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Effective lagrangian and mass generation in supersymmetric QCD
Volume 153B, number 4,5
PHYSICS LETTERS
4 April 1985
E F F E C T I V E L A G R A N G I A N A N D M A S S G E N E R A T I O N IN S U P E R S Y M M E T R I C Q C D W. L E R C H E , D. L O S T x and H. S T E G E R Max- Planck- Institut fur Physik und Astrophysik - Werner Heisenberg Institut fur Physik, FOhringer Ring 6, D-8000 Munich 40, Fed. Rep. Germany
Received 3 January 1985
We calculate the electromagnetic mass shift of composite Goldstone bosons and quasi-Goldstone fermions in supersyrmnetric QCD for the case where supersymmetry is explicitly broken by a photino mass term. We use an effective lagrangian approach incorporating the analogues of the p and A 1 vector mesons together with vector meson dominance.
In the framework o f composite models, quarks and leptons are regarded as (nearly) massless bound states o f preons. In a specific approach [1], the origin of their masslessness is a supersymmetric Goldstone mechanism: the observed fermions are interpreted as N = 1 supersymmetric partners [quasi-Goldstone fermions (QGFs)] of Goldstone bosons, arising from a spontaneous breakdown of a global internal symmetry G to some subgroup H. The observed finite mass shifts are due to small symmetry perturbations. Of course, to achieve a realistic mass spectrum, supersymmetry has to be broken. This can be done either spontaneously or explicitly. The latter possibility is particularly intriguing since the coupling to supergravity leads to explicit breaking terms quite naturally [2]. Such breaking terms are soft and of well defined general form, so one can determine their effects on the mass spectrum quite easily [3]. A detailed analysis based on effective lagrangian method has been presented in ref. [4]. On the other hand, in ref. [5] the masses of QGFs have been computed in complete analogy to the electromagnetic pion mass shift calculation in nonsupersymmetric QCD [6]. Supersymmetry was broken explicitly only by photino mass term. The advantage of this approach is that it takes the heavy excited spectrum into account. In ref. [5], the masses of charged Goldstone 1 Sektion Physik, University of Munich, Munich, Fed. Rep. Germany. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
bosons and QGFs are given in terms of the masses of the 0 and A 1 vector mesons, m 2 = ~u2A~2),
m F = ½/aA(/a2),
(1)
where A0/2) = (2a/Tr)( [ m 2 / ( m 2 - #2)] ln(m2/gt2) 2
2
- [ m o / ( m A - gt2)] l n ( m 2 [ l a 2 ) ) .
(2)
Here/a is the supersymmetry breaking photino mass parameter, and it was implicitly assumed that m2A = 2m 2. In the nonsupersymmetric limit gt -->oo with m2A = 2m 2 one obtains m 2 = (4t~/Z,)m 2 In 2,
(3)
which is precisely the conventional electromagnetic pion mass shift as derived in refs. [6,7], except the factor 4 instead o f 3. (This is due to an additional supersymmetric contribution.) Now, the results (1)have been obtained by using the two supersymmetric versions [8] of Dashen's formula [9], which are valid even for nonperturbative bound states. However, in contrast to the bosonic case, the derivation o f the fermionic Dashen's formula requires unbroken supersymmetry. That is there could in principle appear additional contributions to rn F . The purpose of this paper is to support the results (1) by calculating m 2 and m F using a supersymmetric generalization of the effective lagrangian approach [7, 10]. That is, we compute the masses perturbatively us261
Volume 153B, number 4,5
PHYSICS LETTERS
ing an effective lagrangian containing the chiral Goldstone superfields ,r as well as the corresponding p and A 1 vector meson superfields. As toy model we consider two flavor SQCD with G = SU(2)L X SU(2)R. We assume G is broken to H = SU(2)V while U(1)e m C SU(2)V is gauged. Supersymmetry is explicitly broken by a photino mass tenn. To construct the effective lagrangian we follow the approach of ref. [ 11 ]. The nonlinear action for the chiral Goldstone superfields *r = rrara,a = 1 ... 3 alone is given by [12] r = ~ f ~2 TrfdSzM+M,
peff
4 April 1985
gauge l~break,
= l-'inv.
