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1 Unarticle Examle in D The Harvard community ha made thi article oenly available. Pleae hare how thi acce benefit you. Your tory matter. Citation Publihed Verion Acceed Citable Link Term of Ue Georgi, Howard, Yevgeny Kat Unarticle Examle in D. Phyical Review Letter 101, doi: /phyrevlett Augut 17, 018 5:34:51 PM EDT htt://nr.harvard.edu/urn-3:hul.intreo: Thi article wa downloaded from Harvard Univerity' DASH reoitory, and i made available under the term and condition alicable to Oen Acce Policy Article, a et forth at htt://nr.harvard.edu/urn-3:hul.intreo:dah.current.term-ofue#oap (Article begin on next age)

2 PRL 101, (008) P H Y S I C A L R E V I E W L E T T E R S Unarticle Examle in D Howard Georgi* and Yevgeny Kat + Center for the Fundamental Law of Nature, Jefferon Phyical Laboratory, Harvard Univerity, Cambridge, Maachuett 0138, USA (Received 7 July 008; ublihed 5 Setember 008) We dicu what can be learned about unarticle hyic by tudying imle quantum field theorie in one ace and one time dimenion. We argue that the exactly oluble D theory of a male fermion couled to a maive vector boon, the Sommerfield model, i an intereting analog of a Bank-Zak model, aroaching a free theory at high energie and a cale-invariant theory with nontrivial anomalou dimenion at low energie. We contruct a toy tandard model couling to the fermion in the Sommerfield model and tudy how the tranition from unarticle behavior at low energie to free article behavior at high energie manifet itelf in interaction with the toy tandard model article. DOI: /PhyRevLett PACS number: Tk, Kk, j The term unarticle hyic wa coined by one of u to decribe a ituation in which tandard model hyic i weakly couled at high energie to a ector that flow to a cale-invariant theory in the infrared [1,]. In thi cla of model, one may ee urriing effect from the roduction of unarticle tuff [3] in the cattering of tandard model article. Studying uch model force u to confront ome intereting iue in cale-invariant theorie and effective field theorie. It i imortant to remember that unarticle hyic i not jut about a cale-invariant theory. There are two other imortant ingredient. A crucial one i the couling of the unarticle field to the tandard model. Without thi couling, we would not be able to ee unarticle tuff. Alo imortant i the tranition in the Bank-Zak theory [4] from which unarticle hyic emerge from erturbative hyic at high energie to cale-invariant unarticle behavior at low energie. Thi allow u to find wellcontrolled erturbative hyic that roduce the couling of the unarticle ector to the tandard model. Without thi tranition, the couling of the tandard model to the unarticle field would have to be ut in by hand in a comletely arbitrary way, and much of the henomenological interet of the unarticle metahor would be lot. In thi Letter, we exlore the hyic of the tranition from unarticle behavior at low energie to erturbative behavior at high energie in a model with one ace and one time dimenion in which the analog of the Bank-Zak model i exactly olvable. Thi will enable u to ee how the tranition take lace exlicitly in a imle incluive cattering roce (Fig. 1). We begin by decribing our analog Bank-Zak model and it olution. It i a D model of male fermion couled to a maive vector field. We call it the Sommerfield model becaue it wa olved by Sommerfield [5] in 1963 [6]. Next, we decribe the high-energy hyic that coule the Sommerfield model to our toy tandard model, which i imly a maive calar carrying a global Uð1Þ charge. In the infrared, the reulting interaction flow to a couling of two charged calar to