The Seesaw ehanis By obert. Klauber www.quantufieldtheory.info 1 Bakground It ay see unusual to have suh low values for asses of neutrinos, when all other partiles like eletrons, quarks, et are uh heavier, with their asses relatively losely grouped. Given that partiles get ass via the Higgs ehanis, why, for exaple, should the eletron neutrino be 10 5 ties or ore lighter than the eletron, up and down quarks. That is, why would the oupling to the Higgs field be so any orders of agnitude less? One ight not be too surprised if the Higgs oupling were zero, giving rise to zero ass. One ight likewise not be too surprised if the oupling resulted in asses on the order of the Higgs, or even the GUT, syetry breaking sale. Consider the quite reasonable possibility that after syetry breaking, two types of neutrino exist, with one having zero ass (no Higgs oupling) and the other having (large) ass of the syetry breaking sale. As we will see, it turns out that reasonable superpositions of these fields an result in light neutrinos (like those observed) and a very heavy neutrino (of syetry breaking sale, and unobserved). Fundaental ath Conept Underlying the Seesaw ehanis Consider a real, two diensional spae with a atrix (tensor) expressed in one set of orthonoral basis vetors (pried) for that spae as 0 0 ɶ = 0 100. (1) Now, if we onsider a new set of basis vetors, rotated by an angle φ fro the original basis, then the atrix oponents hange, of ourse, and an be found by osφ sinφ 0 0 osφ sinφ = sinφ osφ 0 100 sinφ osφ 100sin φ 100 osφ sinφ =. 100 osφ sinφ 100 os φ Note how this atrix looks if φ is sall, say φ =, with osφ =.99939 and sin φ =.03490..1 3.488 = 3.488 99.878, (3) and we get the upper left diagonal ter alost 3 orders of agnitude saller than the lower right ter, whih is approxiately the sae as the original suh ter. The off diagonal ters equal the geoetri ean of the diagonal ters, i.e., ( 100 os φ )( 100 sin ) upper left ter, but signifiantly saller that the lower right one. () φ, and are not as sall as the The fundaental point is that by starting with a atrix of for like (1), and transforing to another basis, whih is rotated by a sall angle fro the original, we get a atrix of for like (3).
3 ira vs ajorana ass Ters in the agrangian We don t know a great deal, experientally, about neutrino ass, but on general theoretial grounds, two distint lasses of neutrino ass ters are allowed in the agrangian of eletroweak interations. These are alled ira and ajorana ass ters. Note that ajorana ass ters have nothing to do with the ajorana representation in spinor spae. One an use any representation for the fields of whih ajorana and ira ass ters are oposed. Neither do ajorana ass ters iply the assoiated partiles/fields are ajorana ferions, of whih you ay have heard. ajorana ferions are their own anti-partiles. ore on this in Set. 5. For now, we will assue that both ira and ajorana ass ters ontain only ira type partiles (in any representation we like.) The ira ass ters, whih are the usual ters dealt with in introdutory quantu field theory (QFT), have for ( ) +, (4) and ajorana ass ters, whih ay look unfailiar to the uninitiated, have for ( ) ( ) 1 1 + +, (5) where sub/supersripts and designate left or right hand hirality, and the supersript represents harge onjugation. That is, destroys a H hiral neutrino and reates a H antineutrino, reates and destroys, reates and destroys (does sae as ), destroys and reates (does sae as ), and for subsript, interhange everywhere above. Note that the subsript always refers to partiles. For a non onjugated field, no overbar eans destroys partiles, overbar eans reates partiles, and antipartile ations for the sae field are just reversed fro partile ations (partile antipartile, H H, destroy reate). Charge onjugating a field has the sae effet on partile/antipartile and reation/destrution as an overbar (overbar is effetively a oplex onjugate transpose [plus a γ 0 ultipliation]). That is, the overbar and the supersript have the sae effet. The harge onjugation erely lets us have the overbar (row) operator effet in a non overbar (olun) vetor. In fat, the sybol is used by soe for the ter of (5), with siilar hanges for other ters, where one ust keep in ind for suh notation that inner produt in spinor spae is iplied, even though there is no obvious transpose ter (row vetor on left) in. Note that the first ter in (4) destroys a H partile and reates a H one. The Feynan diagra for this ter shows a H partile disappearing at a point and a H partile appearing. Thus weak (hiral) harge is not onserved, as a H neutrino has +1/ weak harge and a H neutrino has zero weak harge. epton nuber, however, is onserved, as we started with a neutrino (not an anti-neutrino) and ended up with a neutrino.
