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Order-ħ finite corrections to a supersymmetric mass formula
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
ORDER-/i FINITE CORRECTIONS TO A S U P E R S Y M M E T R I C M A S S FORMULA
F. F E R U G L I O 1 International School for Advanced Studies, 1-34100 Trieste, Italy
and J.A. HELAYEL-NETO International Centre for Theoretical P,~v'sics, 1-34100 Trieste, Italy Received 18 December 1985
Finite one-loop corrections to the mass formula Str M 2 = 0 are derived for the broken non-symmetric phase of the O'Raifeartaigh model with the help of supergraph methods extended to the case of broken global supersymmetries.
In supersymmetric field theories with spontaneous breaking of supersymmetry there are, for a fairly large class of renormalizable models, mass sum rules relating the masses of the bosonic (mB) and fermionic (mF) states, which take the form [ 1] StrM 2= ~ states
mi-
~
m2=0.
(1)
states
This mass relation imposes rigid constraints in the construction of realistic models for fundamental interactions based on the fermion-boson symmetry; however, for local supersymmetry and explicitly softly broken global supersymmetry, its generalizations are phenomenologically acceptable [2,3]. At the classical level, the above supertrace mass formula can be derived either by direct inspection of the component-field mass matrices or by taking advantage of a superfield formalism. In the latter approach, by performing a superfield one-loop calculation of the effective potential, all one has to do is to isolate its quadratically divergent part, since its coefficient is nothing but the tree-level value of Str M 2. In the framework of perturbation theory, Str M 2 gets calculable finite corrections in terms of the eventually renormalized parameters of the model under consideration. Some of its quantum properties have been firstly investigated in a work by Girardello and Iliopoulos [4], whose analysis has not taken into account all orders of the supersymmetry breaking parameter. More recently, Girardello, Pernici and Zanon [5] have calculated the oneloop corrections to this mass formula including, however, all orders in the breaking parameters and for a class of models which should exhaust the possible ways of breaking global supersymmetry in renormalizable theories. In their analysis of the O'Raifeartaigh model [6], the authors of ref. [5] have considered the situation in which all physical fields have vanishing tree-level vacuum expectation values (the so-called symmetric phase). However, there is still the possibility of having a non-symmetric phase, for which not only the auxiliary component, but also the physical ones acquire non-zero vacuum expectation values at the classical level. The latter has been considered in ref. [7] for establishing the extension of supergraph techniques [8] to include the cases where global superDipartimento di Fisica dell'Universith and Istituto Nazionale di Fisica Nucleare, Sezione di Padova. 1-35100 Padua, Italy.
228
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
symmetry is spontaneously broken, in a way to account for all orders in the breaking parameters, for a given fixed order in a loopwise expansion. In possess of the self-energy functions for the O'Raifeartaigh model in its broken non-symmetric phase [7], we would like to f'md out the way the tree-level vacuum expectation values of the physical scalars modify the Str M 2 formula derived in ref. [5]. Referring again to ref. [7], it would be worthwhile to recall the quantities in terms of which our final expressions will be given: (i) x is the vacuum expectation value of the physical component relevant for our analysis; for the broken nonsymmetric phase, it satisfies the equation m 2 + 2g~ + 2g2x 2 = 0,
(2)
where ~, m and g are the parameters of the O'Ralfeartaigh model [6] and 2g~ < - m 2 is the relevant region for the phase we are treating. (il) A is the relevant vacuum expectation value for the set of auxiliary fields, A = -m2/2g.
