Massless N=1 super-sinh-Gordon: form factors approach

Physics Letters B 575 (2003) 131–136 www.elsevier.com/locate/physletb

Massless N = 1 super-sinh-Gordon: form factors approach Bénédicte Ponsot Service de Physique Théorique, Commissariat à l’énergie atomique, L’Orme des Merisiers, F-91191 Gif-sur-Yvette, France Received 25 August 2003; accepted 11 September 2003 Editor: L. Alvarez-Gaumé

Abstract The N = 1 super-sinh-Gordon model with spontaneously broken supersymmetry is considered. Explicit expressions for formfactors of the trace of the stress energy tensor Θ, the energy operator , as well as the order and disorder operators σ and µ are proposed.  2003 Elsevier B.V. All rights reserved. PACS: 11.25.Hf; 11.55.Ds

1. Introduction The SShG model can be considered as a perturbed super Liouville field theory, which Lagrangian density is given by 2 2 1 ¯ ¯ 1 ¯ + iµb2ψ ψ¯ ebφ + πµ b e2bφ (∂a φ)2 − (ψ∂ ψ + ψ ∂ψ) 8π 2π 2 with the background charge Q = b + 1/b. This model is a CFT with central charge  3
cSL = 1 + 2Q2 . 2 The super-sinh-Gordon model is (1 + 1)-dimensional integrable quantum field theory with N = 1 supersymmetry. We consider the Lagrangian

L=

1 1 ¯ + 2iµb2ψ ψ¯ sinh bφ + 2πµ2 b2 cosh2 bφ. ¯ ψ¯ + ψ ∂ψ) (∂a φ)2 − (ψ∂ 8π 2π In this model the supersymmetry is spontaneously broken [1]: the bosonic field becomes massive, but the Majorana fermion stays massless and plays the role of goldstino. In the IR limit, the effective theory for the goldstino is to the lowest order the Volkov–Akulov Lagrangian [2] L=

¯ − ¯ ψ¯ + ψ ∂ψ) LIR = (ψ∂

4 ¯ + ···, (ψ∂ψ)(ψ¯ ∂¯ ψ) M2

E-mail address: [email protected] (B. Ponsot). 0370-2693/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2003.09.042

(1)

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B. Ponsot / Physics Letters B 575 (2003) 131–136

where supersymmetry is realized nonlinearly. The irrelevant operator along which the super-Liouville theory flows ¯ which is the lowest dimension nonderivative into Ising is the product of stress-energy tensor T T = (ψ∂ψ)(ψ¯ ∂¯ ψ), operator allowed by the symmetries. The dots include higher-dimensional irrelevant operators. The scattering in the left–left and right–right subchannels is trivial, but not in the right–left channel. The following scattering matrices were proposed in [1] sinh(θ − θ  ) − i sin πν , ν ≡ b/Q. sinh(θ − θ  ) + i sin πν For the right and left movers, the energy momentum is parametrized in terms of the rapidity variables θ and θ  M −θ  θ 0 1 by p0 = p1 = M ). The mass scale of the theory M −2 is equal to 2 sin πν. The form 2 e (and p = −p = 2 e factors1 Fr,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl ) are defined to be matrix elements of an operator between the vacuum and a set of asymptotics states. The form factor bootstrap approach [3–5] (developed originally for massive theories, but that turned out to be also an effective tool for massless theories [6,7]) leads to a system of linear functional relations for the matrix elements Fr,l ; let us introduce the minimal form factors which have neither poles nor zeros in the strip 0 < Im θ < π and which are solutions of the equations fα1 α2 (θ ) = fα1 α2 (θ + 2iπ)Sα1 α2 (θ ), αi = R, L. Then the general form factor is parametrized as follows: SRR (θ ) = SLL (θ  ) = −1,

SRL (θ − θ  ) = −

α Fr,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl )

=



fRR (θi − θj )

1
i
r  l 

fRL (θi − θj )

i=1 j =1



fLL (θi − θj )Qr,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl ),

1
i
and the function Qr,l depends on the operator considered. The RR and LL scattering formally behave as in the massive case, so annihilation poles occur in the RR and LL subchannel. This leads to the residue formula Resθ12 =iπ Fr,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl )  = 2Fr−2,l (θ3 , . . . , θr ; θ1 , θ2 , . . . , θl )

1−

r  j =3

SRR (θ2i )

l 

 SRL (θ2 − θk )

,

(2)

k=1

and a similar expression in the LL subchannel. It is important to note that these equations do not refer to any specific operator.

2. Expression for form factors The minimal form factors read explicitly: θ fRR (θ ) = sinh , 2

fLL (θ  ) = sinh

and fRL (θ ) =

1 2 cosh θ2

∞ exp

θ , 2


  θ dt cosh 12 − ν t − cosh 12 t cosh t 1 − . t sinh t cosh t/2 iπ

0

The latter form factor has asymptotic behaviour when θ → −∞  2    A (A )2 θ 2 2θ + B + AA θ + , f (θ ) ∼ eθ/2 1 + (A + A θ )eθ + e 2 2 1 We refer the reader to [6] for a discussion on form factors in massless QFT.

