Left-right symmetry, KL-KS mass difference and a possible new formula for neutrino mass

Volume 121B, number 5

PHYSICS LETTERS

10 February 1983

LEFT-RIGHT SYMMETRY, KL--K s MASS DIFFERENCE AND A POSSIBLE NEW FORMULA FOR NEUTRINO MASS M.K. PARIDA and C.C. H A Z R A Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India and Post-Graduate Department of Physics, Sambalpur University, Jyoti Vihar, Burla 768017, India Received 19 July 1982

It is shown that a two-step symmetry breaking of SU(2) R in left-right symmetric SU(2) L X SU(2) R × U(1)B_ L yields a Majorana neutrino mass independent of the heavy M R but inversely proportional to the lighter MZR. This fits the low energy phenomenology like the standard model and neutrino masses in the same order as the experimental upper limits without violating the constraint arising out of the KL-K S mass difference even if the symmetry breaking pattern is embedded in GUTs like SO(10) or SU(16).

Since the last few years there have been several attempts to propose theories for finite neutrino mass [1,2] as suggested b y the laboratory limits, mue < 60 eV, mvu < 0.5 MeV, mvr < 250 MeV. Mohapatra and Senjanovic [2] have shown that a spontaneous symmetry breaking o f l e f t - r i g h t symmetry (LRS) [3] in SU(2)L × SU(2)R × U(1)B_ L with minimal Higgs representations gives a Majorana mass to the neutrino of the ith generation

m v i ~ m2 /MR ,

mNi ~ g R ,

(1)

where m i , M R and mvi(mNi ) are the charged lepton, right-handed charged gauge boson and the left (right)handed Majorana neutrino masses, respectively. Constraints imposed b y charged and neutral current data allow a light M R ~ ( 2 - 3 ) M L simultaneously with a light second neutral gauge boson [4,5] consistent with low mass parity restoration [ 6 - 8 ] even if the LRS descends from GUTs like SO(10) [9] or SU(16) [10], provided that Xto = sin20w is allowed to be larger (Xw = 0 . 2 7 - 0 . 2 8 ) than the standard model value [11 ]. Here M L (MZL) and MZR denote the masses o f left-handed charged (neutral) and righthanded neutral gauge bosons, respectively. Such low values o f M R leads to the neutrino mass spectrum in the same order [12] as the laboratory limits .1

mve ~ l O e V ,

mvu ~ lOO k e V ,

rnvr-~ lOOMeV,(2 )

0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

Neutrinoless double beta decay and other lepton flayour changing processes have been suggested to be possible candidates where the effects o f such Majorana neutrinos could be detected [2,14]. But, more recently, it has been shown [15] that manifest LRS (0 L = 0R) and K L - K s mass difference imply a stringent constraint, M R > 1.6 TeV, when spontaneous LRS breaking by minimal Higgs is embedded in SO(10) it turns out that [7] i f M R > 1.6 TeV, t h e n M R = 1 0 4 109 GeV for ×co = 0 . 2 6 - 0 . 2 3 and similar results appear to hold in SU(16) also [8]. Thus the K L - K s mass-difference constraint and manifest LRS, besides reling out low-mass parity restoration, reduces the neutrino masses by at least two orders o f magnitude for the same reasonable choices of Yukawa couplings of Mohapatra and Senjanovic [2]. The neutrino mass falls far below the upper limits b y five to seven orders of magnitude tending to be undetectably small if, in addition, according to the present line o f thinking, the LRS breaking is embedded in GUTs like SO(10) [9] or SU(16) [10] and Xco = 0 . 2 3 - 0 . 2 5 , or even as

4:1 The cosmological bound in the present case can be evaded as done by Roncadelli and Senjanovic [12]. Although another way has been suggested to evade cosmological bounds with larger M R ~- 106 GeV, this involves extra assumptions and other problems as noted by the authors [ 13 ].

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large as 0.26. One way to allow low-mass parity restoration and satisfy the K L - K s mass difference has been suggested by destroying manifest LRS [16], but our approach here is different. The purpose o f the present letter is to show that a two-step breaking o f SU(2)R o f LRS gives mui m2/MZR , with MZR >~ 200 GeV and the values o f M R satisfying the constraints implied by charge and neutral current data and the K L - K S mass difference with manifest LRS even when it is embedded in SO(10) or SU(16). The new formula gives a mass spectrum similar to (2). Our point o f view is that although LRS manifests itself through the relation 0 L = OR, at lower energies, the left- and the right-handed coupling constants are different as a result o f LRS breaking at a mass scale M R ~> 1.6 TeV. At first we obtain the formula without using any GUTs and considering the LRS breaking pattern SU(2)L Z SU(2)R X U(1)B_ L