3 Soewhat siilarly, the first ter in (5) reates two H neutrinos out of the vauu and thus also does not onserve weak harge. But, iportantly, it does not onserve lepton nuber (whih the ira ters do.) We started with zero neutrinos and ended up with two neutrinos. Wholeness Chart 1. Weak Charge and epton Nuber Conservation ira ass ters ajorana ass ters Conserves weak harge? No No Conserves lepton nuber? Yes No atheatially, harge onjugation of the field, where C is the harge onjugation operator, an be expressed as * T T = C = iγ = C = iγ, (6) whih needs soe study in spinor spae to fully understand, but doing so would lead us astray fro the task at hand. With all this in ind, we an then express (4) and (5) in ters of a ass atrix as (where h.. eans heritian onjugate of the prior ter) with 1 ass = ( ) + h.. (7) ters =. (8) (As we will see, the atrix in (8) is the neutrino spae analog of the atrix in (3) of Setion.) Heritian onjugates of fields are as follows h.. h.. h.. h.., so (7) beoes (taking the oplex onjugate transpose of the first ter in (7) for the seond) 1 1 ass = ( ) ( ), (9) ters whih, for our identifiation of the effets of v and (both destroy H partiles and reate H antipartiles), and, and, and, yields (4) plus (5). 4 See-sawing Suppose, as suggested earlier, that Higg s or GUT syetry breaking only gave ajorana ass to neutrinos. That is, oupling to the Higgs (or Higges) was not done in a way that led to ira ass ters. So, the ass atrix would be diagonal, unlike (8), of for
and our agrangian ass ters would look like 4 0 ɶ = 0, (10) 1 ass = ( N ) ɶ + h.., (11) ters N where we have represented the fields diretly oupled to the Higgs by ( N) T. In other words, and N are the ass eigenstates for our neutrinos. On the other hand, the weak eigenstates and (and their onjugates) of (7), whih are linear superpositions of and N, interat diretly via the weak fore, and represent what we detet in weak interation experients (ignoring in this ontext the fat that has zero weak harge and does not so interat.) Finding (10) fro (8) is just an eigenvalue proble, with and the eigenvalues. That is, we ould think of our fields in two different, but essentially equivalent, ways: 1) a ix of ajorana and ira ass ters with the olun vetor of fields in (7), or ) pure ajorana ass ters assoiated with the ass atrix of (10), whose assoiated fields are represented by the different olun vetor ( N) T. Heuristially, finding ( N) T fro ( ) T an be thought of as rotating our basis vetors in an abstrat spae until we find an alignent giving the fields vetor the oponents ( N) T. Assuing that is the ase in the real world (we have no way of knowing via experients to date), what would the ass atrix (10) look like in order to give us the kind of asses (either or perhaps ) that we see? eeber we are looking for a reason why neutrino ass is so uh lower than that of other partiles. That reason posits that the field oponents of the vetor in (11) are the ones diretly oupled to the Higgs field. It works best if the ass = 0, as that eans there is no Higgs oupling for the field, but there is suh oupling for the N. (And (10) then beoes the analog of (1) in Setion.) Note that if we took 0, but <<, we would still be left with our original proble, whih is why is one ass so uh saller than the others?. Having zero ass is easier to explain (no oupling) than extreely low ass (extreely sall oupling.) 4.1 The Shortut Analysis Given the treatent of Setion, we an iediately draw onlusions about the agnitudes of the four oponents of (8), given (10) with the upper left oponent equal to zero and the ( N) T basis being lose to the ( ) T basis. That is, we have the ass hierarhy we need, >> > 0, (1) where the ira ass is the geoetri ean of the left and right ajorana asses, the diagonal oponents of (8). That is, =. (13) Note that for given value of, a higher value for vie versa. This is the reason for the nae see-saw ehanis. eans a lower the value for and
4. The Foral Eigenvalue Analysis 5 The siple dedue by analogy ethod of the prior setion allows us to see, relatively easily, the essene of the see-saw ehanis. But to fully quantify it, we need the following ore rigorous analysis. The harateristi equation for the eigenvalue proble solution of (8) is with eigenvalues, ( )( ) ( ) λ λ 0 =, (14) ( ) ( ) 4( ) λ 1 1 1, = + ± +. (15) For λ 1 = = 0, we ust have the inus sign in (15) and whih, not surprisingly, is the sae as (13). =, (16) Then, we would have, with the plus sign in (15) for λ =, λ = = 0 1 λ = = + We ll work out the eigenvetor N (i.e., for λ ) expressed in the ( ) T basis and leave the sipler ase eigenvetor (i.e., for λ 1 ) for the reader. Fro the eigenvalue proble for (7) and (8), with the eigenvalue λ of (17) we get the two equations This yields and an eigenvetor ( ( )) ( ) + + = 0 ( ) + + = 0.. (17) (18) =, (19) N =. (0) Soe are is needed to note that the top oponent here is really the field with the frational fator indiating the size of the field opared to the field. That is, N is really a superposition of the two fields, suh that if has a oeffiient of one in that superposition, then the field has a oeffiient of /. In other words, in (18), the sybol really stands for the oeffiient (effetively, the agnitude) of the field, not the field itself (whih the loation in the olun vetor denotes.)