+
g a u g e = ( + Tr fd6z[Wa(VL)W l?inv. + WcZ(VR)Wc~(VR )] +
+
12(1 _
(VL)
h.c.) +
2)-1
X Tr f dSz M + exp(gVR)M exp(--gVL) ,
(4) 17break - m2 Tr f dSz [exp(gVL) + exp(--gVR) ] . (9)
where M = exp(i x/~ rt/f,r). The global SU(2)L current superfields JL are defined by the noninvariance of the SQCD action under an infinitesimal local SU(2)L transformation with chiral superparameters A L :
aA, XF sQc° = i T r f d S z (AL - AL)JL
(5)
(the same applies for JR too). The currents transform as
JL = exp(--iAL)JL exp(iAL), J~t = exp(-iAR)JR exp(i~'R),
(6)
while the Goldstones obey M = exp(--iAR) M exp(iAL).
use of (8). peff describes the interactions of Goldstone super1 2 as generalfields with vector superfields of mass ~m ization of the nonsupersymmetric case. However, F ett involves in addition the scalar components of the superfields which feel the usual "D"-potential V. To assure a supersymmetric ground state V = 0 at (VL) r_ ( V R ) = (11")= 0 we derive as constraint ~f2(1 -/32) -1 = m2/g 2.
(7)
We like to include JL and JR as dynamical fields in our effective action perf. The constraint is that [,eft should obey the same local Ward identity (5). Since the currents do not transform like gauge fields, we redefine variables according to the supersymmetlic generalization of the current field identities [ 11 ]
JL = (m2/g2) exp(gVL)' JR =-(m2[g2)exp(-gVR)" (8) Here m is some scale, g2 the effective strong coupling constant and VL, VR are left- and tight-handed gauge fields [the negative value of (JR) follows from (5) and (6)]. Hence one gets as simplest effective action [the factor (1 -/32)-1 is explained below]
262
J
Here W~(VL), Wa(VR) are the usual chiral fields strength spinors corresponding to VL and VR. F elf is invariant under global SU(2)L X SU(2)R transformations. The noninvariance of 1"break under local SU(2)L X SU(2)R transformations reproduces precisely (5) by
(10)
This relation seems model dependent and is due to our specific choice of I"eff. Therefore we treat in the following m andf~r still as independent. Next define J V _- ~t ( J L + J R ),
J A -- ~1 ( J L -- J R ),
(11)
as well as supersymmetric analogues of the O and A 1 vector mesons, 1
P.1
p = ] ( V L + VR) , A - i ( V L - VR).
(12)
This implies the current field identities between J v and p as well asJ A and A' are rather complicated: J v = i (x m 2/,g2) (exp [g(p + g')] - exp [-g(p - g')] ) 1 2 ~g,t p} + ...) = (m2/gE)(gp + -~g
(13)
Volume 153B, number 4,5
PHYSICS LETTERS
JA = ~ (m2/g2)(exp[g(O +A')] + exp[-g(p - A ' ) ] )
1-,em-
= (m2/g2)(1 +gA' + _~g2(p2 + A,2) + ...), (13 cont'd) implying ( J r ) = 0 and (JA) = m2/g 2. In the WessZumino gauge, (13) contains the conventional nonsupersymmetric current field identifies as the 00 components. Substituting the redefined fields (12) back into F eff there occurs a mass mixing term g r mix =i 1 : ~ - - Tr f d S z (~ 7r)A'. (14)
Diagonalization leads to redefining A'= A + i~/vc2 m)(lr
-
if),
(16)
The masses of the vector mesons are now m Z = m 2, P
m 2 = m 2 ( 1 - / 3 2 ) -1.
1 (fd6z(l+21aO2)Wc'(V)W(V)+h.c.) 4e 2
±16
fdSz (D 2 V)(/) 2 V).