MR>>ML
> SU(2)R × U(1)R X U(1)B_ L

, SU(2)L × O ( 1 ) y {AR)4:0 (3)

> U(1)e m • ~ 0:# ~0

We use the Higgs representationsXL(I, 0, 0), XR(0 , I, 0), AL(I,0, 2), AR(0, i, 2) and ~(½,1,0), where the quantities inside brackets denote their respective SU(2)L × SU(2)R X U(1)B_ L quantum numbers, possessing the vacuum expectation values,

0)

X 0

(AL) =

(00) VL 0

'

(5)

which is easily satisfied since I3R = B - L = 0 for X R . In the second step, for MZR >~ ML, the condition AI3L ~-- 0 is also satisfied and (5) is satisfied since I3R = +- 1 a n d B - L = - 2 for A R. Unlike the earlier cases [2] where SU(2)R × U(1)B_ z is broken in one step, here eq. (5) relates the breaking of U(1)R and U(1)B_ L local symmetries. X L and X R, because o f their zero 13 quantum numbers, do not contribute to the masses of Z R and Z L. Although XR, XL, A R, A L and ~ all contribute to the W~ and W~ masses and AR, A L and q~to the Z R and Z L masses, the hierarchy in (3) can be achieved by choosing

vx2>)> V2>~ k2 >2>vX2~ v 2 , '


VX

'

(0 00) VR

'

It m a y be noted that in refs. [5] and [17] since only X R is present, but not X L, there is no LRS prior to symmetry breaking. More imporatnt is that the triplets like Axe and A R are absent in ref. [17] and the neu356

trino does not get a Majorana mass term there. Also the model cannot produce n - f i oscillations by the Higgs exchange mechanism o f Marshak and Mohapatra [2]. Also the Higgs contribution to the evolution equations which plays a significant role in deciding the value L = ge(ML)/gR(ML) at low energies has been neglected [17]. In the present work these deficiencies have been removed and there exists LRS before spontaneous symmetry breaking. As a result o f the chain (3) asymmetry is introduced at low energies at least in two different ways: (i) unequal masses o f the left- and right-handed gauge bosons, (ii) unequal coupling constants at low energie s gL (Me) ~ gR (ML). But we assume, as in e arlie r cases [4,15] the attractive feature like manifest LRS [4] to be preserved. We follow the same line of arguments advanced in ref. [2]. In the first step of symmetry breaking at mass scales M R >~ML, AI3L ~ 0 and only SU(2)R is broken but local B - L symmetry remains unbroken. Charge conservation condition gives AXI3R ~ - ½ A ( B - r ) ,

MZR>ML

ML

10 February 1983

(6)

and k '2 ~ k 2 * 2 for simplicity. It is easy to check that

MR~gVXR,

MZR~gVR,

ML~MZL~gk,

(7)

where to compute orders of magnitudes we have assumed for simplicity g e ~ gR ~ g which is justified

+2 It can be shown that with (6), the WL WR mixing angle is ~ kk'/V~( which is already a small quantity and the condition k '2 ~ k 2 is thus needed only for the sake of simplicity.

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according to the data analyses [5,18] and the present work as discussed later. The triplets X L and X R do not contribute to the fermion masses or the Majorana neutrino mass since for them B - L = 0. Only A R and A L with appropriate B - L contribute to the Majorana neutrino mass. Then following Mohapatra and Senjanovic [2] and neglecting intergenerational mixings we have

large. T h e vanishing o f neutrino mass in this case is a result o f the V - A limit o f neutral currents. The neutrino mass does not tend to vanish if only the charge current approaches the V - A limit is as is possible in the present m o d e l Using the general phenomenological bound [18] ,MZR ~ 200 GeV, we have MNi ~> 200 GeV and a similar neutrino mass spectrum as (2). Choosing VL X2 ~ V2 ~ 0 and VX2 >~ V2 and using formulas similar to those developed in ref. [9] we find that the charge and neutral current parameters [19] like the standard model as shown in table 1, for different values oft? R = (k 2 + k ' 2 ) / V 2 and L2 and X~o • For the parameters given in table 1 we get M L = 7 4 - 7 8 GeV, MZL = 8 8 - 9 3 GeV, MZR = 2 5 0 4 0 0 GeV and M R > 1.5 TeV. It may be pointed out that independent of arguments regarding the K L - K S mass difference as advanced here, the formulas in (10) hold true whenever the present Higgs representations are taken and condition (6) is satisfied. We now show that the symmetry breaking pattern (3) can be embedded in SO(10) yielding low MZR ( ~ M L ) , h i g h M R > 1 TeV for several low energy values o f L 2 and Xto. Consistent with LRS and twostep SU(2)R breaking, the simplest chain is

