Volume 143B, number 4, 5, 6 PHYSICS LETTERS 16 August 1984

ABSENCE OF RADIATIVE MASS S H I F T S FOR C O M P O S I T E G O L D S T O N E S U P E R M U L T I P L E T S

T.E. C L A R K

Department of Physics, Purdue University, West Lafayette, IN 47907, USA

and

S.T. LOVE

Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, 1L, 60510, USA

Received 26 March 1984

A supersymmetric Dashen type formula is used to prove that there are no radiative mass corrections for Goldstone pion supermultiplets due to gauge interactions. In particular, for unbroken supersymmetry the charged to neutral pion mass difference is shown to vanish to all orders in electromagnetism.

When the strong dynamics of a theory results in the spontaneous breakdown of its global symmetry group G to a subgroup H, Goldstone boson pions arise in one to one correspondence with the broken generators. If, in addition, the underlying lagrangi- an contains explicit symmetry breaking terms, these pions will, in general, acquire a non-zero mass. Using the currents, j~, associated with the sponta- neously broken symmetries as interpolating fields for the Goldstone bosons, %, the current field identity

~ = 2 -L,m~,% (1)

can be used in conjunction with the broken Ward identities (WI) to yield information about the pion masses and interactions. In particular, the Dashen formula [1]

stants, f,,, to the symmetry breaking terms in the lagrangian. If the breaking terms in &°are invariant under H, then the number of independent masses and decay constants is the same as the number of H irreducible representations formed by the pions. If, however, terms in £P also explicitly break H, then there will be mass splittings within the vari- ous irreducible representations which depend upon the form of the explicit breaking.

For the case of QCD with two massless flavors, the global chiral symmetry group SU(2) × SU(2) is spontaneously broken by quark condensate forma- tion down to the vector subgroup SU(2)v. Adding SU(2)v invariant up and down quark masses to the lagrangian results in the ~r+, % acquiring a common mass given via Dashen's formula in terms of the current quark mass, m q , and quark con- densates as

( f~,m],)2 f d4x(OIT ~(x)~5(O)l O)

= ( 2 )

where Qs, is the charge obtained from the current j5~,, relates the pion masses, rn,~, and decay con-

2 2 f£ m= = -- /'nq((U.U)o ~- ( d d ) o ) . (3)

When the electromagnetic interactions are intro- duced, however, the charged pions, ~r+, will acquire additional mass corrections which will split them from the % mass [2]. Once again, Dashen's for-

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mula can be used to obtain this electromagnetic mass shift as

8m2+ = m2+ _ m2o

= _ f~-2 ((01[ 05_[ Q5 +,-£-al ] [ 0)

--<0l[Qso, [050, ~ l ] ]0>), (4)

where Qs~ = 2-1/2(Qs, - iQs~) and Qs0 = Qs ; This mass splitting can be calculated to lowest order in the Q E D fine structure constant a by using an effective one-photon exchange electro- magnetic lagrangian which is proport ional to the T product of two electromagnetic currents. The charge commutators in eq. (4) then t ransform this lagrangian into a difference between the third component of the vector and axial-vector two-point functions. Finally, by using the current propaga- tors spectral decomposi t ion [3], the mass splitting can be obtained in terms of a and the p-meson mass as

8m~+ = (3a /2~r) In 2m 2. (5)

The purpose of the present note is to demon- strate that in supersymmetric (SUSY) theories, explicit global symmetry breaking due to gauge interactions yields vanishing radiative mass correc- tions for the Goldstone multiplets. Hence there is no pion electromagnetic mass splitting in SUSY Q C D as long as SUSY remains unbroken. The vanishing of 8rn~+ in SUSY Q C D has been estab- lished to lowest order in a by Lerche et al. [4]. Using the superspace techniques of the present paper, this result is shown to be true to all orders in a in a more economical fashion.