6 Note also that, up to here, we have ignored the Heritian onjugate half of (7), whih we will have to inlude. So our true N will also inlude that, and is, in ters of the fields theselves, rather than as a two oponent vetor, expressed as Siilarly, the other eigenvetor is found to be If we now assue (to be justified below) N = ( + ) + ( + ). (1) = ( + ) ( + ). () >>, (3) then N is oposed alost entirely of (and its siilar sibling ), fro (17) and (3) is very heavy, and is thus effetively sterile. Conversely, an be thought of as oposed alost entirely of N. Siilarly, is oposed alost entirely of (and ), and onversely, is alost entirely oposed of the weightless. Fro (16), one sees that for a given value of, a higher value for eans a lower the value for, and vie versa, and thus, the nae see-saw ehanis. Further, fro (1) and (), the higher the value for, the ore N and. Approahed in a different way, given and, will be the geoetri ean of those two asses, and will generally be loser to the lower of the two. (If = 100 and = 1, then = 10.) Further, if (3) holds, fro (16), we have and fro (17), Thus, the ass hierarhy appears naturally as The 0 (but not 0), (4). (5) >> > 0. (6) These results ath those of the sipler approah of Setion 4.1. >> Assuption An astute reader, who hadn t read Setions and 4.1, ight question if we have gained anything. We originally sought a reason why the known ira ass is so sall opared to other asses. We got that via the eigenvalues analysis above, but in the proess, we had to ake another, seeingly arbitrary, assuption (3). With this assuption, we appear erely to be substituting one ass hierarhy proble for another. That is, we now have to ask why turns out to be so uh saller than. The answer is this. If we start with the ass atrix (10) with one field having zero ass (unoupled to Higgs partile(s)),
7 0 0 ɶ = 0, (7) and do a slight rotation in the spae of ( N) T, we end up with a atrix like (8) with the harateristi (3), whih served as our initial assuption, but whih is justified if we started with (7). Our assuption boils down siply to assuing a sall transforation. 5 istintion between ajorana ass Ters, Partiles, and epresentation The adjetive ajorana is applied to three distintly different things, whih we need to distinguish between. The first use ost people see of this ter is for one of three representations of ira atries and spinors. The three representations are ira-pauli (the Standard ep), Weyl, and ajorana. As noted at the beginning, this use of ajorana has nothing to do with the ajorana ass ters of this artile. Everything in this artile an be done in any one of the three representations. Herein, we so far have been dealing with the seond use of the ter regarding ajorana vs. ira type ass ters in the agrangian, i.e., (4) and (5). The neutrinos and ira atries in these ters an be represented by any one of the three representations above. The third use of the ter refers to type of neutrino. A ajorana partile is defined as a partile that is its own antipartile. A ira partile, on the other hand, has an antipartile that is distintly different fro it. Typially, in alost all of one s study of QFT, one deals with ira type partiles. Neutrinos are the only partiles that an be either ira or ajorana types. All other ferions are known, fro experient, to be ira ferions. No experients to date (Jan 01) have been able to deterine if neutrinos are ajorana or ira partiles. ouble beta deay experients ay one day be able to do this. As an aside, ajorana partiles are easiest to handle atheatially in the ajorana representation. Note that the neutrinos we deal with in our ass ters an be either ira or ajorana neutrinos, but both type ass ters would need to involve the sae partile type. Fro (4) and (5), we see that the partiles in eah type ter are represented by the sae sybols, i.e., they represent the sae partile type (ira or ajorana) in both type ass ters (ira and ajorana). In suary, we an have ajorana representation in spinor spae (it or one of other reps an be used for any of below) ajorana vs ira ass ters in agrangian (both together an be used with either partile type below) ajorana vs ira type partiles (ajorana is its own antipartile) 6 Coents on epton Nuber Conservation With regard to ajorana vs. ira type ass ters in the agrangian, we saw (see Wholeness Chart 1, pg. 3) how both types of ters do not onserve weak harge. We also saw that the ajorana ass ters lead to non-onservation of lepton nuber, whereas the ira ass ters lead to onservation of lepton nuber. These results were speifially for ira neutrinos in both types of ass ter, where ira neutrinos have a lepton nuber +1, and ira antineutrinos have a lepton nuber of 1.