(20)
Supersymmetry is violated due to the explicit 0 dependence. I"em leads to the photon propagator [13] iA~(1,2 ,q2 ,p) = iA~(1,2,q 2) + iA'r(l, 2, q 2, p), (21) where iA~(l, 2, q2) = _ q - 2 612' i~'r (1,2, q 2, p) = q-2 [p(A B + AB) + P 2(AC + AC)] 512'
(15)
which introduces additional Goldstone-vector interactions also in the kinetic terms. The (1 - f12)-I factor in F elf secures that the Goldstone fields have canonical kinetic terms after this diagonalization if
=gfJm.
4 April 1985
(17)
AB = [8(q2 +/a2)] -1 Da~202D, AC = _[32q2(q2 +02)] -1D,~B202~2D2~#,~, 612 = (01 -- 02)2(01 -- 02 )2. A~ represents the supersymmetry preserving part; its contribution vanishes because of the nonrenormalization theorem~ [14]. Therefore we omit A~, i.e. we set A~ = A'r. For the A propagator we use
To achieve a rr+-% mass shift, electromagnetism has to be included. This is done substituting Tr(JL - J R ) in (9) by
iA:b(l, 2, q2) = [q2 + m 2 ] -1
Tr(JL exp(e Vr 3) - JR exp(--e Vr 3))
while for P we take an effective propagator reflecting vector dominance,
= (m2/g 2) Tr(exp [g(p + A')] exp(e Vr3)
X (1 - {D2,D2)/16m2) 6126ab,
i Aab P')'P (1,2,
+ exp [-g(p - A')] exp(-eVr3)),
(18)
which now breaks in addition global SU(2)V invariance. The expansion of pbreak contains a mixing term
2(e/g)m 2
f d8zo°v,
(19)
just as the nonsupersymmetric case. Of course, due to this mixing neither p 0 nor V corresponds to the physical particles. However, for lowest order processes it suffices to take as electromagnetic interaction only (19). This is precisely the idea of P vector dominance. Now, in order to generate mass at all one has to break supersymmetry. This is done by a photino mass term parametrized by p. Hence one adds
(22)
q 2 ,p)--- -iAoA'tAPS a36b3
= -(e2/g2)i[m2/(q 2 + m2)] 2Aq'6a36b3 ,
(23)
which is still transverse:
(Da~2 DoJ8q2) AP'rP = AP'YP, D 2Ap'~p = D 2 A p ' Y P = O.
(24)
Now we are in the position to calculate the pion's mass shift using p dominance. That is we calculate F mass = f d 6 z 7r+Tr_M(x,O) + h.c.
(25)
by considering all tree diagrams for p07r --*p07r and closing the pOTp0 loop with the propagator (23). Then the bosonic and fermionic masses are given by 263
Volume 153B, number 4,5
m2B
PHYSICS LETTERS
f d2OM(O,O),m v = M ( 0 , 0 = 0 )
=
(26)
(we do not consider here bosonic mass terms of the form zr*rr). The relevant vertices contributing to M contained in r elf are of the form rrrrp,zmpp, ~rTrpDC~D2Dotp, rrADe~ 2Deep and permutations. Owing to the transversality of AO~o (24) many terms like the {D 2,/~2) part o f AA vanish. The contributing diagrams are displayed in fig. 1. They all have the same 0-structure. In detail one obtains
M(O, O) = ~ ~
4
F(q2)G(q 2, 0),
(27)
(2~r)" where
[e2m4/i(q 2 + m2) 2]
r ( q 2) =
X [1/q 2 + [32/m2
-- (j32/m2) q2/(q2 + m 2 ) ] ,
(28)
and G(q 2, O) is basically the photon propagator closed in 0-space,
G(q2,01)= f d2OI S'X1,2,q2,U)I2= 1.
=(o4rO[m2/(m~- rno2)l 2 °2 X { [mp/(m -t
[m21(m 2 p,~ A
p2)]
_p2)]
ln(m2/p2) ln(m2A/p2)).