.1~ mass = ~. [hsi(VLvTCvi -- VRNTCNi) +hlik] +h.c. '

(8)

mui,~,(h5i3"+¼h2i/h5i)k2/VR,

mNi=h5iVR,

(9)

where the hi's are Yukawa couplings. We have checked that the minimisation of the left-right symmetric potential in this case yields four different constraints, two of which include the condition VL = 7 k 2 / V R which has been used in (9). We also get another condition Vx = 3"xk2/V x , where the parameter 3'x, like 7 is a ratio of Higgs self couplings. The necessity of a light Higgs doublet also arises in a similar way [2]. With the same reasonable assumption as h l i ~ h5i h i and 3' ~ hli/h5i 2 2 and using (7) in (9) we have

mvi ~, aFlm2/MZR ,

mNi ~ a i g z R ,

10 February 1983

(10)

where a i = hi/g ~ O(1). The small v i - N i mixing angle

ei ~ - mi/Mz R" Thus, according to the present model, the neutrino can have a finite non-vanishing mass even if M R is very large so long as MZR is not too

Table 1 Calculated values of neutral current parameters for v X 2 / v ~ > 102 and VX = VL = k '= 0 as defined in the text. WSG refers to the standard model results and KLLWrefers to the results of ref. [19]. r~R

L2

×co

eL(u)

eR(U)

eL(d)

eR(d)

gv

gA

1.15C1(d)Cl(U) + Cl (u) + 0.5C 1(d)

C2(H) - 0.5C2(d)

0.1

1.9 1.6 1.21

0.23 0.245 0.27

0.349 0.34 0.325

-0.15 -0.16 -0.179

-0.437 -0.432 -0.424

0.062 0.067 0.074

-0.023 0.002 0.048

-0.5 -0.5 -0.5

0.205 0.213 0.227

-0.341 0.383 -0.298

0.015 -0.013 0.058

0.2

1.9 1.6 1.21

0.23 0.245 0.27

0.352 0.343 0.33

-0.147 -0.155 -0.169

-0.451 -0.447 -0.439

0.049 0.053 0.059

-0.006 0.017 0.058

-0.5 -0.5 -0.5

0.205 0.214 0.228

-0.366 -0.341 -0.3

-0.0'6 -0.015 0.06

0.347 0.351 +,0.034

-0.153 -0.18 ± 0.02

-0.424 0.415 +,0.046

0.076 -0.011 +,0.046

-0.04 0.043 -+0.063

-0.5 0.243 -0.545 0.24 +,0.056 +-0.068

-0.384 -0.45 +,0.12

0.06 0.23 -+0.38

WSG KLLW

0.23

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Volume 121B, number 5

PHYSICS LETTERS

MG SO(10) ~ SU(2)L X SU(2)R X U(1)B_ r X SU(3)C

mR , SU(2)L X U(1)R X U(1)B_ L X SU(3)C MZR , SU(2)L X U(1)y X SU(3)C ML , U(1)e m X SU(3)C,

(11)

having one step more than the case considered in ref. [7] as we put MZR ~ML- In the first step SO(10) may be broken by a {45} of Higgs. The Higges XL, X R C (45} and AL, AR, ~ C {126} of SO(10). Retaining their contributions, the evolution equations [20] reduce to the following three simultaneous equations 1/a s = 3/8a - (11/8rr)(x + 30y/11),

(12a)

X~o = 3 _ 1 la(3x + 26y/11),

(t2b)

1 = L 2 - 1 lax/(37rxco) ,

(12c)

where x = In (MR/ML) and y = In (MG/ML). A n important feature o f this chain which has not been observed earlier is that the unification constraints besides restricting allowed values o f M G, M R and X~o , f r o m above and below also give upper and lower bounds o f L 2 . The low energy ratio L 2 is not such a free parameter here as in ref. [17] since there are three equations for three variables, x, y and L 2 . Solving these equations we obtained allowed ranges of ×w, L 2 , the grand unification mass M G and M R for the strong coupling constant a s = 0.12 and the fine structure constant a = 1/128.5 as shown in table 2. Thus the unification constraints allow the values 0.23 ~<×w ~<0.28 corresponding to 1.9 ~>L 2/> 1.07.