(3)
Once the tree-approximation vacuum expectation values are set, the superfields of the model are all shifted and the modified super-Feynman rules are derived which account for all orders in x and A. By using them, one can compute the order - f i contributions to the effective action (see section 4 of ref. [7] ) and therefore derive the one-loop corrected equations of motion for the auxiliary fields of the theory, which is an essential step for the computation of the physical-field mass matrices. It is worthwhile remarking that the elimination of the auxiliary fields through their classical equations of motion does n o t account for all possible contributions of O(h) to the physical masses; one has indeed to use the equations of motion for the auxiliaries by including one-loop corrections. Having obtained these equations, one replaces the auxiliary fields in the component-field effective action by using them, retain only the terms which are of O(h) and finally the poles of the one-loop corrected connected two-point functions can be read off, yielding the bosonic and fermionic mass matrices. Three facts should be observed here: (i) the one-loop corrected fermionic mass matrix has a zero eigenvalue, as it should be, since the goldstino remains massless to all orders of perturbation theory. (ii) The scalar field which at the tree-level is massless, and this is a common feature of spontaneously broken models with only chiral superfields [9], acquires a finite mass given by m 2 = [ m 2 / ( m 2 + 4g2x2)] IC(p 2 = 0)l,
(4)
where C(0) is a momentum-space one-loop integral defined by C(0)
4 2 6 4 f dak k 2 1 1 = -- g d(~)4 (k 2 + / 2 ) 2 (k 2 + ~2)2 _ 64 '
(5)
with 62 = 2gA and/12 = m 2 + 492x 2 . This result is a manifestation of the mechanism of spontaneous mass generation in broken supersymmetric models first pointed out by Bardeen, Piguet and Sibold in ref. [10]. Cli0 Finally, our explicit computations show that the infinities introduced into the one4oop corrected Str M 2 exactly cancel each other without even the need o'f performing wave-function renormalizations, so that Str M 2 turns out to be automatically Finite. This should not be regarded as a new conclusion coming from our calculations, but should rather be seen as a good check of our superspace computations, since, as already mentioned in ref. [5], Str M 2 should be automatically finite to all orders, as only wave-function renormalizations are required in the model [ 11]. Finally, having obtained all physical masses as poles of the one-loop corrected connected two-point functions, 229
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
we can present our result for the O(h)-corrections to the mass formula of eq. (1): Str M 2 = (g2/47r2)m2 one -loop X { - 2 + [(1 + 2;)2(1 + 1/~) log(1 + ~ ) - 2X 2 logl~l + (1 - ~)2(1 - 1/~) log(1 - Z ) ] )
+ (g4/4n2)x2 [-x/~zr + 8~ log(1 + E) + 4(3 - 2 ~ ) log(1 - E) + 8(1 - ~ ) I ( - 1 - 2;, 2; 1 , - 1 - ~ , 1) - 8(1 - 1!3)I(- 1 + ~ , 2; 1 , - 1 + N, 1)
- 4(2-I(-1
~ ) I ( - 1 - ~ , 2 ; 1 + Z , - 1 - X, 1) + 4 ( 2 +5-:,,2- 2~; 1 +N,-1
+{I(-1 - Z,2-
~ ) I ( - 1 + Y~, 2; 1 + ~ , - 1
+ ~, 1 - g)+ 3I(-1-
2Z;1 +Z,-1
- Z, 1 - Z ) -
23,2+2~; 1- g,-1
sI(-1 + Z,2+2Z;1-
+ E, 1) - Y,, 1 + 1!3)
Z , - 1 +~3, 1 + Z ) ] ,
(6)
where ~ = 6 2//a2 = 2gA/(m 2 + 4g2x 2) is such that IZl < 1 for the non-symmetric phase, and
1
aI t + a 2
aI
I(al, a2; b 1 , b2, b3) = f dt log 0 blt2 + b2t + b 3 2bl
bI + b2 + b3
b2
b3
2b 1
0j
dt bl t2 + b2t 1 + b3
(7)
The reason we have left the integrals l(al, a2; b l , b2, b3) in the corrected mass formula is that they may assume different analytic forms in the domain of validity o f the parameter ~ . Moreover, their replacement by the possible analytic forms they take does not bring any substantial simplification. To conclude, we remark that the additional contributions with respect to the result o f ref. [5] is just proportional to a factor x 2 and we notice that in the case x = 0 our expression reproduces the results of Girardello et al., if we follow their prescription of taking the real part o f the logarithm. We would like to express our gratitude to Franco Legovini and Luciano Girardello for discussions and suggestions. J.A.H.-N. would also like to acknowledge Professor Abdus Salam, the IAEA and UNESCO for hospitality at ICTP.
References [1 ] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [2] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150; N. Sakai, Z. Phys. C l l (1981) 153. [3] R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. l19B (1982) 343; J. Polchinsky and L. Susskind, Phys. Rev. D26 (1982) 3661. [4] L. GirardeUo and J. Iliopoulos, Phys. Lett. 88B (1979) 85. [5] L. Girardello, M. Pernici and D. Zanon, Phys. Rev. D29 (1984) 318. [6] L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. [7] F. Feruglio, J.A. Helay~l-Neto and F. Legovini, Nucl. Phys. B249 (1985) 533. [8] M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429. [9] B. Zumino, Spontaneous breaking of supersymmetry, in: Proc. Heisenberg Symp. on Unified theories of elementary particles, Lecture Notes in Physics, Vol. 160 (Springer, Berlin, 1982) p. 137. [10] W.A. Bardeen, O. Piguet and K. Sibold, Phys. Lett. 72B (1977) 231. [11] J. Iliopoulos and B. Zurnino, Nucl. Phys. B76 (1974) 310.