(32)
In the nonsupersymmetric case t2 is a free parameter which is empirically found to be/~2 = 1/2, implying m 2 = 2m 2. Hence we recover precisely the results (1) of ref. [5] where to saturate the spectral functions mA 2 = 2m p2 has been assumed implicitly. That is, the use of the fermionic Dashen's formula is justified in this case, though supersymmetry is broken explicitly. On the other hand, in the effective lagrangian (9) parameters are constrained by (10) in order that the potential vanishes. This fixes/32 = 2/3 and leads to m 2 = 3m 2. However, as remarked above, this relation may be model dependent. We thank R.D. Peccei for suggesting the problem and S.T. Love for comments.
fd20 G(q 2, 0) = 4u2(q 2 +/a2) -1 , G(q 2, 0) = 2p(q 2 + / / 2 ) - 1 ,
(30)
one computes from (27) m F = ~laA(.#2),
(31)
with V
V
'°°';'0
,0,
Y
ql ~
II' A
~
Fig. 1. Supergraph contributions to M. The cross denotes the supersymmetry breaking photino mass. 264
= 4 fJ ~d4q F(q2)(q 2 + p 2 ) - I
(29)
Since [5]
m 2 = p2A0a2),
A ~ 2)
4 April 1985
[1] W. BuchmOller, S.T. Love, R.D. Peccei and T. Yanagida, Phys. Lett. l15B (1982) 233. [2] L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982) 65; R. Barbieri, S. Ferrara and C. Savoy, Phys. Lett. l19B (1982) 343. [3] D. LOst, Nucl. Phys. B239 (1984) 508; W. Lerche and D. Lttst, Nucl. Phys. B244 (1984) 157. [4] A. Masiero and G. Veneziano, Split light composite supermultiplets, CERN preprint TH-3950/84. [5] W. Lerche, Phys. Lett. 149B (1984) 361. [6] T. Das, G. Guralnik, V. Mathur, F. Low and J. Young, Phys. Rev. Lett. 18 (1967) 759. [7] G.C. Wick and B. Zumino, Phys. Lett. 25B (1967) 479. [8] G. Veneziano, Phys. Lett. 128B (1983) 199; G.M. Shore, Nucl. Phys. B231 (1984) 139; T. Clark and S. Love, Nucl. Phys. B232 (1984) 306; W. Lerche, Nucl. Phys. B246 (1984) 475. [9] R.F. Dashen, Phys. Rev. 183 (1969) 1245; D3 (1971) 1879. [10] B.W. Lee and H.T. Nieh, Phys. Rev. 166 (1967) 1507. [ 11 ] K. Konishi, Dynamical supersymmetry breaking and effective lagrangian in an SU(5) model, Pisa preprint IFUP TH-11/84 (1984); E. Guadagnini and K. Konishi, Effective gauge symmetry in supersymmetric confining theories, Pisa preprint (1984). [12] W. Lerche, Nucl. Phys. B238 (1984) 582. [13] J. Helay~l-Neto, Phys. Lett. 135B (1984) 78. [14] W. Lerche, R.D. Peccei and V. Vi~nji~, Phys. Lett. 140B (1984) 363.
PHYSICS LETTERS
4 April 1985
E F F E C T I V E L A G R A N G I A N A N D M A S S G E N E R A T I O N IN S U P E R S Y M M E T R I C Q C D W. L E R C H E , D. L O S T x and H. S T E G E R Max- Planck- Institut fur Physik und Astrophysik - Werner Heisenberg Institut fur Physik, FOhringer Ring 6, D-8000 Munich 40, Fed. Rep. Germany
Received 3 January 1985
We calculate the electromagnetic mass shift of composite Goldstone bosons and quasi-Goldstone fermions in supersyrmnetric QCD for the case where supersymmetry is explicitly broken by a photino mass term. We use an effective lagrangian approach incorporating the analogues of the p and A 1 vector mesons together with vector meson dominance.