an unarticle field with a nontrivial anomalou dimenion. We aly the oerator roduct exanion to the olution of the Sommerfield model to find the correlation function of the low-energy unarticle oerator. Finally, we tudy the imlet unarticle roce hown in Fig. 1 in which two toy tandard model calar diaear into unarticle tuff. Becaue we have the exact olution for the unarticle correlation function, we can ee reciely how the ytem make the tranition from low-energy unarticle hyic to the high-energy hyic of free article. The anwer i imle and elegant. The ectrum of the model conit of unarticle tuff and maive boon. A the incoming energy of the tandard model calar i increaed, the unarticle tuff i alway there, but more and more maive boon are emitted, and the combination become more and more like the free-fermion cro ection. The Sommerfield(-Thirring-We) model [6,7] i the Schwinger model [8] with an additional ma term for FIG. 1. A diaearance roce =08=101(13)=131603(4) Ó 008 The American Phyical Society

3 PRL 101, (008) P H Y S I C A L R E V I E W L E T T E R S the vector boon [9]: L ¼ ði@ 6 ea6 Þ 1 4 F F þ m 0 A A : (1) We are intereted in thi theory ince it i exactly olvable and (unlike the Schwinger model) ha fractional anomalou dimenion. In articular, we are intereted in the comoite oerator O 1 () becaue, in the low-energy theory, it cale with an anomalou dimenion. The olution for all fermion Green function in the model can be written down exlicitly, in term of roagator for free fermion, and for male and maive calar field with ma m: m ¼ m 0 þ e : (3) The hyical ma m lay the role in thi model of the unarticle cale U from Ref. [1], etting the cale of the tranition between free article behavior at high energie and unarticle behavior at low energie. Exlicitly, the calar roagator are [10] ðxþ ¼ Z d e ix ðþ m þ i ¼ i K 0ðm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ iþ; (4) DðxÞ ¼ Z d e ix ðþ þ i ¼ i x 4 ln þ i x : (5) 0 The n-oint function for the O and O field can then be contructed uing the oerator roduct exanion. We will decribe all of thi in detail in a earate ublication [11,1]. Here we will imly write down and ue the reult for the -oint function in oition ace i O ðxþ h0jtoðxþo BðxÞ ð0þj0i ¼ 4 ð x þ iþ ; (6) where BðxÞ¼ex i 4e m f½ðxþ ð0þš ½DðxÞ Dð0ÞŠg e ¼ ex m ½K 0ðm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þiþþlnðm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þiþš ; with (7) ¼ e E =: (8) In the hort-ditance limit (jx j1=m ), BðxÞ!1, and one obtain free-fermion behavior. In the large-ditance limit (jx j1=m ), K 0 doe not contribute, o BðxÞ i jut a ower of x, and the -oint function i roortional to an unarticle roagator [13] where i O ðxþ!i U ðxþ ¼ 1 4 ðmþ a ð x ; (9) 1þa þ iþ a e m ¼ 1 1 þ m : (10) 0 =e Thu at large ditance and low energie, the comoite oerator O cale with dimenion 1 þ a, correonding to an anomalou dimenion of a for 1.For0 <m 0 < 1, a i fractional, which lead to unarticle behavior. In momentum ace iaðaþ i U ðþ¼ ðmþ a inðaþ ð iþ a ¼ AðaÞ Z 1 ðmþ a dm ðm Þ a i 0 M þi ; (11) where the function AðaÞ inðaþ ð aþ 1þa (1) ð1 þ aþ i oitive in the range relevant to our model ( 1 <a< 0). Since Im U ðþ ¼ AðaÞ ðmþ a ð Þð Þ a ; (13) the unarticle hae ace i U ðþ ¼ AðaÞ ðmþ a ð0 Þð Þð Þ a : (14) To generate a couling to a toy tandard model, we aume that the very high-energy theory include the interaction L int ¼ ½ ð1 þ 5 Þ þ ð1 5 ÞŠþH:c: ¼ ð 1 þ 1 ÞþH:c: (15) that coule the fermion of the unarticle ector to a neutral comlex calar with ma m m that lay the role of a tandard model field. The interaction i mediated by the heavy fermion with ma M m; =m and the ame couling to A a. The theory ha a global Uð1Þ ymmetry with charge þ1 for, 1, and, 1 for,, and 1, and 0 for. Integrating out, we obtain L int ¼ h ðo þ O Þ; h M : (16) The comoite oerator O defined in () ha charge under the global Uð1Þ ymmetry