230
PHYSICS LETTERS
27 March 1986
ORDER-/i FINITE CORRECTIONS TO A S U P E R S Y M M E T R I C M A S S FORMULA
F. F E R U G L I O 1 International School for Advanced Studies, 1-34100 Trieste, Italy
and J.A. HELAYEL-NETO International Centre for Theoretical P,~v'sics, 1-34100 Trieste, Italy Received 18 December 1985
Finite one-loop corrections to the mass formula Str M 2 = 0 are derived for the broken non-symmetric phase of the O'Raifeartaigh model with the help of supergraph methods extended to the case of broken global supersymmetries.
In supersymmetric field theories with spontaneous breaking of supersymmetry there are, for a fairly large class of renormalizable models, mass sum rules relating the masses of the bosonic (mB) and fermionic (mF) states, which take the form [ 1] StrM 2= ~ states
mi-
~
m2=0.
(1)
states
This mass relation imposes rigid constraints in the construction of realistic models for fundamental interactions based on the fermion-boson symmetry; however, for local supersymmetry and explicitly softly broken global supersymmetry, its generalizations are phenomenologically acceptable [2,3]. At the classical level, the above supertrace mass formula can be derived either by direct inspection of the component-field mass matrices or by taking advantage of a superfield formalism. In the latter approach, by performing a superfield one-loop calculation of the effective potential, all one has to do is to isolate its quadratically divergent part, since its coefficient is nothing but the tree-level value of Str M 2. In the framework of perturbation theory, Str M 2 gets calculable finite corrections in terms of the eventually renormalized parameters of the model under consideration. Some of its quantum properties have been firstly investigated in a work by Girardello and Iliopoulos [4], whose analysis has not taken into account all orders of the supersymmetry breaking parameter. More recently, Girardello, Pernici and Zanon [5] have calculated the oneloop corrections to this mass formula including, however, all orders in the breaking parameters and for a class of models which should exhaust the possible ways of breaking global supersymmetry in renormalizable theories. In their analysis of the O'Raifeartaigh model [6], the authors of ref. [5] have considered the situation in which all physical fields have vanishing tree-level vacuum expectation values (the so-called symmetric phase). However, there is still the possibility of having a non-symmetric phase, for which not only the auxiliary component, but also the physical ones acquire non-zero vacuum expectation values at the classical level. The latter has been considered in ref. [7] for establishing the extension of supergraph techniques [8] to include the cases where global superDipartimento di Fisica dell'Universith and Istituto Nazionale di Fisica Nucleare, Sezione di Padova. 1-35100 Padua, Italy.
228
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
symmetry is spontaneously broken, in a way to account for all orders in the breaking parameters, for a given fixed order in a loopwise expansion. In possess of the self-energy functions for the O'Raifeartaigh model in its broken non-symmetric phase [7], we would like to f'md out the way the tree-level vacuum expectation values of the physical scalars modify the Str M 2 formula derived in ref. [5]. Referring again to ref. [7], it would be worthwhile to recall the quantities in terms of which our final expressions will be given: (i) x is the vacuum expectation value of the physical component relevant for our analysis; for the broken nonsymmetric phase, it satisfies the equation m 2 + 2g~ + 2g2x 2 = 0,
(2)
where ~, m and g are the parameters of the O'Ralfeartaigh model [6] and 2g~ < - m 2 is the relevant region for the phase we are treating. (il) A is the relevant vacuum expectation value for the set of auxiliary fields, A = -m2/2g.