In the framework o f composite models, quarks and leptons are regarded as (nearly) massless bound states o f preons. In a specific approach [1], the origin of their masslessness is a supersymmetric Goldstone mechanism: the observed fermions are interpreted as N = 1 supersymmetric partners [quasi-Goldstone fermions (QGFs)] of Goldstone bosons, arising from a spontaneous breakdown of a global internal symmetry G to some subgroup H. The observed finite mass shifts are due to small symmetry perturbations. Of course, to achieve a realistic mass spectrum, supersymmetry has to be broken. This can be done either spontaneously or explicitly. The latter possibility is particularly intriguing since the coupling to supergravity leads to explicit breaking terms quite naturally [2]. Such breaking terms are soft and of well defined general form, so one can determine their effects on the mass spectrum quite easily [3]. A detailed analysis based on effective lagrangian method has been presented in ref. [4]. On the other hand, in ref. [5] the masses of QGFs have been computed in complete analogy to the electromagnetic pion mass shift calculation in nonsupersymmetric QCD [6]. Supersymmetry was broken explicitly only by photino mass term. The advantage of this approach is that it takes the heavy excited spectrum into account. In ref. [5], the masses of charged Goldstone 1 Sektion Physik, University of Munich, Munich, Fed. Rep. Germany. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
bosons and QGFs are given in terms of the masses of the 0 and A 1 vector mesons, m 2 = ~u2A~2),
m F = ½/aA(/a2),
(1)
where A0/2) = (2a/Tr)( [ m 2 / ( m 2 - #2)] ln(m2/gt2) 2
2
- [ m o / ( m A - gt2)] l n ( m 2 [ l a 2 ) ) .
(2)
Here/a is the supersymmetry breaking photino mass parameter, and it was implicitly assumed that m2A = 2m 2. In the nonsupersymmetric limit gt -->oo with m2A = 2m 2 one obtains m 2 = (4t~/Z,)m 2 In 2,
(3)
which is precisely the conventional electromagnetic pion mass shift as derived in refs. [6,7], except the factor 4 instead o f 3. (This is due to an additional supersymmetric contribution.) Now, the results (1)have been obtained by using the two supersymmetric versions [8] of Dashen's formula [9], which are valid even for nonperturbative bound states. However, in contrast to the bosonic case, the derivation o f the fermionic Dashen's formula requires unbroken supersymmetry. That is there could in principle appear additional contributions to rn F . The purpose of this paper is to support the results (1) by calculating m 2 and m F using a supersymmetric generalization of the effective lagrangian approach [7, 10]. That is, we compute the masses perturbatively us261
Volume 153B, number 4,5
PHYSICS LETTERS
ing an effective lagrangian containing the chiral Goldstone superfields ,r as well as the corresponding p and A 1 vector meson superfields. As toy model we consider two flavor SQCD with G = SU(2)L X SU(2)R. We assume G is broken to H = SU(2)V while U(1)e m C SU(2)V is gauged. Supersymmetry is explicitly broken by a photino mass tenn. To construct the effective lagrangian we follow the approach of ref. [ 11 ]. The nonlinear action for the chiral Goldstone superfields *r = rrara,a = 1 ... 3 alone is given by [12] r = ~ f ~2 TrfdSzM+M,
peff
4 April 1985
gauge l~break,
= l-'inv.
+
g a u g e = ( + Tr fd6z[Wa(VL)W l?inv. + WcZ(VR)Wc~(VR )] +
+
12(1 _
(VL)
h.c.) +
2)-1
X Tr f dSz M + exp(gVR)M exp(--gVL) ,
(4) 17break - m2 Tr f dSz [exp(gVL) + exp(--gVR) ] . (9)
where M = exp(i x/~ rt/f,r). The global SU(2)L current superfields JL are defined by the noninvariance of the SQCD action under an infinitesimal local SU(2)L transformation with chiral superparameters A L :
aA, XF sQc° = i T r f d S z (AL - AL)JL
(5)
(the same applies for JR too). The currents transform as
JL = exp(--iAL)JL exp(iAL), J~t = exp(-iAR)JR exp(i~'R),
(6)
while the Goldstones obey M = exp(--iAR) M exp(iAL).
use of (8). peff describes the interactions of Goldstone super1 2 as generalfields with vector superfields of mass ~m ization of the nonsupersymmetric case. However, F ett involves in addition the scalar components of the superfields which feel the usual "D"-potential V. To assure a supersymmetric ground state V = 0 at (VL) r_ ( V R ) = (11")= 0 we derive as constraint ~f2(1 -/32) -1 = m2/g 2.
