking of continuous global symmetries

Physics Letters B 268 ( 1991 ) 75-80 North-Holland

PHYSICS LETTERS B

On spontaneous versus explicit breaking of continuous global symmetries ¢r M. Olechowski 1 Institut fur Theoretische Physik der UniversiRitHeidelberg, Philosophenweg 16, W-6900 Heidelberg, FRG

Received 8 July 1991

Pseudo-Goldstone boson masses are considered in models with very slightly explicitly broken continuous global symmetries. These masses are compared with the size of the explicit symmetry breaking and with the masses of other scalars. The lepton number violation in some extensions of the standard model is discussed as an example.

Many extensions of the standard model contain continuous global symmetries [ 1-6 ]. Spontaneous breaking o f any of such symmetries (let us call it H ) produces, due to the Goldstone theorem, one massless s c a l a r - a Goldstone boson (GB). But massless scalars have not been observed in nature. There are mainly two ways to avoid the conflict with experiment. Firstly, one may try to make the GB invisible by reducing its coupling to the ordinary matter. Secondly, one may give a mass to the GB by explicit breaking o f H. Such a nearly massless scalar is called a pseudo-Goldstone boson (PGB) [ 7 ]. Its mass rnpcB is typically much smaller than masses o f other scalars in a model. One could naively expect that mpG a is of the same order o f magnitude as parameters describing the explicit breaking o f H. In a recent paper [ 8 ] Lusignoli, Masiero and Roncadelli compared this naive expectation with one specific example - the triplet majoron model with continuous global lepton number symmetry [4 ]. The authors o f ref. [ 8 ] find that the PGB mass is, unexpectedly, many orders o f magnitude larger than the size of the explicit lepton number violation. They also show an example in which mpG B does not depend at all on the size o f the explicit breaking. ~r Supported by Deutsche Forschungsgemeinschaft. i On leave of absence from Institute for Theoretical Physics, Warsaw University, ul. Ho~'a69, PL-00-681 Warsaw, Poland.

In the present letter we discuss the PGB masses in a more general case. We compare these masses with two quantities: the size o f the explicit symmetry breaking and the masses of other (no P G B ) scalars. We find it rather typical that mpGa is not of the same order as the size o f the explicit symmetry breaking. Usually the PGBs are lighter than other scalars but very often they are much heavier than naively expected. We also find that, unlike in ref. [ 8 ], it is impossible that mpGa does not depend on the size o f the explicit breaking. However, this discrepancy is caused by the different definition o f PGB. We illustrate our considerations with the lepton n u m b e r violation in some extensions of the standard model. We will investigate models in which the scalar potential can be written in the form v ( o , ~) = Vo(O) + ~vB ( ¢ ) ,

(1)

where ~ is a real vector o f scalar fields and e is some small parameter. We denote the m i n i m u m o f V by g2(e) and that of Vo by 12o. The v a c u u m expectation values (VEVs) o f scalar fields are <(gj>a(,)=vj(E),

ao=Vj,

(2)

where #j are components o f ~. The potential V is invariant under a continuous symmetry group G while Vo is invariant under a larger group Go. The symmetry group Go is explicitly broken to G by interactions in VB. We choose e in such a way that parameters in Vu are typically o f the same order o f magnitude as

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those in Iio. Go is slightly broken because e is very small. But how small should be the parameter E? We will treat e Va as a perturbation. Hence, we will calculate values of some quantities at the minimum t2(e) of the full potential Vby expanding around their values at t2o in a series in e. We assume that the higher order terms of this expansion are small and can be neglected. Thus, we demand that 12(e) is "close" to ~o for small e. There are two kinds of situations when this is not the case: (a) Vo has two (or more) nearly degenerate local minima. It is possible that t2 (e) is close to one of these minima for e < eo and to the other for E> %. In such a case, the position of the vacuum in the field space is not a continuous function of E for E= e0. Thus, expanding around 12o may lead to wrong results (the sum of a few first terms in the expansion may be far from the full result). Hence, we demand that e is smaller than the smallest eo for which the position of 12(e) changes discontinuously. (b) Vo has exactly degenerate minima. Of course, this always happens when some symmetries are broken spontaneously. Knowing t2o, we can obtain another minimum of Vo acting on 12o with the generators of the spontaneously broken symmetries. However, we want t2o to be close to t2(e) for small ~. The simplest choice is I2o= limI2(e),

vj= limvj(e)

¢~0

e~O

.