(3)

B. Ponsot / Physics Letters B 575 (2003) 131–136

133

where A = (1 − 2ν) cos πν − 1 + 2i sin πν, A = − π2 sin πν, B = 12 (cos 2πν − 1). The logarithmic contributions come from resonances. The residue condition (2) written in terms of the function Qr,l reads Resθ12 =iπ Qr,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl ) = Qr−2,l (θ3 , . . . , θr ; θ1 , θ2 , . . . , θl ) × (−)r−1 (2i)l+r−1   l l r  

  1  r+l  sinh(θ2 − θk ) + i sin πν − (−1) sinh(θ2 − θk ) − i sin πν . × sinh θ2j j =3

k=1

(4)

k=1

Let us introduce now the functions SRR 2i , = φ(θij ) ≡ fRR (θij )fRR (θij + iπ) sinh θij

φ(θij ) ≡

SLL  fLL (θij )fLL (θij

+ iπ)

=

2i sinh θij

as well as Φ(θi − θj ) ≡

SRL (θi − θj ) fRL (θi − θj )fRL (θi

− θj


 = −2i sinh(θi − θj ) − i sin πν ,

+ iπ)

and
 i − θj ) ≡ Φ(θi − θj + iπ) = 2i sinh(θi − θj ) + i sin πν . Φ(θ We assign odd Z2 -parity to both right and left movers (ψR → −ψR , ψL → −ψL , φ → φ) and even (odd) parity to right (left) movers under duality transformations (ψR → ψR , ψL → −ψL , φ → −φ). 2.1. Neveu–Schwarz sector: trace of the stress-energy tensor The operator Θ has nonzero matrix elements on (even, even) number of particles. The first form factor is determined by using the Lagrangian: Q2,2 = −4πM 2 . We introduce the sets S = (1, . . . , 2r) and S  = (1, . . . , 2l), and propose  ) Q2r,2l (θ1 , θ2 , . . . , θ2r ; θ1 , θ2 , . . . , θ2l   

 2  k − θi ), = −4πM Φ(θ φ(θik ) φ(θik ) Φ(θi − θk ) T ∈S, T  ∈S  , i∈T , #T = r−1 #T  = l−1 k∈T

i∈T  , k∈T

i∈T  , k∈T

i∈T , k∈T

where T , T are respectively subsets of S and  S, the notation ‘#’ stands for ‘number of elements’, and by definition T = S\T , T = S  \T  . Let us show that this representation does indeed satisfy the residue condition (4): only two cases will contribute to this computation, namely when 1 ∈ T , 2 ∈ T and 2 ∈ T , 1 ∈ T. It amounts to evaluate the residue at θ12 = iπ of the quantity:      2 − θi ) Φ(θ φ(θ12 ) φ(θ1k ) φ(θi2 ) Φ(θ1 − θk ) i∈T −{1}

k∈T−{2}

+ φ(θ21 )



φ(θ2k )

k∈T−{1}

× −4πM 2



 i∈T −{2}



i∈T 

k∈T

φ(θi1 ) 

U ∈S−{1,2}, T  ∈S  , i∈U,  #U = r−2 #T  = l−1 k∈U



Φ(θ2 − θk )

k∈T

φ(θik )

 i∈T  , k∈T



1 − θi ) Φ(θ



i∈T   φ(θik )



i∈U, k∈T

Φ(θi − θk )

 i∈T  ,  k∈U

k − θi ). Φ(θ

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 ); the evaluation of the residue at θ = iπ of the The last line is nothing but Q2r−2,2l (θ3 , θ4 , . . . , θ2r ; θ1 , θ2 , . . . , θ2l 12 term into brackets gives explicitly 

2r 2l 2l   

  1 2r+2l−1 2r−2  2l  (2i) sinh(θ2 − θk ) − i sin πν − (−1) sinh(θ2 − θk ) + i sin πν , (−1) sinh θ2j j =3

k=1

k=1

and Eq. (4) is satisfied. As a remark, we would like to note that the leading infrared behaviour of F2,2 is given by T T, which defines the direction of the flow in the IR region. To determine the subleading IR terms that appear in the expansion (1), one uses the asymptotic development for fRL given by Eq. (3). For example (up to the logarithmic terms): fRL (θ1 − θ1 )fRL (θ1 − θ2 )fRL (θ2 − θ1 )fRL (θ2 − θ2 )    2  2   A A θ1 +θ2 −θ1 −θ2 θ1 −θ1 2θ1 −2θ1 θ1 −θ2 2θ1 −2θ2 +B e +B e + + 1 + Ae 1 + Ae ∼e 2 2      2  2 A A     + B e2θ2 −2θ1 1 + Aeθ2 −θ2 + + B e2θ2 −2θ2 . × 1 + Aeθ2 −θ1 + 2 2 The terms into brackets give   2