Table 2 Allowed values of mass scales, L2 and Xto with SO(10) unification constraints considered in the text. Xto

L2

MR (GeV)

MG (GeV)

0.23 0.24 0.245 0.25 0.26 0.27 0.28

1.9 1.71 1.61 1.53 1.36 1.21 1.07

1012 101° 2x 109 2x 108 3 x 106 6× 104 103

1013 5× 1013 1014 2x 1014 1015 5x 1015 2 x 1016

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10 February 1983

It may be noted that there is only one allowed value o f L 2 for each value of ×to- We have checked that in increasing the member of steps in (3) by one more, allows many values of×to for each value of L 2. N o t e that this chain does not allow L 2 = +1.0. In the allowed range the proton life time varies between 1023_ 1036 yr. For the stability of the proton it is necessary that X~o > 0.245 (MG/> 1014 GeV). Similar results can be obtained if we replace SO(10) in (11) by SU(16) remembering that XL, X R C 255 and AL, AR, ¢ C 136 of SU(16). Various other chains of symmetry breaking in SO(10) and SU(16) embedding two steps of symmetry breaking of SU(2)R would allow different regions o f L 2 and intermediate and grand unification mass scales. Details of these investigations will be reported elsewhere [21 ]. According to this model parity restoration, if any, would be possible only in the TeV and higher mass-scale region. We now calculate the K L - K S mass difference by using two different coupling constants gL and gR but by using 0 L = 0 R = 0C where 0 C is the Cabbibo angle. With the values of masses and other parameters chosen in ref. [15] we have in the four-quark model Am K ~ 0.32 X 10-14(1 - 430fl/L 2) GeV, where [3 = M L2/ M R 2 which gives the lower bound M R/> 1.6 TeV/L ,

(13)

showing that the bound depends upon the low energy ratio L of the coupling constants. It is clear from table 2 that SO(10) [and similarly SU(16)] and manifest LRS can easily satisfy (13) forL 2 > 1.07 and Xto < 0.28. Since now M R > 1.6 TeV the r/parameter in neutrinoless double beta decay [2,14] is reduced by at least three orders of magnitude for ruNe 200 GeV and the branching ratios for/2 ~ e + 7 and ju- + A(Z) ~ e - + A(Z) each by at least two orders of magnitude and that for/2- + A(Z) -~ e - + A(z - 2 ) by at least one order as compared to the estimation made in ref. [ 14]. This would make it extremely difficult and in some cases impossible to detect the effect of Majorana neutrinos by these processes. Even then substantial lepton number violation might be observed in processes by a possible dominant Higgs exchange contribution of Mohapatra and Vergados [22]. In view of these the only possible ways of detecting the effects of Majorana neutrinos appear to be through neutral current interactions [23].

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One o f us (M.K.P.) expresses his gratitute to Professor J.C. Pati for constant e n c o u r a g e m e n t and thanks him and Professor R.N. Mohapatra and Dr. U. Sarkar for useful discussions. The authors are grateful to Professor T. Pradhan for providing facilities at the Institute o f Physics where this w o r k was c o m p l e t e d .

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[8] A. Raychaudhuri and U. Sarkar, Phys. Rev. D, to be published. [9] H. Georgi, in: Particles and fields 1974; Proc. Williamsburg meeting of the Division of particles and fields of the APS, ed. C.E. Carlson (AIP, New York, 1975); H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193. [10] J.C. Pati, A. Salam and J. Strathdee, Nuovo Cimento 264 (1975) 77; Nucl. Phys. B185 (1981) 445. [ 11 ] S.L. Glashow, Nucl. Phys. 22 ( 1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle theory. Relativistic groups and analyticity (Nobel Symposium No. 8) ed. N. Swartholm (Wiley, New York, 1968). [12] M. RoncadeUi and G. Senjanovic, Max Planck Institute preprint, MPI-PAE/PTh 16/81. [13] A. Masiero and M. Roncadelli, Phys. Rev. D25 (1982) 2612. [14] Riazuddin, R.E. Marshak and R.N. Mohapatra, Phys. Rev. D24 (1981) 1310. [15] B. BeaU, M. Bander and A. Soni, Phys. Rev. Lett. 48 (1982) 848. [ 16] A. Dutta and A. Raychaudhuri, Calcutta University preprints CUPP/82-7 and CUPP/82-8. [17] S. Rajpoot, Phys. Lett. 108B (1982) 303. [18] V. Barger, E. Ma and K. Whisnant, Phys. Rev. Lett. 48 (1982) 1589. [19] J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. [20] H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451; T. Appelquist and J. Carrazone, Phys. Rev. D11 (1975) 2856. [21] M.K. Parida and C.C. Hazra, in preparation. [22] B. Kayser and R.E. Shrock, Phys. Lett. l12B (1982) 137. [23] S.P. Rosen, Phys. Rev. Lett. 48B (1982) 842.

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