(3)

We are interested in models in which the symmetries in Go/G are also broken spontaneously. But what does it mean in the case with the explicit breaking? Usually one says that a symmetry H is broken spontaneously if the potential (or rather the full lagrangian) is invariant under H while the vacuum is not. In our case Vis not invariant under Go/G so, strictly speaking, these symmetries cannot be spontaneously broken. However, when the explicit breaking is small we can adopt the definition that the symmetry is spontaneously broken if it is spontaneously broken in the limit of vanishing explicit breaking. In the case of the potential (1) this means that Go/G is spontaneously broken in the minimum of 1Io. This definition means also that PGBs are those scalars that be76

3 October 1991

come real GBs in the limit of vanishing explicit breaking a~ We are ready now to consider our two problems. Let us start with the first: what is the ratio between the PGB mass and the size of the explicit symmetry breaking. One can answer this question without doing any calculation. Let us assume that the main contribution to mmB comes from an interaction in eVa which is k-linear in the scalar fields. The coefficient of such a term is of the form E/~a-k, where/1 is some mass parameter. Thus, the size of the explicit breaking is given by the quantity ~ta~ l/(4-k)/l. The scalar masses squared are given in the tree approximation by the second derivatives oft.he scalar potential evaluated at the minimum. Thus, the leading (in E) contribution to the PGB mass is of the form

m2poB=O( E~4--kvk-2) =O(~4--kv k-2)

,

(4)

where v denotes vacuum expectation values of some fields (it may be a product of different VEVs). From this we find

1--k/2 rnp°a =O((~)/~a

)"

(5)

The behaviour of the above ratio in the limit of very small explicit breaking (/~B--.0) depends simply on the dimension of the symmetry breaking interaction. mpoB is much bigger than aB if k = 3 which means that the global symmetry is broken by an interaction threelinear in the scalar fields ~2. This happens in the triplet majoron model considered in refs. [4,8 ].Thus, it is not surprising that the majoron mass is much bigger than the size of the explicit symmetry breaking, especially when this breaking is very small. The PGB mass is of the order of/~B when the symmetry is broken by bilinear interactions. And finally, mpoB <
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lars. The answer is again very simple if the typical scalar mass squared, m~, is of order 2v 2 and all VEVs are of the same order of magnitude. Then,

3 October 1991

0 = ( 0i V ) a . ~

=~(0iVn)ao+ Y, (0g Vo)~oSvj+O(C).

(ll)

J

mpon = O

(6)

/72 S

V

"

However, the situation is not so simple in models with a hierarchy of VEVs. This is the case in models with lepton number violation [4-6]. Usually the VEVs which break the lepton number conservation are much smaller than those breaking the standard SU2×UI gauge symmetry. We have to do some calculation in order to understand what happens in such a case. Let us start with the symmetric potential Vo. The scalar mass matrix evaluated at the minimum f2o has one zero eigenvalue,

T~vjO,,

(8)

where matrix M is defined by

M~#=- X (Tav)i(Tav),,

o= Y~ [(03kVo)~o(r~v),+ (0~Vo)~or~, + ( 0~kVo)~o T~ ] (T'~vb = ~ (03~Vo)ao(T'~v)i(Tav)j ij

+ ~ (OaikVo)ao(T~Tav)i,

(12)

i

is a consequence of the invariance of Vo under the transformations generated by T ~, T p. Using eqs. ( 11 ) and ( 12 ) we w r i t e / / i n the following form:

(7)

for each spontaneously broken symmetry (generated by T~). The Goldstone boson fields are given by linear combinations of those zero eigenvalues. If some of the spontaneously broken symmetries are also explicitly broken by Fa (or more generally: by any interaction in the considered model) then the corresponding GBs get masses and become PGBs. The mass squared matrix for the light bosons (GBs and PGBs) is given by [ 9 ] rr/a2B= - [M-IHM-I]a#,

The second,

(9)

+ ~ (OsVB)~o(T~TPv)i)+O(e2).