−θ  
θ A     θ2 −θ2 1 1 + B e2θ1 −2θ1 + e2θ1 −2θ2 + e2θ2−2θ1 + e2θ2 −2θ2 e + 1+A e +e +e 2
        + A2 e2θ1 −θ1 −θ2 + eθ1 +θ2 −2θ1 + eθ1 +θ2 −2θ2 + e2θ2 −θ1 −θ2 + 2eθ1 +θ2 −θ1 −θ2 + · · · =1+

2 A B 2 −1 + A L2−1  −2 + · · · , L−1 L + 4 L−2 L L−1 2 4 M 2M M

where L−1 = eθ1 + eθ2 and L−2 = e2θ1 + e2θ2 . So the next irrelevant operator appearing in (1) is T 2 T 2 (up to derivatives). 2.1.1. Form factors of the energy operator  The number of left movers and right movers is odd. Let S = (1, . . . , 2r + 1), S  = (1, . . . , 2l + 1). The lowest form factor is Q1,1 = 1. We propose  Q2r+1,2l+1(θ1 , θ2 , . . . , θ2r+1; θ1 , θ2 , . . . , θ2l+1 )   

  k − θi ). φ(θik ) φ(θik ) Φ(θi − θk ) Φ(θ = T ∈S, T  ∈S  , i∈T , #T = r #T  = l k∈T

i∈T  , k∈T

i∈T , k∈T

i∈T  , k∈T

The proof that this expression satisfies Eq. (4) is the same as above. 2.2. Ramond sector 2.2.1. Order operator σ It has nonvanishing matrix elements when the sum of left movers and right movers is odd. Let S = (1, . . . , 2r + 1), S  = (1, . . . , 2l). The lowest form factors are Q1,0 = Q0,1 = 1. We propose:  Q2r+1,2l (θ1 , θ2 , . . . , θ2r+1; θ1 , θ2 , . . . , θ2l )

     k − θi ). = Φ(θ φ(θik ) φ(θik ) Φ(θi − θk ) T ∈S, T  ∈S  , i∈T , #T = r #T  = l k∈T

i∈T  , k∈T

i∈T , k∈T

i∈T  , k∈T

B. Ponsot / Physics Letters B 575 (2003) 131–136

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2.2.2. Disorder operator µ It has nonvanishing matrix elements when the sum of left and right movers is even. As it is explained in [9], there is an additional minus sign in front of the product of S matrices in the residue condition (2). • The number of left and right movers are both even. Let S = (1, . . . , 2r), S  = (1, . . . , 2l) and the lowest form factor Q0,0 = 1. We propose:  Q2r,2l (θ1 , θ2 , . . . , θ2r ; θ1 , θ2 , . . . , θ2l ) 

  1 1    k − θi ); φ(θik )e 2 θki φ(θik )e 2 θik Φ(θi − θk ) = (−i)r+l Φ(θ T ∈S, T  ∈S  , i∈T , #T = r #T  = l k∈T

i∈T  , k∈T

i∈T , k∈T

i∈T  , k∈T

• The number of left and right movers are both odd.  Let S = (1, . . . , 2r + 1), S  = (1, . . . , 2l + 1). The lowest form factor is Q1,1 = e(θ1 −θ1 )/2 . We propose:  Q2r+1,2l+1(θ1 , θ2 , . . . , θ2r+1; θ1 , θ2 , . . . , θ2l+1 )  

  1 1    k − θi ). Φ(θ φ(θik )e 2 θik φ(θik )e 2 θki Φ(θi − θk ) = (−i)r+l T ∈S, T  ∈S  , i∈T , #T = r #T  = l k∈T

i∈T  , k∈T

i∈T , k∈T

i∈T  , k∈T

Let us note that the exponentials will be responsible for the additional minus sign in the residue condition (4). 2.2.3. Remarks • One can check that in the IR, the form factors of the operator O = σ + µ satisfy the cluster property like an exponential of a Bose field [8], for example: Or,l (θ1 , θ2 , . . . , θr ; θ1 , θ2 , . . . , θl ) ∼ O1,0 (θ1 )Or−1,l (θ2 , . . . , θr ; θ1 , θ2 , . . . , θl )

for θ1 → −∞.

• The expressions for the form factors of σ and µ give the expected leading IR behaviour [4,9,10]:  θij θij IR (θ , θ , . . . , θ ; θ  , θ  , . . . , θ  ) ∼ Fr,l 1 2 r 1 2 i
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B. Ponsot / Physics Letters B 575 (2003) 131–136

Finally, the representations we provide for the functions Qr,l are in principle general enough3 to provide results for other massless models flowing to the Ising model, but where the S-matrix has a more complicated structure of resonance poles [14].

Acknowledgements I am grateful to Al.B. Zamolodchikov for suggesting the problem. Discussions with D. Bernard, V.A. Fateev, G. Mussardo, H. Saleur, F.A. Smirnov and especially G. Delfino are acknowledged. Work supported by the CEA and the Euclid Network HPRN-CT-2002-00325.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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3 Their architecture is very similar to the one found for the form factors of the operator eαφ in the bosonic sinh-Gordon model in [11],

Eq. (61)—let us recall that another representation in terms of determinant formula was first proposed in the prior work [8].