As is the case for ordinary QCD, we assume that the strong dynamics of SUSY Q C D results in the spontaneous breakdown of the global chiral symmetry to a vector subgroup, while leaving the supersymmetry unbroken ,1. It follows f rom the SUSY Noether theorem [6] that for each symmetry generator of G (labelled by the index A) there corresponds a vector superfield current JA(x, O, 0). For each of the broken generators (labelled by the

~:1 The question of whether chiral symmetry breaking con- densates form in SUSY theories is still an open one [5].

index i), SUSY dictates the existence of a Gold- stone chiral multiplet ~rg consisting of the Gold- stone boson and its supersymmetric partners. The currents can be used as interpolating superfields for the ~r~ with the current-field identity taking the form [6]

-- 1DDJ, = ½m~ f , ,~ (no sum on i) . (6)

In analogy to the ordinary case, we desire a SUSY Dashen type formula which expresses the pion superfield masses, rn , as the double varia- tion of a SUSY chiral lagrangian. When the ex- plicit global symmetry breaking term is a SUSY chiral mass term, such a formula, linear in rn,~,, was previously obtained independently by Vene- ziano [7] and the present authors [6]. In this paper, we shall extend the formula to also allow for global symmetry breaking arising from gauge in- teractions. We first construct the Noether currents.

A general superfield action can be written as

I= fdVLv(~,, ¢, v )+ f dS Ls(*)

+ / d g Z s ( , ~ ) , (7)

with E)aL s = 0 = D~L s and dV = d4x d 2 0 d20, d S = d4x d 2 0 , d S = d4x d 2 0 . Here (~) ¢h are underly- ing (anti-)-chiral matter superfields and V are vec- tor superfields associated with the gauged symme- tries of the theory. Alternatively, defining the (anti-) chiral lagrangians as

L = - ~ D D L v + L s ,

Z, = _ 1 D D L v + Z s ,

m

DaL = 0 (s)

D~7, = 0 '

it follows that

I=fdsL+fdgZ (9)

Varying the (anti-) chiral lagrangian leads to the (anti-) chiral Noether theorems

~AZ = - -',iDDJA + ~A~,8I/~, -- ~DD[~V~I/~V], ~L = ¼iDDJ~ + ( S X / ~ ) 8 .~ - ~ [ 8~V~X/SV ],

(10)

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where the current superfield is given by

= ½i( 3A~OLv/3 ~ -- (3Lv/3,)8.4 , G

-[- E [ ( ~<a,SAV ) -- ( -- ) lal+ perm`a) (,~)

x3#( L,aLv/ ,o,v)l ) (11)

The symbol .@¢~) represents a produ__ct of [a[ of the SUSY covariant derivatives D~ or Da,

-@(,~) = D , , . . . D a . , tal = n , ( 1 2 )

while ~ T corresponds to the product in trans- posed order, with perm(a) being the number of permutations required to transpose order the origi- nal sequence. Finally F,(~) dictates a summation over all possible number and configurations of such products.

By combining the current-field identity eq. (6) with the zero momentum WI for the two super- space point function of DD JA, we arrive at the desired SUSY Dashen formula. To obtain this WI, we use the chiral Noether theorem to write

- ( 0 I T ( - ¼ DDJ.4 ( 1 ) ) ( - ~ D D J . (2))10 5

= (0[T(SAL(1) 8~L(2))l 0)

- (0IT 8.4 L (1) [ 31/3, (2)] 8s , (2)10)

+ ½ (01T 8 A L (1) DD[ 8~ VaL v/a V ] (2) 10)

+ i(0[T[ 31/3, (1)] 3.4, (1)( - ~DD 4 (2))105

- ~i(OIT(DD[3.4VOLv/aV](1))

× ( - ¼DY)Je (2))10>. (13)

Unbroken SUSY, however, dictates that the (O, 0) dependence of two superspace point functions take a specific form. Letting M and N be genetic super- fields, SUSY requires that

(0IT M(a)N(2)10)

=exp[i(O,o~'02-02o~'01)32,]

X (0IT M(0)N(2 - 1)10). (14)

Moreover, if ( N) N is (anti-) chiral so that (DaN

= 0) -DaN = 0, it follows that

N ( x , 0, 0 ) = exp(-iOo~'OO,)N(x, O, O) (15)