(13)

The PGB masses are given by the nonzero eigenvalues of (8). One can immediately notice one thing: these masses depend not only on the size of the explicit breaking (parameters in Vs) but also on the size of the spontaneous breaking (through the VEVs determined by Vo). Now we will apply the above equations to some extensions of the standard model with lepton number violation. Let us assume that there are p electrically neutral, complex scalar fields:

i

and is necessary to orthonormalize the states (7). Matrix 1/is equal to

H~p= - E (O~V)a(~)(T'~v)i(TPv)j ij

× (Tav)i(TBv)j+O(e 2) ,

(10)

where 6v~vk(*)- Vk and we used the notation 0~...-= 0"/00~0j... We need two relations to simplify the above equation. One is the extremum condition for V:

• i=vi+pi+irli,

i = 1 .... , p ,

(14)

where vi is a real VEV and pi (t/i) is a real (imaginary) component of @. There are two spontaneously broken U~ symmetries: one gauge symmetry, Uiz, which is a subgroup of the standard SU2 × UI and one global symmetry, U[, of the lepton number conservation. We denote the charges of the ~, field under the U z and U~ transformations by qZ and q~, respectively. Now we can write matrices (9) and ( 13 ) in the form P i

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The off-diagonal terms may be even smaller if v~ is the U z invariant. On the other hand, Hzz and /-/EL are at most O(evz4 ) and the three first terms in the

I1,~ = ~ ( ~ qTqfvivj ( 0.,,~ 2 V~ ) m 6

sum (16) where a, fl=Z, L and we neglected contributions of higher orders in E. The mass (squared) matrix for the two light bosons (massless for E=0) can be calculated from eq. (8). One of these bosons remains massless and is eliminated by the Higgs mechanism. The other is the PGB of the broken lepton number conservation - the majoron. The square of its mass, m E , is equal to the trace of (8): m~ = -Tr [M-lI1M-J -

×

] q?q~

qTq]vivj

m 2 = - (M~zZHzz + M ~ l l L z

+ M ~z2IIzL + M ~ llLL )

are not bigger than O(Ev.2 ). The last contribution to the sum (21) is of the order

--M~I1LL =O(~(V~-1 ( Op~VB)~0 -

(

0~,,~ VB)ao)

) •

det(M2)=(~

(24)

We assumed that the massful parameters in Va are of the order vz, thus

(qZvi)2)(~/ (qLvi) 2)

mZ ~ev~-l (O~VB)~o=O(ev;Iv 3) >>O(ev 2) .

- - ( ~ qZiqI(v2) 2 .

(18)

The precise value of m~t depends in detail on the model but we are able to estimate it when there is a hierarchy of VEV:

vz>>v~,

(19)

where v~ is the largest VEV breaking U z ( q Z ~ 0 usually the VEV of the standard Higgs doublet) and v~is the largest VEV which breaks U~. We always can make v~ and v~ real using appropriate U z and U~ rotations (we also can easily generalize eqs. ( 15 ) - ( 17 ) for complex VEVs). The determinant o f M 2 is of the order v~2v~2 and the matrix M -2 has the form

~ ~ [°(Vz~) - = ( , O ( v ; ~) 78

(23)

which means that ( 0p~Va )ao ~ P3-'v~ •

where

(22)

There are no charges qZ and qL in the above formula because they cancel in the leading term. The second term in (22) is again at most O(E~ ) but the first one may be bigger if (0p~ VB) is of the order Vz 3 . This happens if there is a U ~ violating interaction in lib which is linear in p~ and couples p~ only to the scalar with the largest VEV: VB ~ 1~3-'pQp~ ,

(17)

(21)

°(Vz~)~ O(v~~)}"

(20)

(25)

If the potential VBdoes not contain terms of the form (23) than m ~ = O ( e v 2) or may be even smaller in some specific cases. However, in all simple models with lepton number violation the condition (23) is fulfilled. We will now discuss briefly three models of this type [4-6,8 ]. In all these models the gauge U z symmetry is broken by the VEV of the Higgs doublet, ~o. Thus, our Pz is just the real part of the neutral component of this doublet. The transformation properties ofp~ are different in each model but its VEV is always much smaller than v~= (~0). Triplet majoron model [4,8]. There is only one scalar with nonzero lepton n u m b e r - the triplet d transforming as (1; 1, - 2 ) under S U 2 X U r X U ~ . There is also only one U~ violating interaction: VB =p~ ~Tiz2 (zA*)~+ h.c.