(7)
We like to include JL and JR as dynamical fields in our effective action perf. The constraint is that [,eft should obey the same local Ward identity (5). Since the currents do not transform like gauge fields, we redefine variables according to the supersymmetlic generalization of the current field identities [ 11 ]
JL = (m2/g2) exp(gVL)' JR =-(m2[g2)exp(-gVR)" (8) Here m is some scale, g2 the effective strong coupling constant and VL, VR are left- and tight-handed gauge fields [the negative value of (JR) follows from (5) and (6)]. Hence one gets as simplest effective action [the factor (1 -/32)-1 is explained below]
262
J
Here W~(VL), Wa(VR) are the usual chiral fields strength spinors corresponding to VL and VR. F elf is invariant under global SU(2)L X SU(2)R transformations. The noninvariance of 1"break under local SU(2)L X SU(2)R transformations reproduces precisely (5) by
(10)
This relation seems model dependent and is due to our specific choice of I"eff. Therefore we treat in the following m andf~r still as independent. Next define J V _- ~t ( J L + J R ),
J A -- ~1 ( J L -- J R ),
(11)
as well as supersymmetric analogues of the O and A 1 vector mesons, 1
P.1
p = ] ( V L + VR) , A - i ( V L - VR).
(12)
This implies the current field identities between J v and p as well asJ A and A' are rather complicated: J v = i (x m 2/,g2) (exp [g(p + g')] - exp [-g(p - g')] ) 1 2 ~g,t p} + ...) = (m2/gE)(gp + -~g
(13)
Volume 153B, number 4,5
PHYSICS LETTERS
JA = ~ (m2/g2)(exp[g(O +A')] + exp[-g(p - A ' ) ] )
1-,em-
= (m2/g2)(1 +gA' + _~g2(p2 + A,2) + ...), (13 cont'd) implying ( J r ) = 0 and (JA) = m2/g 2. In the WessZumino gauge, (13) contains the conventional nonsupersymmetric current field identifies as the 00 components. Substituting the redefined fields (12) back into F eff there occurs a mass mixing term g r mix =i 1 : ~ - - Tr f d S z (~ 7r)A'. (14)
Diagonalization leads to redefining A'= A + i~/vc2 m)(lr
-
if),
(16)
The masses of the vector mesons are now m Z = m 2, P
m 2 = m 2 ( 1 - / 3 2 ) -1.
1 (fd6z(l+21aO2)Wc'(V)W(V)+h.c.) 4e 2
±16
fdSz (D 2 V)(/) 2 V).
(20)
Supersymmetry is violated due to the explicit 0 dependence. I"em leads to the photon propagator [13] iA~(1,2 ,q2 ,p) = iA~(1,2,q 2) + iA'r(l, 2, q 2, p), (21) where iA~(l, 2, q2) = _ q - 2 612' i~'r (1,2, q 2, p) = q-2 [p(A B + AB) + P 2(AC + AC)] 512'
(15)
which introduces additional Goldstone-vector interactions also in the kinetic terms. The (1 - f12)-I factor in F elf secures that the Goldstone fields have canonical kinetic terms after this diagonalization if
=gfJm.