4 PRL 101, (008) P H Y S I C A L R E V I E W L E T T E R S In a 4D unarticle theory, the interaction correonding to (16) would tyically be nonrenormalizable, becoming more imortant a the energy increae. That doe not haen in our D toy model. But we can and will tudy the roce in Fig. 1 in the unarticle limit and learn omething about the tranition region between the ordinary article hyic behavior at energie large comared to m and the unarticle hyic at low energie. To that end, we conider the hyical roce þ! Sommerfield tuff hown in Fig. 1: Becaue coule to O, we can obtain the total cro ection for thi roce from the dicontinuity acro the hyical cut in the O - oint function. Thi i analogou to the otical theorem for ordinary article roduction. For momenta P 1 and P, thi i ¼ ImMðP 1;P! P 1 ;P Þ ; (17) with ¼ P, P ¼ P 1 þ P, and imðp 1 ;P! P 1 ;P Þ¼ ih O ðpþ: (18) In the unarticle limit ( m), uing (13) or directly the hae ace (14), we find the fractional ower behavior exected with unarticle roduction: ¼ AðaÞ h 1 ðmþ a 1 a : (19) On the other hand, in the free article limit aroriate for high energie m, we have BðxÞ!1 in (6), and then ¼ h 4 1 ; (0) which i the cro ection for þ! þ 1. Since we have the exact olution, we can tudy the tranition between the two limit by writing (6) for arbitrary x a i O ðpþ¼ X1 ð 4aÞ n Z d U n! ðþ i Uð U Þ Y n d i ðþ ið iþ ðþ P U Xn j : i¼1 j¼1 () Thi decribe a um of -oint diagram in which the incoming momentum P lit between the unarticle roagator and n maive calar roagator. Each i aociated with the roagation of a free maive calar field, o thi give the dicontinuity ðpþ¼ AðaÞ ðmþ a X 1 Y n i¼1 ð 4aÞ n n! Z d U ðþ ð0 U Þð U Þð U Þa d i ðþ ð i m Þð 0 i Þ ðþ P U Xn j¼1 j : (3) ffiffi For <Nm, only the firt N term in (3) (thoe involving the roduction of fewer than N maive boon) contribute, and (18) decribe the roduction of unarticle ffiffi tuff lu between 0 and N 1 maive boon. For < m, we have ure unarticle behavior. A we go to higher energie, the unarticle tuff i alway reent, but the emiion of more and more maive boon build u the incluive reult for free-fermion roduction. Thi haen quickly if a i mall but very gradually for a cloe to 1. One can eaily obtain exlicit reult in the cae of mall a, when only the firt few term in the exanion contribute. The leading correction in a come from n ¼ 1: ð1þ ¼ að mþ ln m þ Oða Þ; (4) which give the total hae ace a i O ðxþ ¼i U ðxþ ex½ 4iaðxÞŠ ¼ i U ðxþ X1 ð 4aÞ n ½iðxÞŠ n : (1) n! At ditance not large comared to 1=m, the higher term in the um in (1) become relevant. Notice that, in (1), we have exanded in a only the term involving the maive boon roagator. Thi i critical to our reult. It would be a mitake to exand i U in ower of a. Thi would introduce uriou infrared divergence becaue the male boon roagator i ick in 1 þ 1 dimenion [14]. The model decribe not maive and male boon but rather maive boon and unarticle tuff. In momentum ace, we obtain FIG.. Phae ace for the ffiffi diaearance roce in Fig. 1 a a function of the energy (in unit of m) for a ¼ 0:

5 PRL 101, (008) PHYSICAL REVIEW LETTERS ¼ 1 ln a m e E ffiffi þ ð mþ ln þ Oða Þ: m (5) ffiffi For energie >m, thi exreion reduce to ¼ 1 þ Oða Þ; (6) that i, the free-fermion reult (0). Thu, for jaj 1 there i a dicontinuity in d=d ffiffi at ¼ m, where a tranition occur from ure unarticle behavior below energy m to ure free-fermion behavior above m (ee Fig. ) [19]. To thi order, the free-fermion behavior i a um of the unarticle and the maive calar contribution. For larger value of jaj, higher ower of a mut be included in (5) to aroximate the free-fermion regime. Since each new maive calar give a contribution with only one additional ower of a,ifai cloe to 1, the freefermion behavior i aroached very lowly. In fact, the limit a! 1 i ingular, and it correond to the Schwinger model (m 0 ¼ 0). A i often the cae, it i not trivial to obtain a gauge theory a the limit of a theory with a maive vector boon. The unarticle tuff i abent ince Að 1Þ ¼0, and the ectrum include only a maive boon with m ¼ e =. The cae of the Schwinger model ha been tudied in Ref. [0,1]. We find thi icture of the unarticle cale U ¼ m in the Sommerfield model very atifying. There i a cloe analog between the way m enter in the roce of Fig. 1 and the way the dimenional tranmutation cale QCD enter in incluive rocee in QCD. In QCD, the hyical tate are hadron, tyically with mae of the order of QCD unle they are rotected by ome ymmetry (like the ion). But in the total e þ e cro ection into hadron (to ick the imlet and mot famou examle) at high energy E, the um over hyical tate reroduce the arton model reult with calculable correction of order 1= lnðe= QCD Þ. We have hown that the roce of Fig. 1 in the Sommerfield model work the ame way, with the hyical tate being the maive boon and unarticle tuff. We are grateful to C. Cordova, V. Lyov, P. Petrov, A. Sajjad, and D. Simmon-Duffin for dicuion. Thi reearch i uorted in art by the National Science Foundation under Grant No. PHY *georgi@hyic.harvard.edu + kat@hyic.harvard.edu [1] H. Georgi, Phy. Rev. Lett. 98, 1601 (007). [] H. Georgi, Phy. Lett. B 650, 75 (007). [3] We refer unarticle tuff to unarticle for the hyical tate, becaue it i not clear to u what the noun unarticle i uoed to mean and certainly not clear whether it hould be ingular or lural. [4] T. Bank and A. Zak, Nucl. Phy. B196, 189 (198). [5] Georgi Ph.D. advior and Schwinger tudent. [6] C. M. Sommerfield, Ann. Phy. (N.Y.) 6, 1 (1964). [7] W. E. Thirring and J. E. We, Ann. Phy. (N.Y.) 7, 331 (1964). [8] J. S. Schwinger, Phy. Rev. 18, 45 (196). [9] We will ue convention in which g ¼ diagð1; 1Þ and 5 ¼ diagð1; 1Þ. [10] K 0 i the modified Beel function of the econd kind, and x 0 i an arbitrary contant that will cancel out in the following. For y!1, K 0 ðyþ ffiffiffiffiffiffiffiffiffiffiffi =ye y! 0. For y! 0, K 0 ðyþ ¼ lnðy=þ E þ Oðy Þ, where E ¼ 0 ð1þ 0:577 i Euler contant. Note that ð0þ Dð0Þ ¼ði=Þ lnðe E x0 m=þ i finite. [11] H. Georgi and Y. Kat (to be ublihed). [1] There we will alo include more comlete reference to the unarticle and Sommerfield model literature. [13] Here and below, we incororate a dimenional factor of 1=ðmÞ a in the unarticle roagator o it matche moothly onto the O roagator. [14] Thi tatement ha a long hitory in the mathematical hyic literature, going back at leat to Ref. [15]. For an early ummary in Englih, ee [16]. See alo [17]. It i alo worth noting that Ref. [15] introduce the notion of infraarticle an aroach to continuou ma rereentation of the Poincaré grou, which of coure include unarticle tuff. See [18], where we learned of thi intereting early reference. [15] B. Schroer, Fortchr. Phy. 11, 1 (1963). [16] J. Tarki, J. Math. Phy. (N.Y.) 5, 1713 (1964). [17] S. R. Coleman, Commun. Math. Phy. 31, 59 (1973). [18] B. Schroer, arxiv: [19] The linear aroximation (5) i not valid for m due to large ln ffiffi, but we have the exact exreion (14). [0] A. Caher, J. B. Kogut, and L. Sukind, Phy. Rev. Lett. 31, 79 (1973). [1] A. Caher, J. B. Kogut, and L. Sukind, Phy. Rev. D 10, 73 (1974)