8 However, what if the neutrinos we are dealing with in experient are atually ajorana neutrinos? Then neutrinos and anti-neutrinos would have the sae lepton nuber, sine they are the sae partile. But this nuber would have to be its own negative, sine quantu nubers for antipartiles have opposite sign of those for partiles. Zero is the only nuber that works, so we ould onlude that ajorana partiles have lepton nuber zero. Therefore, for ajorana neutrinos in both types of ass ters, all interations solely fro ass ters of either for will result in no hange of lepton nuber. So, if we are dealing with ajorana neutrinos, the No we have in the last row, last olun of Wholeness Chart 1 will hange to a Yes. Prior to this, we had been assuing we were working with ira neutrinos. However, onsider a typial interation suh as n p + e + = for ajorana neutrino (8) where what we usually onsider a ira anti-neutrino with lepton nuber 1, is now a ajorana neutrino with lepton nuber 0. Thus, we started with a neutron having zero lepton nuber, but end up with produts having a net +1 lepton nuber (fro the eletron in (8)). We onlude that even though ajorana neutrinos in the agrangian ass ters (both ira and ajorana ass ters) will not lead to lepton nuber violation, interations of ajorana neutrinos will. Thus, we will have lepton nuber non-onservation for i) ira neutrinos if, and only if, ajorana ass ters exist in the agrangian or ii) ajorana ferions regardless of what ass ters are in the agrangian. 7 Possible Physial Senarios There are three possible senarios, assuing both neutrino types exist. Possibilities for both ira and ajorana neutrinos existing in nature 1) ira and ajorana ferions both interat weakly, and what we see in experients is a blend of both. (Not onsidered likely by ost.) ) Only ira neutrinos interat weakly, and we don t ever see ajorana neutrinos in any experients. 3) Only ajorana neutrinos interat weakly, and we don t ever see ira neutrinos in any experients. Possibilities if only one type exists in nature 4) ira neutrinos exist, but no ajorana ones. 5) ajorana neutrinos exist, but no ira ones. If the See-Saw ehanis is True If the see-saw ehanis exists, then we have both type ass ters of (4) and (5), and with >> > 0, (9) and for whih we ould have, in one senario, ira neutrinos represented by and if only ira neutrinos exist. Alternatively, we ould instead have ajorana neutrinos represented by those sybols. In either ase, our interation ters would inlude the sybols and, along with interediate vetor boson fields.
9 For uh larger than, the ass ter would not play a role in the theory at energy levels of the present day. So we would effetively see neutrinos, be they ira or ajorana neutrinos, as having ass, i.e., as having ass of the ajorana ass ters in. 8 Suary of See-Saw ehanis See-saw ehanistheory The (oon textbook) treatent overed in Setion 4. began with a general, non-diagonal ass atrix, looked at finding the ass eigenvalues of that atrix, and exained the relationships engendered between the asses. However, looking at it soewhat in reverse, as in Setion 4.1, an be helpful pedagogially. That is, start with the ass eigenstates fields and N, the ones oupled diretly to the Higgs field (with having zero oupling), and the diagonal ass atrix (7). The weak eigenstates fields and (and their harge onjugation fields) are superpositions of the and N fields. We then ask If we transfor ( N) T into the ( ) T, what would the transfored ass atrix look like? Well, if nature has hosen to ake this a slight transforation (a sall rotation in the spae of the fields), whih is reasonable, then we would get a ass atrix with a very sall upper left diagonal ter, a very large lower right diagonal ter, and off diagonal ters whih are eah the geoetri ean of the diagonal ones, as in (16). We would have a see-saw relation between the asses, and ould readily have neutrino asses of the order observed. For a greater rotation in fields spae, the greater would be the see-saw effet (bigger and lower ), and also the greater the value of. The neutrinos we see in experients ould be either ira or ajorana types, though either type would have both fors for ajorana and ira ass ters in the agrangian. - For further pedagogi explanations of topis in quantu field theory by the sae author, see www.quantufieldtheory.info.