4 April 1985
(17)
AB = [8(q2 +/a2)] -1 Da~202D, AC = _[32q2(q2 +02)] -1D,~B202~2D2~#,~, 612 = (01 -- 02)2(01 -- 02 )2. A~ represents the supersymmetry preserving part; its contribution vanishes because of the nonrenormalization theorem~ [14]. Therefore we omit A~, i.e. we set A~ = A'r. For the A propagator we use
To achieve a rr+-% mass shift, electromagnetism has to be included. This is done substituting Tr(JL - J R ) in (9) by
iA:b(l, 2, q2) = [q2 + m 2 ] -1
Tr(JL exp(e Vr 3) - JR exp(--e Vr 3))
while for P we take an effective propagator reflecting vector dominance,
= (m2/g 2) Tr(exp [g(p + A')] exp(e Vr3)
X (1 - {D2,D2)/16m2) 6126ab,
i Aab P')'P (1,2,
+ exp [-g(p - A')] exp(-eVr3)),
(18)
which now breaks in addition global SU(2)V invariance. The expansion of pbreak contains a mixing term
2(e/g)m 2
f d8zo°v,
(19)
just as the nonsupersymmetric case. Of course, due to this mixing neither p 0 nor V corresponds to the physical particles. However, for lowest order processes it suffices to take as electromagnetic interaction only (19). This is precisely the idea of P vector dominance. Now, in order to generate mass at all one has to break supersymmetry. This is done by a photino mass term parametrized by p. Hence one adds
(22)
q 2 ,p)--- -iAoA'tAPS a36b3
= -(e2/g2)i[m2/(q 2 + m2)] 2Aq'6a36b3 ,
(23)
which is still transverse:
(Da~2 DoJ8q2) AP'rP = AP'YP, D 2Ap'~p = D 2 A p ' Y P = O.
(24)
Now we are in the position to calculate the pion's mass shift using p dominance. That is we calculate F mass = f d 6 z 7r+Tr_M(x,O) + h.c.
(25)
by considering all tree diagrams for p07r --*p07r and closing the pOTp0 loop with the propagator (23). Then the bosonic and fermionic masses are given by 263
Volume 153B, number 4,5
m2B
PHYSICS LETTERS
f d2OM(O,O),m v = M ( 0 , 0 = 0 )
=
(26)
(we do not consider here bosonic mass terms of the form zr*rr). The relevant vertices contributing to M contained in r elf are of the form rrrrp,zmpp, ~rTrpDC~D2Dotp, rrADe~ 2Deep and permutations. Owing to the transversality of AO~o (24) many terms like the {D 2,/~2) part o f AA vanish. The contributing diagrams are displayed in fig. 1. They all have the same 0-structure. In detail one obtains
M(O, O) = ~ ~
4
F(q2)G(q 2, 0),
(27)
(2~r)" where
[e2m4/i(q 2 + m2) 2]
r ( q 2) =
X [1/q 2 + [32/m2
-- (j32/m2) q2/(q2 + m 2 ) ] ,
(28)
and G(q 2, O) is basically the photon propagator closed in 0-space,
G(q2,01)= f d2OI S'X1,2,q2,U)I2= 1.
=(o4rO[m2/(m~- rno2)l 2 °2 X { [mp/(m -t
[m21(m 2 p,~ A
p2)]
_p2)]
ln(m2/p2) ln(m2A/p2)).
(32)
In the nonsupersymmetric case t2 is a free parameter which is empirically found to be/~2 = 1/2, implying m 2 = 2m 2. Hence we recover precisely the results (1) of ref. [5] where to saturate the spectral functions mA 2 = 2m p2 has been assumed implicitly. That is, the use of the fermionic Dashen's formula is justified in this case, though supersymmetry is broken explicitly. On the other hand, in the effective lagrangian (9) parameters are constrained by (10) in order that the potential vanishes. This fixes/32 = 2/3 and leads to m 2 = 3m 2. However, as remarked above, this relation may be model dependent. We thank R.D. Peccei for suggesting the problem and S.T. Love for comments.
fd20 G(q 2, 0) = 4u2(q 2 +/a2) -1 , G(q 2, 0) = 2p(q 2 + / / 2 ) - 1 ,
(30)
one computes from (27) m F = ~laA(.#2),
(31)
with V
V
'°°';'0
,0,
Y
ql ~
II' A
~
Fig. 1. Supergraph contributions to M. The cross denotes the supersymmetry breaking photino mass. 264
= 4 fJ ~d4q F(q2)(q 2 + p 2 ) - I
(29)
Since [5]
m 2 = p2A0a2),
A ~ 2)
4 April 1985
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