(3)
Once the tree-approximation vacuum expectation values are set, the superfields of the model are all shifted and the modified super-Feynman rules are derived which account for all orders in x and A. By using them, one can compute the order - f i contributions to the effective action (see section 4 of ref. [7] ) and therefore derive the one-loop corrected equations of motion for the auxiliary fields of the theory, which is an essential step for the computation of the physical-field mass matrices. It is worthwhile remarking that the elimination of the auxiliary fields through their classical equations of motion does n o t account for all possible contributions of O(h) to the physical masses; one has indeed to use the equations of motion for the auxiliaries by including one-loop corrections. Having obtained these equations, one replaces the auxiliary fields in the component-field effective action by using them, retain only the terms which are of O(h) and finally the poles of the one-loop corrected connected two-point functions can be read off, yielding the bosonic and fermionic mass matrices. Three facts should be observed here: (i) the one-loop corrected fermionic mass matrix has a zero eigenvalue, as it should be, since the goldstino remains massless to all orders of perturbation theory. (ii) The scalar field which at the tree-level is massless, and this is a common feature of spontaneously broken models with only chiral superfields [9], acquires a finite mass given by m 2 = [ m 2 / ( m 2 + 4g2x2)] IC(p 2 = 0)l,
(4)
where C(0) is a momentum-space one-loop integral defined by C(0)
4 2 6 4 f dak k 2 1 1 = -- g d(~)4 (k 2 + / 2 ) 2 (k 2 + ~2)2 _ 64 '
(5)
with 62 = 2gA and/12 = m 2 + 492x 2 . This result is a manifestation of the mechanism of spontaneous mass generation in broken supersymmetric models first pointed out by Bardeen, Piguet and Sibold in ref. [10]. Cli0 Finally, our explicit computations show that the infinities introduced into the one4oop corrected Str M 2 exactly cancel each other without even the need o'f performing wave-function renormalizations, so that Str M 2 turns out to be automatically Finite. This should not be regarded as a new conclusion coming from our calculations, but should rather be seen as a good check of our superspace computations, since, as already mentioned in ref. [5], Str M 2 should be automatically finite to all orders, as only wave-function renormalizations are required in the model [ 11]. Finally, having obtained all physical masses as poles of the one-loop corrected connected two-point functions, 229
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
we can present our result for the O(h)-corrections to the mass formula of eq. (1): Str M 2 = (g2/47r2)m2 one -loop X { - 2 + [(1 + 2;)2(1 + 1/~) log(1 + ~ ) - 2X 2 logl~l + (1 - ~)2(1 - 1/~) log(1 - Z ) ] )
+ (g4/4n2)x2 [-x/~zr + 8~ log(1 + E) + 4(3 - 2 ~ ) log(1 - E) + 8(1 - ~ ) I ( - 1 - 2;, 2; 1 , - 1 - ~ , 1) - 8(1 - 1!3)I(- 1 + ~ , 2; 1 , - 1 + N, 1)
- 4(2-I(-1
~ ) I ( - 1 - ~ , 2 ; 1 + Z , - 1 - X, 1) + 4 ( 2 +5-:,,2- 2~; 1 +N,-1
+{I(-1 - Z,2-
~ ) I ( - 1 + Y~, 2; 1 + ~ , - 1
+ ~, 1 - g)+ 3I(-1-
2Z;1 +Z,-1
- Z, 1 - Z ) -
23,2+2~; 1- g,-1
sI(-1 + Z,2+2Z;1-
+ E, 1) - Y,, 1 + 1!3)
Z , - 1 +~3, 1 + Z ) ] ,
(6)
where ~ = 6 2//a2 = 2gA/(m 2 + 4g2x 2) is such that IZl < 1 for the non-symmetric phase, and
1
aI t + a 2
aI
I(al, a2; b 1 , b2, b3) = f dt log 0 blt2 + b2t + b 3 2bl
bI + b2 + b3
b2
b3
2b 1
0j
dt bl t2 + b2t 1 + b3
(7)
The reason we have left the integrals l(al, a2; b l , b2, b3) in the corrected mass formula is that they may assume different analytic forms in the domain of validity o f the parameter ~ . Moreover, their replacement by the possible analytic forms they take does not bring any substantial simplification. To conclude, we remark that the additional contributions with respect to the result o f ref. [5] is just proportional to a factor x 2 and we notice that in the case x = 0 our expression reproduces the results of Girardello et al., if we follow their prescription of taking the real part o f the logarithm. We would like to express our gratitude to Franco Legovini and Luciano Girardello for discussions and suggestions. J.A.H.-N. would also like to acknowledge Professor Abdus Salam, the IAEA and UNESCO for hospitality at ICTP.
References [1 ] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403. [2] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150; N. Sakai, Z. Phys. C l l (1981) 153. [3] R. Barbieri, S. Ferrara and C.A. Savoy, Phys. Lett. l19B (1982) 343; J. Polchinsky and L. Susskind, Phys. Rev. D26 (1982) 3661. [4] L. GirardeUo and J. Iliopoulos, Phys. Lett. 88B (1979) 85. [5] L. Girardello, M. Pernici and D. Zanon, Phys. Rev. D29 (1984) 318. [6] L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. [7] F. Feruglio, J.A. Helay~l-Neto and F. Legovini, Nucl. Phys. B249 (1985) 533. [8] M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429. [9] B. Zumino, Spontaneous breaking of supersymmetry, in: Proc. Heisenberg Symp. on Unified theories of elementary particles, Lecture Notes in Physics, Vol. 160 (Springer, Berlin, 1982) p. 137. [10] W.A. Bardeen, O. Piguet and K. Sibold, Phys. Lett. 72B (1977) 231. [11] J. Iliopoulos and B. Zurnino, Nucl. Phys. B76 (1974) 310.
230