(26)

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This interaction is of the form (23); thus, the majoron mass is of the order (25): 2 raM(triplet)

~ --1 2 ~ ~ / 1 V d Vrp •

(27)

The authors of ref. [ 8 ] discuss also an example in which v ~ e/.tl and the PGB mass is independent of the U L breaking parameter. However, in such a case v a - 0 when e - 0 . According to our definition this means that there is no spontaneous symmetry breaking and only the explicit one. It is true that the mass of q~o is (almost) e independent but qao should not be called a PGB because it is not a GB in the E~0 limit. An interesting situation may emerge in a two doublet version of the triplet majoron model. One can always choose a basis in which the second doublet, ¢2, has vanishing VEV: (~2) = 0. It is possible that A couples in Va mainly to q2 and tara is very small. In the special case when A couples only to ~2 the majoron mass calculated from (17) is zero. O f course, this does not mean that there is a massless scalar without a broken symmetry. The nonzero mM is generated by the loop corrections. Doublet majoron model [ 5 ]. There are two scalars transforming nontrivially under global U~: h + ~ (0; 1, 2) and ~ ' ~ (½; ½, 2); but only (p' may develop a VEV. There are two terms in VB which are of the type (23) Va ~ /Z2~+~O' +2~0 +~0~0+~0' +h.c.

(28)

The majoron mass is again much bigger than Evg: m 2 M (doublet)

~e(Iz~+2v~)vg,'v~.

(29)

Singlet-triplet majoron model [6]. Now there are two scalars with nonzero lepton number and both have electrically neutral components: s ~ (0; 0, 2), X~ ( 1; 1, 2). For phenomenological reasons [ 6 ] the VEV of the singlet must be much bigger than that of the triplet. Thus, the majoron is almost equal to the real part of the singlet: SR- Its mass is again large, 2 2 mM(singlet) ~eVs 1 (/./3 +//4V~o) ,

(30)

due to the following interactions in VB: V B = ]./3S R "3r ]-/4SR~0 +~0 •

3 October 1991

the general case. Let us assume that there is a global symmetry generated by T '~ which is only slightly spontaneously broken: I T~vl 2 << Ivl 2 .

(32)

Then

Mg~ = O ( I T ' ~ v 1 - 2 ) >> I v l - e .

(33)

The first term in//,,~ (see eq. ( 13 ) ) is proportional to [T°'vl 2 but the second only to I T°'vl. If an appropriate first derivative of VB at I2o is of order Ivl 2 then

m2cB=O(elT'~vl-llvl3)>>O(elv]2)

.

(34)

The PGB associated with T '~ may be relatively heavy if two conditions are fulfilled. Firstly, the global symmetry generated by T '~ is only slightly spontaneously broken (32). Second, it is explicitly broken by a term in Ea which couples scalars breaking T ~ to fields with large VEVs. Of course, in a specific model this may be changed if there are some cancellations (as for example in the triplet majoron model with two Higgs doublets). In such a case one has to use eqs. (8), (9),

(13). To summarize we have discussed the pseudoGoldstone boson masses in theories with continuous global symmetries which are spontaneously and slightly explicitly broken. We have found that these masses usually are not of the same order of magnitude as the size of the explicit symmetry breaking. They are much larger if the symmetry is explicitly broken by trilinear terms in the potential and are much smaller if the symmetry is broken by linear terms. The pseudo-Goldstone boson masses may be much increased if the symmetry in question is broken by VEVs which are much smaller than other VEVs in the model. This happens for example in models with spontaneously and explicitly broken lepton number conservation - the majoron mass is much bigger than naively expected. I would like to thank Professor S. Pokorski for very useful discussions and Professor M. Schmidt for carefully reading the manuscript.

(31)

The above discussion about the relation between PGB and non-PGB masses can also be repeated for 79

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[4] G.B. Gelmini and M. RoncadeUi, Phys. Lett. B 99 (1981) 411; H. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1981) 297; L.-F. Li, Y. Liu and L. Wolfenstein, Phys. Lett. B 158 ( 1985 ) 45. [5] S. Bertolini and A. Santamaria, Nucl. Phys. B 310 (1988) 714. [ 6] K. Choi and A. Santamaria, University of California preprint UCSD/PTH 91/01 ( 1991 ). [7] S. Weinberg, Phys. Rev. Lett. 29 (1972) 1698; Phys. Rev. D 7 (1973) 2887; H.Georgi and A. Pais, Phys. Rev. D 12 ( 1975 ) 508. [ 8 ] M. Lusignoli, A. Masiero and M. Roncadelli, Phys. Lett. B 252 (1990) 247. [9] S. Weinberg, Phys. Rev. D 7 (1973) 2887.