N(x, O, O) = exp(iOo"OO~,)N(x, 0, 0), (16)

which further restricts the (0, t~) structure of two superspace point functions containing (anti-) chiral superfields. Thus integrating eq. (13) over x 2 and using eqs. (14)-(16), it follows, upon application of the action principle,

(0[ r &~(M/8,)xI0> = i(0lT 8 , 8 x / 8 , 1 0 ) , (17)

that the zero momentum WI takes the form

fd4x2(O IT 8 A L (1) 8 B L (2)I 0)

= ifd4x2<0l T 8s* (2)[ 8/8 , (2)] 8.4 L (1)10). (18)

However, using Noether's theorem and the cur- rent-field identity we recognize the left-hand side of eq. (18) as the zero momentum chiral Goldstone pion propagator so that

f d%(O IT 3,L (1) 3jL (2)10)

1" 2 = ~l(mw, f% ) [ 3 ( 0 1 - - 0 2 ) / m w , ] 8 i . , . (19)

Moreover, since the SUSY WI eqs. (14)-(16) im- ply that

fd G<O[T 88~ (2)[3/8~ (2)] 3~ L (1) [0) = 0, (20)

fdV2(0 IT 8BV(2 ) [ 8 /8 V(2)] 8.4 L (1)10) = 0,

it follows that - ¼ D z D 2 on the tight hand side of eq. (18) is simply the double variation of the chiral lagrangian. We thus arrive at the SUSY Dashen type formula

8ijm,,,f 2 = 4(018~3jL (0)10). (21)

Recalling that L = - ~DD L v + L s and noting that unbroken SUSY dictates that @[Lv[0 ) be superspace (x, 0, 0) independent so that D D (0[Lv[0) vanishes, it follows that

(O[8,3jL (0)[0) = (O[3,3jLs(O) [0). (22)

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Thus only those breaking terms arising from pure chiral fields can contribute to a nonvanishing pion mass in the unbroken SUSY limit. Those explicit symmetry breaking terms arising from gauge inter- actions do not lead to any radiative nontrivial pion mass shifts. Thus for the particular example of SUSY QCD, it follows that after the inclusion of electromagnetic interactions the charged to neutral pion mass difference vanishes to all orders in the electromagnetic coupling.

The absence of perturbative mass shifts for zero tree mass fundamental chiral superfields can be understood in terms of the remarkable nonrenor- malization properties of SUSY theories. We have seen here that even when the chiral superfield arises as a massless bound state of some com- plicated strong interaction dynamics, all radiative corrections to its mass again vanish order by order in the explicit symmetry breaking gauge coupling constant. It thus appears that a nonrenormaliza- tion theorem is also operating at the bound state level [4]. If the Goldstone multiplet is to develop any nonvanishing mass in the absence of SUSY breaking, it can do so only through some nonper-

turbative mechanism giving rise to pure chiral condensates of the form considered in refs. [6] and [7].

This work is supported in part by the US Department of Energy.

References

[1] R.F. Dashen, Phys. Rev. 183 (1969) 1245; D3 (1971) 1879; S. Weinberg, in: Proc. 24th Intern. Conf. on High energy physics (CERN, Geneva, 1969).

[2] T. Das, G. Guralnik, V. Mathur, F. Low and J. Young, Phys. Rev. Lett. 18 (1967) 759.

[3] S. Weinberg, Phys. Rev. Lett. 18 (1967) 507. [4] W. Lerche, R.D. Peccei, and V. Visnjic Phys. Lett. 140B

(1984) 363. [5] T.R. Taylor, G. Veneziano and S. Yankielowicz, Nucl.

Phys. B218 (1983) 493; G. Veneziano, Phys. Lett. 124B (1983) 357; M. Peskin, SLAC-PUB 3061 (1983); A. Davis, M. Dine and N. Seiberg, Phys. Lett. 125B (1983) 487.

[6] T.E. Clark and S.T. Love, Nucl. Phys. B232 (1984) 306. [7] G. Veneziano, Phys. Lett. 128B (1983) 199;

see also G.M. Shore, Nucl. Phys. B231 (1984) 139.

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